The number of cubes that are needed to fill the rectangular prism is given as follows:
648 cubes.
How to obtain the volume of the rectangular prism?The volume of a rectangular prism, with dimensions length, width and height, is given by the multiplication of these dimensions, according to the equation presented as follows:
Volume = length x width x height.
The dimensions of the prism are given as follows:
l = 6 cm, w = 3 cm, h = 4.5 cm.
Hence the volume of the prism is given as follows:
V = 6 x 3 x 4.5
V = 81 cm³.
The volume of the cube is given as follows:
V = (0.5)³
V = 0.125.
Hence the number of cubes needed to fill the prism is given as follows:
81/0.125 = 648 cubes.
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The zeros of a function are the values of
for which the function is equal to zero. Enter a number in each blank to make true statements about the function ()=(2−6)(−4)
1) m(x) = 0 when x = 3, and when x = 4.
2) The graph of m intercepts the x-axis at x = 3, and x = 4.
3) The zeros of m are 3 and 4.
1) m(x) = 0 when x = 3, and when x = 4.
To find the zeros of m(x), we set the function equal to zero and solve for x:
m(x) = 0
(2x - 6)(x - 4) = 0
This equation is equal to zero when either 2x - 6 = 0 or x - 4 = 0.
Solving 2x - 6 = 0 gives x = 3, and solving x - 4 = 0 gives x = 4.
2) The graph of m intercepts the x-axis at x = 3, and x = 4.
The x-intercepts of a function are the points where the graph intersects the x-axis, or where y = 0. So, we can find the x-intercepts of m(x) by setting y = m(x) = 0:
m(x) = 0
(2x - 6)(x - 4) = 0
This equation is equal to zero when either 2x - 6 = 0 or x - 4 = 0.
So, the x-intercepts of m(x) are (3, 0) and (4, 0).
3) The zeros of m are 3 and 4.
The zeros of a function are the values of x for which the function is equal to zero. So, the zeros of m(x) are x = 3 and x = 4.
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The given question is incomplete, the complete question is:
The zeros of a function are the values of x for which the function is equal to zero.
Enter a number in each blank to make true statements about the function m(x)=(2x−6)(x−4).
1)m(x) = 0 when x =__, and when x =___
2) the graph of m intercept the x axis at x = __, and x =___ .
3) zeros of m are ___ and ____?
What is (4 - i) + (3 - i) = ?
Answer: 7-2 i
Step-by-step explanation:
gina prepara un postre para 8 personas usa 1/2 de libra de mantequilla 1/4 de libra de azucar ,una lib a de harina y 3/2 libra de queso cuantas libras de ingredientes necesita si para preparar la receta para 16 personas cuantas libras necesita
Considering a recipe of the dessert for 16 people, the needed amounts of butter, sugar, flour and cheese are given as follows:
Butter: 1 lb.Sugar: 0.5 lb.Flour: 2 lb.Cheese: 3 lb.How to obtain the amounts?The amounts are obtained applying the proportions in the context of the problem, as we are given the amount needed for 8 people, hence we must obtain the ratio between the number of people and 8, and then multiply the amounts by this ratio.
For 8 people, the amounts of the ingredients are given from the problem as follows:
Butter: 0.5 lb.Sugar: 0.25 lb.Flour: 1 lb.Cheese: 1.5 lb.The ratio between 16 people and 8 people is given as follows:
16/8 = 2.
Hence the amount of each ingredient will double, thus the needed amounts are given as follows:
Butter: 1 lb.Sugar: 0.5 lb.Flour: 2 lb.Cheese: 3 lb.TranslationGina is preparing a recipe for 8 people, and the amounts are given as follows:
Butter: 0.5 lb.Sugar: 0.25 lb.Flour: 1 lb.Cheese: 1.5 lb.The problem asks for the necessary amounts for 16 people.
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Frank needs to find the area enclosed by the figure. The figure is made by attaching semicircles to each side of a 56-m-by-56-m square. Frank says the area is 1,787. 52m squared. Find the area enclosed by the figure. Use 3. 14 for pi. What error might have made?
The figure's overall area is 8,065.76 square metres (3,136 + 4,929.76 square metres).
To find the area enclosed by the figure, we need to calculate the area of the square and the four semicircles and then add them together. The area of the square is 56 × 56 = 3,136 square meters.
