The given situation can be written in an algebraic way:
Say x the width of the rectangle and y its height.
- The width of a rectangle is 8 inches less than its length:
x = y - 8
- The area of the rectangle is 9 square inches:
xy = 9
In order to find the values of y and x, you first replace the expression
x = y - 8 into the expression xy = 9, just as follow:
[tex]\begin{gathered} xy=9 \\ (y-8)y=9 \end{gathered}[/tex]you apply distribution property, and order the equation in such a way that you obtain the general form of a quadratic equation:
[tex]\begin{gathered} (y-8)y=9 \\ y^2-8y=9 \\ y^2-8y-9=0 \end{gathered}[/tex]Next, you use the quadratic formula to solve the previous equation for y:
[tex]y=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]here you have a = 1, b = -8 and c = 9. By replacing these values you obtain:
[tex]\begin{gathered} y=\frac{-(-8)\pm\sqrt[]{(-8)^2-4(1)(-9)}}{2(1)}=\frac{8\pm\sqrt[]{64+36}}{2} \\ y=\frac{8\pm\sqrt[]{100}}{2}=\frac{8\pm10}{2}=\frac{8}{2}\pm\frac{10}{2}=4\pm5 \end{gathered}[/tex]Hence, you have two solutions for y:
y1 = 4 + 5 = 9
y2 = 4 - 5 = -1
You select only the positive solution, because negative lengths do not exist in real life. Hence, you have y = 9.
Finally, you replace the value of y into the expression x = y - 8 to obtain x:
[tex]\begin{gathered} x=y-8 \\ x=9-8 \\ x=1 \end{gathered}[/tex]Hence, the width and length of the given recgtangle are:
width = 1 in
length = 9 in
Choose the correct answer below
The book is not the same story or the movie is not the same story.
What is De Morgan's law?The intersection of two sets' complements is the complement of the union of two sets, and the intersection of two sets' complements is the complement of the intersection of two sets. They are referred to as De Morgan's laws. These have the name De Morgan after the mathematician.De Morgan's laws are a pair of transformation rules that can both be used as rules of inference in propositional logic and Boolean algebra. They have the name of the 19th-century British mathematician Augustus De Morgan.When attempting to demonstrate that the NAND gate is equivalent to an OR gate with inverted inputs and the NOR gate is equivalent to an AND gate with inverted inputs, we can employ De Morgan's theorems.To learn more about De Morgan's law refer to:
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Find mCBD. the number might be a bit blurry but it is 192
Circle is 360 degrees.
Arc DB = 360 - 192 = 168°
The measure of angle CBD is half the measure of Arc DB.
Thus,
[tex]\begin{gathered} \angle\text{CBD}=\frac{1}{2}(168) \\ =84\degree \end{gathered}[/tex]F(x) = 5-7x find f(-3)
Answer:
26
Step-by-step explanation:
Just plug -3 in where ever you see x
[tex]f(x)=5-7x\\f(-3)=5-7(-3)\\f(-3)=5+21\\f(-3)=26[/tex]
Use the positions of the numbers on the number line to compare them.Select the two true inequalities.A. 3/4 < 4/5B. 0.85 > 4/5C. 3/4 > 4/5D. 0.85 < 4/5
Answer:
Explanation:
Given:
0.85,4/5, 3/4
To easily compare the given numbers, we simplify each number first and plot them on the number line:
Therefore, the two true inequalities are:
[tex]\frac{3}{4}<\frac{4}{5}[/tex]and
[tex]0.85>\frac{4}{5}[/tex]
HELP PLEASE will give BRAINLIEST!!! You are setting up a zip line in your yard. You map out your yard in a coordinate plane. An equation of the line representing the zip line is
y = 3/2x +6. There is a tree in your yard at the point (6, 2). Each unit in the coordinate plane represents 1 foot. Approximately how far away is the
tree from the zip line? Round your answer to the nearest tenth.
Answer:
Hello lovely. Assume that the attached graph represents your situation, with the red line representing the zip line and the blue dot representing the tree. The tree is at point (6, 2). You will need to choose a reference point to calculate the distance between the tree and the zip line. We'll use the point (0, 6), or the y intercept
To calculate the distance between two points, we use the formula d=√((x2 – x1)² + (y2 – y1)²).
Substitute
d=√((0 – 6)² + (6 – 2)²).
Simplify
d=√((-6)² + (4)²).
d=√(36 + 16).
d = √52
The distance is approximately equal to 7.2 feet
Simplify and give answer as positive exponentkoa) x4. x-7xb)k4
To simplify the expression, we need to use an exponent propertie
[tex]a^n\cdot a^m=a^{n+m}[/tex]Then, we can see that in this case a = x, n = 4 and m = -7
So now we must replace the values
[tex]x^4\cdot x^{-7}=x^{4-7}=x^{-3}[/tex]Need help with my math please..
Answer:
i can't read this very well
Write a situation for this equation
1.5 < 1.67
The inequality equation is correct the way it is in the form 1.5 < 1.67 and will continue to be correct if 1.5x < 1.67 where x is
negative numberx less than or equal to 1What are inequalities?Inequalities as used in mathematics refers to the symbol that is used to related the values in the left hand side and the values at the right hand side of the expression
The symbol used in the inequality expression are
less than or equal togreater than or equal toless thangreater thanThe given expression is less than and read as 1.5 is less than 1.67
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1. Which scatter plot could have a trend line whose equation is y - 3x + 10 (A) 60 60 40 40 20 20 0 y M 10 20 0 10 20 D . 12 60 8 40 4 29 0 10 220 0 10 10 20
Explanation
Given the trend line equation that defines a scatter plot
We will have to substitute the values of x = 2.5,5,7.5,10,15,20 and check the graphs
So, when x =2.5
[tex]\begin{gathered} y=3(2.5)+10=7.5+10=17.5 \\ y=17.5 \end{gathered}[/tex]when x=5
[tex]\begin{gathered} y=3(5)+10=15+10 \\ y=25 \end{gathered}[/tex]When x= 7.5
[tex]y=3(7.5)+10=32.5[/tex]When x =10
[tex]\begin{gathered} y=3(10)+10=40 \\ y=40 \end{gathered}[/tex]If we check all the values obtained to the graph, we will discover that the best option will be
Option B is more correct
Because most of the points conform to the trend line equation
If 16 is added to a number, the result is 35 less than twice the number. Find the number.
Let us represent the number as x:
16 is added to a number is represented as:
[tex]16+x[/tex]The result been 35 less than twice the number is represented as:
[tex]2x-35[/tex]Combining the above expression together to find the number will be:
[tex]16+x=2x-35[/tex]Simplifying further:
[tex]\begin{gathered} 16+35=2x-x \\ 51=x \\ \end{gathered}[/tex]The number, therefore, is 51
Find the 11th term of the arithmetic sequence -5x- 1, -8x + 4, -11 x+ 9, ...
Recall that an arithmetic sequence is a sequence in which the next term is obtained by adding a constant term to the previous one. Let us consider a1 = -5x-1 as the first term and let d be the constant term that is added to get the next term of the sequence. Using this, we get that
[tex]a_2=a_1+d[/tex]so if we replace the values, we get that
[tex]-8x+4=-5x-1+d[/tex]so, by adding 5x+1 on both sides, we get
[tex]d=-8x+4+5x+1\text{ =(-8+5)x+5=-3x+5}[/tex]To check if this value of d is correct, lets add d to a2. We should get a3.
Note that
[tex]a_2+d=-8x+4+(-3x+5)=-11x+9=a_3[/tex]so the value of d is indeed correct.
Now, note the following
[tex]a_3=a_2+d=(a_1+d)+d=a_1+2d=a_1+d\cdot(3-1)[/tex]This suggest the following formula
[tex]a_n=a_1+d\cdot(n-1)[/tex]the question is asking for the 11th term of the sequence, that is, to replace the value of n=11 in this equation, so we get
[tex]a_{11}=a_1+d\cdot(10)=-5x-1+10\cdot(-3x+5)\text{ =-5x-1-30x+50 = -35x+49}[/tex]so the 11th term of the sequence is -35x+49
Move the sliders h and k so that the graph of y = r2 gets shifted up 3 units and to the right 2 units. Then type the new function, f(t) in the answer box 3 2 1 4. بنا -2 0 1 2 3 f(x) -1 h = 0.00 -2 K = 0.00 о Don't forget to shift the graph. Using function notation, i.e. f(x) = , enter the function that results from the transformation.
Given the graph of the function:
[tex]y=x^2[/tex]The graph will be shifted 3 units and to the right 2 units
So, the new vertex will be the point ( 2, 3 )
The new function will be:
[tex]f(x)=(x-2)^2+3[/tex]So, we will adjust the slider on the following values:
[tex]\begin{gathered} h=2 \\ k=3 \end{gathered}[/tex]Write the equation of a line in point slope form that goes through the points (7,-5) and (3,8)
Write the equation of a line in point slope form that goes through the points (7,-5) and (3,8)
step 1
Find the slope
m=(8+5)/(3-7)
m=13/-4
m=-13/4
step 2
write the equation in point slope form
so
y-y1=m(x-x1)
we take the point (7,-5)
substitute
y+5=-(13/4)(x-7)If you take the point (3,8)
we have
y-8=-(13/4)(x-3)Carlos is adding insulation to a room he just finished framing in his home. The room is 16ft. by 12ft., and the ceilings are 9ft. tall. There are two windows in the room measuring 5ft. by 6ft. each. How many square feet of insulation does Carlos need?