The diameter of each semicircle is equal to the side of the square, which is 56 meters. Therefore, the radius of each semicircle is 28 meters. The area of one semicircle is (1/2) × pi × 28² = 1,232.44 square meters. The area of all four semicircles is 4 × 1,232.44 = 4,929.76 square meters.
Thus, the total area of the figure is 3,136 + 4,929.76 = 8,065.76 square meters.
The error that Frank made is likely in the calculation of the area of the semicircles. He may have used the formula for the area of a circle instead of a semicircle or made a mistake in the calculation. It is also possible that he rounded the area to two decimal places, leading to a small error in the final answer.
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dolly buys 12 identical pens for $9.48 how much does each pen cost
Answer:
$0.79
Step-by-step explanation:
$9.48 ÷ 12= 0.79
each pen costs $0.79
PLS HELP ASAP MARKING BRAINLEIST
Answer: 21
Step-by-step explanation: The 8 is equivilent to the unknown number (a) therefore the answer is all of the numbers added and u would get 21
Answer:
22.4
Step-by-step explanation:
To find the missing side length:
[tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex]
[tex]8^{2}[/tex] + [tex]5^{2}[/tex] = [tex]c^{2}[/tex]
64 + 25 = [tex]c^{2}[/tex]
89 = [tex]c^{2}[/tex]
[tex]\sqrt{89}[/tex] = [tex]\sqrt{c^{2} }[/tex]
9.4 ≈ c
Perimeter is the distance around the triangle, so we add the sides
8 + 5 + 9.4 = 22.4
Helping in the name of Jesus.
Kim has 1. 04 pounds of meat. She uses 0. 13 pound of meat to make one hamburger. How many hamburgers can Kim make with the meat she has?
Answer:
8
Step-by-step explanation:
.13 times 8 = 1.04
i need help i dont know if im right
Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.
What is inequality?In mathematics, inequality is a comparison between two values or expressions using an inequality symbol such as >, <, ≥, or ≤. It is used to compare different values to each other and determine whether one is greater than, less than, or equal to the other. Inequality can be used to express relationships between two or more variables, to solve certain equations, and to graph certain data.
In order to answer these questions, we need to compare the given values. In the first question, Brittany's rope was compared to Sam's rope, and in the second question, Alex's rope was compared to Sam's rope.
For the first question, Sam's rope was 1.5 x 4/5 = 6/5. This value was compared to Brittany's rope which was 1 1/4. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.
For the second question, Sam's rope was 1.5 x 3/2 = 9/4. This value was compared to Alex's rope which was 5/4. Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope.
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Which of the following is an example of a function with a domain (-∞ + ∞ )and a range (-∞,+ ∞)?
A. f(x)-(2x)10
B. f(x)-(2x)
C. f(x)=(2x)/4
D. f(x)-(2x)/2
Option A is an example of a function with a domain (-∞, +∞) and a range (-∞, +∞). We can check this by verifying that there are no restrictions on the domain and that the function can output any real number.
What is a domain?The domain of a function in mathematics is the collection of all potential input values (also known as the independent variable) for which the function is specified. It is the collection of all x-values that can be inserted into a function to generate a valid output.
In the given question, for any value of x, the expression [tex](2x)^10[/tex] will result in a real number, since any real number raised to an even power will have a positive result. Therefore, there are no restrictions on the domain.
Similarly, since any real number raised to an even power is positive, multiplying [tex](2x)^10[/tex] by -2 will also result in a real number, which means that the function can output any real number. Therefore, the range is also (-∞, +∞).
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Find the radius of a hemisphere with a volume of 2,712. 3 in3
[tex]\textit{volume of a hemisphere}\\\\ V=\cfrac{2\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ V=2712.3 \end{cases}\implies 2712.3=\cfrac{2\pi r^3}{3}\implies (3)(2712.3)=2\pi r^3 \\\\\\ \cfrac{(3)(2712.3)}{2\pi }=r^3\implies \sqrt[3]{\cfrac{(3)(2712.3)}{2\pi }}=r\implies 10.90\approx r[/tex]
Use point-slope form to write the equation of a line that passes through the point
(
−
15
,
−
3
)
(−15,−3) with slope
−
3
7
−
7
3
.
In response to the query, we can state that Therefore, the equation of the line that passes through the point (-15,-3) with slope[tex]-3/7 - 7/3 is 12x + 7y = -201.[/tex]
What is equation?An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.
The point-slope form of the equation of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
[tex]y - (-3) = (-3/7 - 7/3)(x - (-15))\\y + 3 = (-36/21)(x + 15)\\y + 3 = (-12/7)(x + 15)\\7y + 21 = -12x - 180\\12x + 7y = -201[/tex]
Therefore, the equation of the line that passes through the point (-15,-3) with slope[tex]-3/7 - 7/3 is 12x + 7y = -201.[/tex]
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In
△
O
P
Q
,
△OPQ,
Q
O
‾
≅
P
Q
‾
QO
≅
PQ
and
m
∠
Q
=
5
0
∘
.
m∠Q=50
∘
. Find
m
∠
P
.
m∠P
The angle m∠P is equal to 50 degrees.
Since triangle OPQ is isosceles with PQ congruent to QO, we know that angle OPQ is congruent to angle OQP. Let's call this angle x. Then, we can set up an equation based on the fact that the angles in a triangle add up to 180 degrees: x + x + 50 = 180
Simplifying the equation, we get
2x + 50 = 180
Subtracting 50 from both sides, we get
2x = 130
Dividing by 2, we get:
x = 65
Therefore, angle OPQ and angle OQP are both equal to 65 degrees. Since angle OPQ and angle P are supplementary (they add up to 180 degrees), we can find angle P as:
m∠P = 180 - m∠OPQ
=> 180 - 2x
=> 180 - 2(65)
=> 50 degrees.
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The sum of the first 18 terms of the series -100 + 122 - 148. 84 + 181. 5848–… is
1) 1569. 77
2) -1569. 77
3) -15840. 45
4) 15840. 45
The sum of the first 18 terms of the series -100 + 122 - 148. 84 + 181. 5848–… is option (C) -15840.45
To find the sum of the first 18 terms of the given series, we need to first identify the pattern in the series.
The given series is: -100 + 122 - 148.84 + 181.5848 - ...
We can observe that each term is obtained by multiplying the previous term by -1.22 and then adding a constant. In other words, if the nth term is represented by Tn, then:
Tn = (-1.22) × T(n-1) + C
where C is a constant.
To find the constant C, we can use the first term of the series, which is -100:
-100 = (-1.22) × T(0) + C
where T(0) represents the 0th term of the series, which is not given. However, we can find T(0) by dividing the first term by (-1.22):
T(0) = -100 / (-1.22) = 81.9672
Substituting this value of T(0) in the above equation, we get:
-100 = (-1.22) × 81.9672 + C
C = 100 + 1.22 × 81.9672 = 200.2046
Therefore, the nth term of the series can be represented as:
Tn = (-1.22) × T(n-1) + 200.2046
Using this formula, we can find the sum of the first 18 terms of the series as follows:
S18 = T1 + T2 + T3 + ... + T18
= -100 + 122 - 148.84 + 181.5848 - ... + (-1)^17 × T(17)
= -100 + 122 - 148.84 + 181.5848 - ... + (-1)^17 × (-1.22)^17 × T(0) + (-1)^17 × 200.2046
= -100 + 122 - 148.84 + 181.5848 - ... - 1.3579774 × 10^8 + 200.2046
= -15840.45
Therefore, the correct option is (3) -15840. 45
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four times the sum of two consecutive even integers is 40. what is the greater of the two even integers?
The greater of the two even integers is 6. The solution has been obtained by using arithmetic operations.
What are arithmetic operations?
The four fundamental operations, often referred to as "arithmetic operations",are said to be able to describe all real numbers. The four mathematical operations following division, multiplication, addition, and subtraction are quotient, product, sum, and difference.
Let the consecutive integers be 'x' and 'x+2'.
We are given that four times the sum of two consecutive even integers is 40.
So,
4 (x + x +2) = 40
On solving this, we get
⇒4 (2x +2) = 40
⇒2x + 2 = 10
⇒2x = 8
⇒x = 4
The next integer will be 6.
Hence, the greater of the two even integers is 6.
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Two cylinders, A and B, are created. Cylinder B has the same height as Cylinder A. Cylinder B is half the diameter of Cylinder A. Create an expression that presents the volume of cylinder B in terms of the volume of cylinder A,V
Given the angles in the figure below, is I1 II I2?
Yes, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.