Solution
Now
[tex]A=2(16\times12)+2(16\times9)+2(9\times12)-2(5\times6)[/tex][tex]828ft^2[/tex]square feet of insulation Carlos need is
[tex]828ft^2[/tex]What is the value of f(-5) in the piecewise function -3x + 1 when x > 1 f(x) = -2x when x = 1 2x - 1 when x < 1
Answer:
f(-5)=-11
Explanation:
Given the piecewise function:
[tex]f(x)=\begin{cases}{-3x+1,\text{ when }x>1} \\ {-2x,\text{ when }x=1} \\ {2x-1,\text{ when }x<1}\end{cases}[/tex]We want to find the value of f(-5).
When x=-5:
[tex]\begin{gathered} -5<1\implies f(x)=2x-1 \\ \text{ Therefore:} \\ f(-5)=2(-5)-1 \\ =-10-1 \\ =-11 \end{gathered}[/tex]The value of f(-5) is -11.
Write the equation below in standard form and then answer the following questions. If a value is a non-integer type your answer as a decimal rounded to the hundredths place. 4x^2+24x+25y^2+200y+336=0The center of the ellipse is (h,k). h= Answer and k= AnswerThe value for a is Answer . The value for b is Answer .The foci with the positive x value is the point ( Answer, Answer)The foci with the negative x value is the point ( Answer, Answer)
Given:
[tex]4x^2+24x+25y^2+200y+336=0[/tex]Aim:
We need to convert the given equation into the standard form of the ellipse equation.
Explanation:
Consider the standard form of the ellipse equation.
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]Consider the given equation.
[tex]4x^2+24x+25y^2+200y+336=0[/tex][tex]Use\text{ }336=36+400-100.[/tex][tex]4x^2+24x+25y^2+200y+36+400-100=0[/tex][tex]4x^2+24x+36+25y^2+200y+400-100=0[/tex]Take out the common terms.
[tex]4(x^2+6x+9)+25(y^2+8y+16)-100=0[/tex]Add 100 on both sides of the equation.
[tex]4(x^2+6x+9)+25(y^2+8y+16)-100+100=0+100[/tex][tex]4(x^2+6x+9)+25(y^2+8y+16)=100[/tex][tex]4(x^2+2\times3x+3^2)+25(y^2+2\times4y+4^2)=100[/tex][tex]\text{Use (a+b)}^2=a^2+2ab+b^2.[/tex][tex]4(x+3)^2+25(y+4)^2=100[/tex]Divide both sides by 100.
[tex]\frac{4\mleft(x+3\mright)^2}{100}+\frac{25\mleft(y+4\mright)^2}{100}=\frac{100}{100}[/tex][tex]\frac{\mleft(x+3\mright)^2}{25}+\frac{\mleft(y+4\mright)^2}{4}=1[/tex][tex]\frac{\mleft(x+3\mright)^2}{5^2}+\frac{\mleft(y+4\mright)^2}{2^2}=1[/tex][tex]\frac{\mleft(x-(-3)\mright)^2}{5^2}+\frac{\mleft(y-(-4)\mright)^2}{2^2}=1[/tex]The standard form of the given equation is
[tex]\frac{\mleft(x-(-3)\mright)^2}{5^2}+\frac{\mleft(y-(-4)\mright)^2}{2^2}=1[/tex]Compare with the general form of the ellipse equation.
We get h=-3, k=-4, a=5 and b=2.
The centre of the ellipse is h= -3 and k = -4.
The value of a is 5.
The value of b is 2.
We need to find the eccentricity of the ellipse.
[tex]e=\sqrt[]{1-\frac{b^2}{a^2}}[/tex]Substitute b=2 and a =5 in the formula.
[tex]e=\sqrt[]{1-\frac{2^2}{5^2}}=\sqrt[]{1-\frac{4}{25}}=\sqrt[]{\frac{25-4}{25}}=\sqrt[]{\frac{21}{25}}=0.9165[/tex][tex]e=0.9165[/tex]The foci of the ellipse are
[tex]((h\pm a)e,0)[/tex]Substitute h =-3, a=5 and e =0.9165 in the formula.