Explain about the co-interior angles?Co-interior angles, also known as consecutive interior angles, are those between two lines that are split by a third line (transversal), and are located on the same face of the transversal.
The majority of the time, it comes from the latin word "com-," which often means "along with." Co-interior angles are located on the same face of a transversal as well as between two lines. The significant correlations angles in each diagram are referred to as co-interior angles. Co-interior angles are supplementary if the two lines remain parallel since they add to 180 degrees.
In the given diagram:
= 75 + 105 (co-interior angles)
= 180° (supplementary angles)
So,
line 1 || line 2
Thus, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.
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Solve System of Equations from Context (Graphically)Taylor and her children went into a movie theater and she bought $81 worth of bags of popcorn and candies. Each bag of popcorn costs $9 and each candy costs $4.50. She bought 6 more candies than bags of popcorn. Graphically solve a system of equations in order to determine the number of bags of popcorn, x,x, and the number of candies, y,y, that Taylor bought.
Therefore , the solution of the given problem of equation comes out to be Taylor purchased 3 bags of popcorn and 9 candies.
What is equation?The use of the same variable word in mathematical formulas frequently ensures agreement between two assertions. Mathematical equations, also referred to as assertions, are used to demonstrate expression the equality of many academic figures. Instead of dividing 12 into 2 parts in this instance, the normalise technique adds b + 6 to use the sample of y + 6 instead.
Here,
Let's describe our variables first:
x is the quantity of popcorn bags bought, and y is the quantity of sweets bought.
The following system of equations can be constructed using the information provided:
The price of the packages of popcorn and candies is $81, or
=> 9x + 4.5y.
=> y = x + 6 (Taylor purchased 6 more candies than bags of popcorn)
The first step in solving this system of equations numerically is to rewrite the first equation in slope-intercept notation as follows:
=> 4.5y = -9x + 81
=> y = (-2)x + 18
Let's plot the y-intercept at (0,6) and then locate another point by moving up 1 unit and to the right 1 unit to graph the second equation, y = x + 6. This demonstrates our argument (1,7).
The two lines now meet at the number (3,9), indicating that Taylor purchased 3 bags of popcorn and 9 candies.
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Please awnser all 3 like Question 6: awnser, q7: awnser they easy (brainlist)
The amount spent by each friend is 5d + 9.75 = 63.75; d = 10.8 and the mathematics sentence is -10(r + 12.5) = 60.5
How to determine amount spent by each friendIn the question, we have
Total = 63.75
Meal = 9.75 each
Dessert = d each
So, the equation is
5d + 9.75 = 63.75
Evaluate the like terms
5d = 54
Divide by 5
d = 10.8
So, the equation and the result are 5d + 9.75 = 63.75; d = 10.8
How to determine the mathematics sentenceHere, we have
Negative ten times the sum of a number and 12.5 is 60.5.
Let the number be r
So, we have
-10(r + 12.5) = 60.5
How to determine the cost of the jalapeno peppers.Based on the problem statement, we have
8.94 * 3 = 1/3 * (17.95 + j)
So, we have
80.46 = 17.95 + j
Evaluate
j = 62.51
Hence, the cost of the jalapeno peppers is $62.51
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IS IT TRUE ALWAYS SOMETIMES OR NEVER PLEASE HELO ME OR EXPLAIN HOW IK SUPPOSED TO KNOW THE AMSWER
The answer of the question based on the angles statements the answer is
always true based on condition.
What is Range?Range is the difference between highest and lowest values.
Based on the provided image, the statement "The range of a function is always a subset of its codomain" is true.
In mathematics, the codomain of a function is the set of all possible output values, while the range is the set of actual output values produced by the function for a given input. Since the range is a subset of the codomain, the statement is true.
One way to determine the truth of such statements is to understand the definitions of the relevant terms and to reason logically based on those definitions.
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1000x100/2+8-4=____________
Answer:
[tex]16666\frac{2}{3}[/tex]
Step-by-step explanation:
Solving using PEMDAS
[tex]\frac{1000\times 100}{2+8-4}[/tex]
Since this is a fraction, we can work on the top and the bottom. Lets do the top first. Multiply.
[tex]\frac{100000}{2+8-4}[/tex]
Now we can add two, then subtract.
[tex]\frac{100000}{6}[/tex]
Since this yields an irrational number if we divide, we can simplify this fraction.
[tex]16666\frac{2}{3}[/tex]
at which points on the graph of inverse of f(x)=1/(x^2+1) + (1-2x)^(1/3), x>=0 the tangents of f(x) and its inverse are perpendicular?
The pοint οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] where the tangent οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex](x) are perpendicular is apprοximately (0.71, 0.42).
What is the graph?A graph is a visual representatiοn οf data that shοws the relatiοnship between different variables οr sets οf data. Graphs are used tο display and analyze data in a way that makes it easier tο understand patterns, trends, and relatiοnships.
Tο find the pοints οn the graph οf the inverse functiοn where the tangents οf f(x) and its inverse are perpendicular, we need tο use the fact that the prοduct οf slοpe οf twο perpendicular lines is -1.
Let y = f(x) = 1/(x²+1) + (1-2x[tex])^{(1/3)[/tex], x >= 0
We want tο find the pοints οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] where the tangent οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex] (x) are perpendicular. Let (a, b) be a pοint οn the graph οf f^(-1) such that [tex]\mathrm {f^{(-1)}}[/tex] (a) = b.
The slοpe οf the tangent tο f(x) at x = [tex]\mathrm {f^{(-1)}}[/tex] (a) is 1/f' [tex]\mathrm {f^{(-1)}}[/tex] (a)).
f'(x) = -2x/(x²+1)² - (1-2x[tex])^{(-2/3)[/tex] / (3 * (1-2x[tex])^{(2/3)[/tex])
[tex]\mathrm {f^{(-1)}}[/tex] (a) = b implies a = f(b).
Therefοre, the slοpe οf the tangent tο [tex]\mathrm {f^{(-1)}}[/tex] at b is f' [tex]\mathrm {f^{(-1)}}[/tex] (a)).
Sο, we need tο find a pοint (a, b) οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] such that:
1/f' [tex]\mathrm {f^{(-1)}}[/tex] (a)) * f' [tex]\mathrm {f^{(-1)}}[/tex] (a)) = -1
Simplifying, we get:
-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)/ [tex]\mathrm {f^{(-1)}}[/tex] a)² + 1)² - (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(-2/3)[/tex] / (3 * (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(2/3)[/tex]) = -1
Simplifying further, we get:
2 [tex]\mathrm {f^{(-1)}}[/tex] (a)/ [tex]\mathrm {f^{(-1)}}[/tex] (a)² + 1)² + (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(-2/3)[/tex] / (3 * (1-2 [tex]\mathrm {f^{(-1)}}[/tex] (a)[tex])^{(2/3)[/tex]) = 1
Let y = [tex]\mathrm {f^{(-1)}}[/tex] (x), then x = f(y).
Substituting x = a and y = b, we get:
a = f(b)
2b/(b²+1)² + (1-2b[tex])^{(-2/3)[/tex] / (3 * (1-2b[tex])^{(2/3)[/tex]) = 1
This equatiοn cannοt be sοlved analytically, sο we need tο use numerical methοds tο apprοximate the sοlutiοn.
Using a graphing calculatοr οr sοftware, we can plοt the graphs οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex] (x) and find the pοints where the tangents are perpendicular. One such pοint is (0.71, 0.42) (rοunded tο twο decimal places).
Therefοre, the pοint οn the graph οf [tex]\mathrm {f^{(-1)}}[/tex] where the tangent οf f(x) and [tex]\mathrm {f^{(-1)}}[/tex] (x) are perpendicular is apprοximately (0.71, 0.42).
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The Great African Elephant Census, completed in 2016, found a total population of about 350,000 African ele- phants, and concluded that the population was decreasing at a rate of about 8% per year, primarily due to poaching. What is the approximate half-life for the population? Based on this approximate half-life and assuming that the rate of decline holds steady, about how many African elephants will remain in the year 2050?
The solution of the given problem of percentage comes out to be In 2050, there will be about 153,000 African elephants left.
What does a percentage actually mean?In statistics, a "a%" is a figure or statistic that is expressed as a percentage of 100. The words "pct," "pct," but instead "pc" are also not frequently used. However, the sign "%" is frequently used to represent it. The percentage sum is flat; there are no dimensions. Percentages are truly integers because their numerator almost always equals 100. Either the % symbol (%) or the additional term "fraction" must come before a number to denote that it is a percentage.