[tex]((-3\pm5)0.9165,0)[/tex]The foci with a positive x value are the point
[tex]((-3+5)0.9165,0)\text{ =}(1.83,0)[/tex]
[tex](1.83,0)[/tex]
The foci with a negative x value are the point
[tex]((-3-5)0.9165,0)\text{ =}(-7.33,0)[/tex][tex](-7.33,0)[/tex]x- sq root 6 is a factor of x^4-36 true or false
We want to know if (x-sqroot(6)) is a factor of (x^4 - 36)
That's mean:
[tex](x^4-36)=(x-\sqrt[]{6})\text{ P(x)}[/tex]Where P(X) is a polinomial.
In this case, if x = sqroot(6) the polinomail (x^4 - 36) must be zero, that's mean sqroot(6) is a root (or a zero) of (x^4-36).
So, if we evaluate (x^4 - 36) in x=sqroot(6):
[tex](\sqrt[]{6})^4-36=6^2-36=0[/tex]So, the answer is true.
Evaluate each function. Be sure to show your substitutions.h(x) = 7x^2 - 4x-15h(20)
The function is given as,
[tex]h(x)=7x^2-4x-15[/tex]The objective is to determine the value h(20).
This can be obtained by substituting 20 for 'x' in the given expression,
[tex]\begin{gathered} h(20)=7(20)^2-4(20)-15 \\ h(20)=7(400)-80-15 \\ h(20)=2800-95 \\ h(20)=2705 \end{gathered}[/tex]Thus, the value of the given function h(20) is 2705.
Colin is playing a video game. He wins 25 points for each gold coin he finds. His goal is to win more than 200 poijts. He wants to know how many gold coins he needs to find.
25 points for each gold coin
He wants more tha 200 points
Number of coins to get 200 points: = 200/25 = 8
Answer:
He needs to find 8 gold coins or more
>= 8
At an appliance store, if 63 stereos were sold during a one-month period, which of the following must be true?A. At least one stereo was sold on each day of the monthB. Exactly two stereos were sold on the same day during the monthC. At least one stereo was sold on either Monday, Wednesday, or Friday during the monthD. At least three stereos were sold on one day of the month.
Answer:
Alternative D. At least three stereos were sold on one day of the month.
Explanation:
Now, let's evaluate the options:
A. At least one stereo was sold on each day of the month
It is false.
We can not affirm that. For example, all the stereos can be sold on only one day of the month
B. Exactly two stereos were sold on the same day during the month
It is false.
Same explanation as A.
C. At least one stereo was sold on either Monday, Wednesday, or Friday during the month
It is false.
We can not affirm that too. The explanation is the same as for alternative A.
D. At least three stereos were sold on one day of the month.
It is true.
If two stereos are sold every day, for a month of 30 days, 60 stereos are sold. So, on some days 3 or more stereos are sold.
Also, if all the stereos are sold on the same day, more than 3 stereos were also sold.
So, alternative D is correct.
Find the solution(s) to the system of equations represented in the graph.0, −2) and (2, 0) (0, −2) and (−2, 0) (0, 2) and (2, 0) (0, 2) and (−2, 0)
Solution
The solution is the point of intersection.
Therefore, the answer is
[tex](0,2)\text{ and }(-2,0)[/tex]What is the coefficient of the second term in this expression?-k + 10m² - 6 - n² ?
Given the expression:
[tex]-k+10m^2-6-n^{2^{}}[/tex]The second term in the expression means the 2nd term from left to right of an expression.
Here, the second term is 10m².
A coefficient is a number that is being multiplied by the variable.
Therefore, the coefficient of the term 10m² is 10.
q divided by 4 + 8q, for q=8
We have to calculate the value of the expression:
[tex]\frac{q}{4+8q}[/tex]when q = 8.
To calculate this, we replace q with its value and solve as:
[tex]\frac{q}{4+8q}=\frac{8}{4+8\cdot8}=\frac{8}{4+64}=\frac{8}{68}=\frac{2}{17}[/tex]Answer: 2/17
I need help with solving residential plots and correlation vs causation how do I solve a liner model from the data ?