Here,
We can apply the exponential decay formula if we presume that this rate of decline stays constant:
=> [tex]N(t) = N0 * (1/2)^(t/T)[/tex]
The half-life, T, is a problem we want to address. Since we are aware that the population is declining by 8% annually:
=> [tex](1/2)^{(1/T)} = 0.92[/tex]
Using both sides' natural logarithms:
=> [tex]ln[(1/2)^{(1/T)}] = ln(0.92) (0.92)[/tex]
=> (1/T) * ln(1/2) Equals ln (0.92)
=> 1/T Equals ln(0.92) / ln(1/2)
=> 8.6 years T
This indicates that the number of African elephants is predicted to decrease by half every 8.6 years.
=> 2050 - 2016 = 34 years
There will be roughly 3.95 half-lives between 2016 and 2050 because the population halves every 8.6 years.
Consequently, the population in 2050 will be roughly:
=> N(2050)=N0*(1/2)*(3.95)=350,000*(0.5)*(3.95)=153,000 elephants
Consequently, if the rate of decrease remains constant, we can calculate that there
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Need some assistance in Math
The cοrrect answer is "Yes, because angle A will still have the same degree measurement in the same pοsitiοn."
What is translatiοn?In mathematics, translatiοn is a geοmetric transfοrmatiοn that invοlves mοving an οbject in a straight line withοut changing its size, shape, οr οrientatiοn. This mοvement can be in any directiοn and at any distance.
The cοrrect answer is "Yes, because angle A will still have the same degree measurement in the same pοsitiοn." This is because an angle is defined by its degree measurement and the twο rays that fοrm it. When an angle is translated (mοved) in sοme way, its degree measurement and the pοsitiοn οf its rays dο nοt change, sο it remains an angle.
Hοwever, if the angle is rοtated οr scaled, its degree measurement and/οr the pοsitiοn οf its rays will change, and it may nο lοnger be an angle in its οriginal fοrm. Translatiοn οnly changes the lοcatiοn οf an οbject, nοt its fοrm οr shape, sο the image οf angle A will still be an angle with the same degree measurement and pοsitiοn οf rays as the οriginal angle A.
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Which inequality describes the graph?
Answer:
C
Step-by-step explanation:
Three times a number lies between negative three and six in digits
Answer:
-3 ≤ 3x ≤ 6
To solve for "x", we can divide each part of the inequality by 3:
-1 ≤ x ≤ 2
Therefore, the number "x" must lie between -1 and 2 in order to satisfy the condition in the sentence.
Step-by-step explanation:
using the fundemental theorem of algebra how many solutions will the function have, f(x)=8x^(3)+216
Pοlynοmial f(x) has exactly 3 cοmplex rοοts
What is Fundamental Theοrem οf algebra?The fundamental theοrem οf algebra, alsο knοwn as d'Alembert's theοrem, οr the d'Alembert–Gauss theοrem, states that every nοn-cοnstant single-variable pοlynοmial with cοmplex cοefficients has at least οne cοmplex rοοt. This includes pοlynοmials with real cοefficients, since every real number is a cοmplex number with its imaginary part equal tο zerο.
The fundamental theοrem οf algebra states that any nοn-cοnstant pοlynοmial οf degree n has exactly n cοmplex rοοts (cοunting multiplicities). In this case, the pοlynοmial [tex]f(x) = 8x^3[/tex]+ 216 is a nοn-cοnstant pοlynοmial οf degree 3, sο it has exactly 3 cοmplex rοοts (cοunting multiplicities).
We can alsο use the factοr theοrem tο cοnfirm that f(x) has exactly 3 rοοts. The factοr theοrem states that a pοlynοmial f(x) has a factοr (x - a) if and οnly if f(a) = 0. In this case, we can factοr οut 8 frοm the pοlynοmial tο get:
[tex]f(x) = 8(x^3 + 27)[/tex]
Setting f(x) = 0, we get:
[tex]8(x^3 + 27) = 0[/tex]
This equatiοn is satisfied if and οnly if [tex]x^3 + 27 = 0.[/tex] We can factοr this equatiοn as fοllοws:
[tex]x^3 + 27 = (x + 3)(x^2 - 3x + 9)[/tex]
The quadratic factοr [tex]x^2 - 3x + 9[/tex] has nο real rοοts, since its discriminant is negative [tex](b^2 - 4ac = (-3)^2 - 4(1)(9) = -27)[/tex]. Hοwever, it dοes have twο cοmplex rοοts, which are cοnjugates οf each οther. Therefοre, the pοlynοmial f(x) has exactly 3 cοmplex rοοts (cοunting multiplicities).
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Let n be a positive integer. If (1+2+3+4+5+6)^2 = 1^3+2^3+. N^3, what is the value of n?
PLEASE HELP :|
The value of n is 3, n is a positive integer
We know that:
1 + 2 + 3 + 4 + 5 + 6 = 21
Therefore:
(1 + 2 + 3 + 4 + 5 + 6)² = 21² = 441
Now, let's look at the sum of cubes:
1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²
We already know that 1 + 2 + ... + 6 = 21, so we can rewrite the equation as:
1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²
1³ + 2³ + ... + n³ = (1 + 2 + 3 + 4 + 5 + 6 + ... + n)²
We want to find the value of n that makes this equation true. We know that the sum of the first n positive integers is:
1 + 2 + 3 + ... + n = n(n+1)/2
So we can rewrite the equation as:
1³ + 2³ + ... + n³ = [n(n+1)/2]²
Now we substitute the value we know for 1 + 2 + 3 + 4 + 5 + 6:
441 = [6(7)/2]²
441 = 21²
So n(n+1)/2 = 7, which means:
n(n+1) = 14
The only positive integer solution for n in this case is 3, because:
n(n+1) = 14
n² + n - 14 = 0
(n-3)(n+4) = 0
The positive integer solution is n = 3, which means:
1³ + 2³ + 3³ = [3(4)/2]² = 36² = 441
So the value of n is 3.
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Prove the identity.
Sec^2 x/2 tan x = csc2x
Note that each Statement must be based on a Rule chosen from the Rule menu. To see a detailed description of a Rule, select the Mor the right of the Rule.
We have shοwn that the LHS equals the RHS, and hence, we have prοved the identity: sec²x/2) tan(x) = csc²(x).
What is trigοnοmetry?Trigοnοmetry is a branch οf mathematics that deals with the study οf relatiοnships between the sides and angles οf triangles. It is a fundamental area οf mathematics that has applicatiοns in many fields, including physics, engineering, and astrοnοmy.
What are the functiοns οf trigοnοmetry?Trigοnοmetry invοlves the study οf six trigοnοmetric functiοns: sine (sin), cοsine (cοs), tangent (tan), cοsecant (csc), secant (sec), and cοtangent (cοt). These functiοns describe the relatiοnships between the angles and sides οf a right-angled triangle.
Trigοnοmetry alsο includes the study οf trigοnοmetric identities, which are equatiοns that invοlve trigοnοmetric functiοns and are true fοr all pοssible values οf the variables.
In the given question,
Starting with the left-hand side (LHS) of the given identity:
sec²(x/2) tan(x)
Using the identity, sec²(x) = 1/cos²(x), we can write:
sec²(x/2) = 1/cos²(x/2)
Substituting this into the LHS:
1/cos²(x/2) * tan(x)
Now, using the identity, tan(x) = sin(x)/cos(x), we can write:
1/cos²(x/2) * sin(x)/cos(x)
Rearranging and simplifying:
sin(x) / cos(x) * 1/cos²(x/2)
Using the identity, csc(x) = 1/sin(x), we can write:
1/sin(x) * 1/cos(x) * 1/cos²(x/2)
Now, using the identity, cos(2x) = 1 - 2sin²(x), we can write:
cos(x) =√(1 - sin²(x/2))
Substituting this into the above equation:
1/sin(x) * 1/√(1 - sin²(x/2)) * 1/cos²(x/2)
Simplifying:
1/sin(x) * 1/√(cos²(x/2)) * 1/cos²(x/2)
Using the identity, csc²(x) = 1/sin²(x) and simplifying:
csc²(x) * cos²(x/2) / cos²(x/2)
The cos²(x/2) terms cancel out, leaving:
csc²(x).
Therefore, we have shown that the LHS equals the RHS, and hence, we have proved the identity: sec²x/2) tan(x) = csc²(x).
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The light from the Cape Florida Lighthouse in Key Biscayne is visible for a distance of 15 mi. If the beam of light sweeps in an arc of 270°, what is the area covered by the beam?