The image shows point that have a value that is close to zero, so they are small values
In the question, they say that those points represent the residual plot, that means that they represent the error of the linear model
The error is very small, close to zero
So the residual plot shows a non-random pattern, becuase all the point are close to zero
And then the date can be represented by a linear model
So the answer for the left box is "non-random"
And for the right box is "linear"
On the left box:..... non-random
On the right box...... linear
Which statements are true?Select all that apply.A.The slope of AC is equal to the slope of BC.B.The slope of AC is equal to the slope of BD.C.The slope of AC is equal to the slope of line t.D.ECThe slope of line t is equal toAEE.FBThe slope of line t is equal toFDF.The slope of line t is equal to AB
Line t, segments AC, BC and BD are colinear, that is, all of them are in the same line. Then, the true statements are
A, B, and C
The figure below shows two parallel lines, k and f, cut by a transversal. What is the value of x?
A 25
B 35
C 45
D 65
Answer:
x=65 0r in other words D
Step-by-step explanation:
110=2x-20
+20 +20
130=2x
/2 /2
65=x
Determine the effect on the graph of the parent f(x)=x
To answer this question we first graph the parent function
Now we compare the two graphs. We notice that the graph shown is translated two units up.
To translate the graph of function we have to add the ammount we want to translate, then in this case
[tex]g(x)=f(x)+2[/tex]Therefore the answer is J.
The absolute value of 1/4
Answer: 1/4 is the absolute
Step-by-step explanation:
Answer:
1/4
Step-by-step explanation:
Absolute value just means the distance from zero.
Jason enjoys watching the squirrels in his neighborhood park. They eat the red oak acorns. After the city removed 4 diseased red oak trees, the population of squirrels decreased from 105 to 98 in one year. If the population continues to decline at the same rate, how many squirrels will live in the park in 15 years? First, calculate the rate of decay by subtracting the two populations and dividing the difference by the initial population. Then, use the formula A=a0e^kt
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given parameters
[tex]\begin{gathered} Initial\text{ squirrels}=105 \\ Num\text{ber of squirrels after one year}=98 \\ change\text{ in number of squirrels in a year}=105-98=7 \\ chan\text{ge in diseased oak trees}=y-4 \end{gathered}[/tex]STEP 2: Calculate the rate of decay (k)
[tex]\begin{gathered} rate\text{ of decay\lparen k\rparen}=\frac{Final\text{ population-Initial population}}{initial\text{ population}} \\ \text{By substitution,} \\ k=\frac{98-105}{105}=\frac{-7}{105}=-0.06666666\approx-0.0667 \end{gathered}[/tex]STEP 3: Calculate the number of squirrels after 15 years
[tex]\begin{gathered} A=a_0e^{kt} \\ a_0=105 \\ k=-0.0667 \\ t=15 \end{gathered}[/tex]By substitution,
[tex]A=105\cdot e^{-0.0667\times15}[/tex]By simplification,
[tex]\begin{gathered} \mathrm{Apply\:exponent\:rule}:\quad \:a^{-b}=\frac{1}{a^b} \\ =105\times \frac{1}{e^{15\times \:0.0667}} \\ \mathrm{Multiply\:fractions}:\quad \:a\times \frac{b}{c}=\frac{a\:\times \:b}{c} \\ =\frac{1\times \:105}{e^{1.0005}} \\ \mathrm{Multiply\:the\:numbers:}\:1\times \:105=105 \\ =\frac{105}{e^{1.0005}} \\ e^{1.0005}=2.71964 \\ =\frac{105}{2.71964} \\ \mathrm{Divide\:the\:numbers:}\:\frac{105}{2.71964}=38.60803 \\ =38.60803 \end{gathered}[/tex]By approximation, this leaves us with 34 squirrels
S = 2^0 + 2^1 + 2^2 + 2^3 + ...... 2^99a) Show that S can be divided by 15b) Show that S has at least 30 digits
Answer:
Explanation:
Here, we want to show that the sum is divided by 15
From what we have, the given sum is a geometric sequence
The first term is 1
Now, the pattern of ending afterwards will be 2, 4, 6 and 8
This ending keeps repeating itself
This 2,4,6,8 pattern could repeat itself 24 times
So we have a total of 1 + 24(4) = 97 terms
To make it 100, we have the last three terms as 2,4 and 8
So we have the ending number ONLY sum as follows:
1 + 24(2+4+6+8) + 2 + 4 + 8 = 1 + 480 + 14 = 495
We can divide this by 15 and that gives 495/15 = 33
That shows that the sum is divisible by 15
Secondly, we want to show that S has at least 30 digits
We can infer this from the last terms
We can write 2^99 as 2^33 * 2^33 * 2^33
A single 2^33 has a value of 8,589,934,592
That means 10 digits
The other two has 10 digits too
The sum of all possible digits in the largest term is 10 + 10 + 10 = 30
That makes a total of 30
The question states 30 or more
Hence, this is correct