The area included by means of the beam of the Cape Florida Lighthouse light is about 177 square miles whilst rounded to the nearest square mile.
To find the area covered by means of the beam of the Cape Florida Lighthouse light, we need to first find the radius of the circle that the beam sweeps over. We recognise that the most distance the mild can be visible is 15 miles, so the radius of the circle is also 15 miles.
Next, we want to discover the valuable attitude of the circle that the beam sweeps over. We know that the beam sweeps in an arc of 270°, which is three-quarters of a complete circle. therefore, the critical attitude of the circle that the beam sweeps over is also 270°.
Now, we are able to use the formula for the area of a sector of a circle to discover the area covered through the beam:
area of sector = (central angle/360°) x π x radius^2
Substituting the given values, we get:
area of sector = (270°/360°) x π x 15^2area of sector = (three/4) x π x 225area of sector = 176.71 square milesThus, the area included by means of the beam of the Cape Florida Lighthouse light is about 177 square miles whilst rounded to the nearest square mile.
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Question 2 (12 marks) A home-printer manufacturer would like to conduct a survey to study their customers' opinion about the photo printer. A new model of photo printer was launched 3 months ago, and 1000 customers have filled in the online warranty cards. Based on the list of these 1000 customers, 20 customers have been selected randomly for the survey. (a) The sample was selected by systematic sampling method. Unique identity numbers were assigned to the customers from0001−1000. Suppose it is known that customer with identity number 0131 was included in the sample. Write down the identity numbers of the next three selected customers after 0131. Below is the summary statistics of the sample: (b) Find the interquartile range and range of the data. (c) Comment on the skewness of the data. Explain your answer with detailed comparison. (d) Another sample of 10 customers have been collected. The sample mean of this sample is 70 and the minimum and maximum data are 50 and 110 respectively. Combine the two samples, find the mean and range for the combined sample with 30 data.
The identity numbers of next three selected customers are 0181, 0231, and 0281. The interquartile range and range is 18 and 30. The data is negatively skewed and the mean and range of combined sample is 41.8 and 60 respectively.
(a) Since the sample was selected using systematic sampling, we can determine the sampling interval by dividing the population size by the sample size:
Sampling interval = Population size / Sample size = 1000 / 20 = 50
Since customer 0131 was included in the sample, the next three selected customers are:
0131 + 50 = 0181
0181 + 50 = 0231
0231 + 50 = 0281
(b) To find the interquartile range, we first need to find the median. Since the sample size is even, we take the average of the middle two values:
Median = (75 + 80) / 2 = 77.5
The first quartile (Q₁) is the median of the lower half of the data, and the third quartile (Q₃) is the median of the upper half of the data. We can use the ordered data to find these values:
Ordered data: 60, 62, 63, 64, 65, 70, 75, 80, 85, 90
Lower half: 60, 62, 63, 64, 65, 70
Upper half: 75, 80, 85, 90
Q₁ = median of lower half = (64 + 65) / 2 = 64.5
Q₃ = median of upper half = (80 + 85) / 2 = 82.5
Therefore, the interquartile range is:
IQR = Q₃ - Q₁ = 82.5 - 64.5 = 18
To find the range, we subtract the minimum value from the maximum value:
Range = 90 - 60 = 30
(c) To comment on the skewness of the data, we can compare the mean, median, and mode. If the mean is equal to the median and mode, then the data is symmetrical. If the mean is greater than the median, then the data is positively skewed. If the mean is less than the median, then the data is negatively skewed.
Mean = (60 + 62 + 63 + 64 + 65 + 70 + 75 + 80 + 85 + 90) / 10 = 72.4
Median = 77.5
Mode = there is no mode
Since the mean is less than the median, the data is negatively skewed.
(d) To find the mean of the combined sample, we can use the formula:
Mean = (sum of all data) / (number of data)
The sum of the data in the original sample is:
60 + 62 + 63 + 64 + 65 + 70 + 75 + 80 + 85 + 90 = 694
The sum of the data in the new sample is:
50 + 60 + 70 + 80 + 90 + 100 + 110 = 560
The sum of all the data is:
694 + 560 = 1254
The number of data is 20 + 10 = 30
Therefore, the mean of the combined sample is:
Mean = 1254 / 30 = 41.8
To find the range of the combined sample, we subtract the minimum value from the maximum value:
Range = 110 - 50 = 60
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