Find The Circumference of the following circle in terms of Pi. A. 25piB. 50piC 12 5pi
Answers
Solution
Given the circle with 25 yard diameter
Circumference of a circle = 2πr
[tex]\begin{gathered} diameter\text{ =25yrd} \\ d=2r \end{gathered}[/tex]Circumference = πd
[tex]C=25\pi[/tex]Therefore the correct answer = 25pi
Hence the correct answer is Option A
Related Questions
about 23% of people are at a higher risk of stroke due to other medical conditions like high blood pressure. their risk is about 9% of stroke compared with the general population's 3% chance of having a stroke in their lifetime.
Answers
Pls help me!!!!!!!!!
Answers
2. 4+ (-10)
3. 3+(-15)
4.2+5
5. (-10)+(-5)
six teachers share 4 packs of paper equally.how much paper does each teacher get
Answers
Six teachers share 4 packs of paper
Each teacher gets
[tex]\frac{4}{6}=\frac{2}{3}\text{ = 4 sixths of a pack, option C}[/tex]The answer is option C
2. The area A of a rectangle is represented by the formula A = Lw, where Lis the length and wis the width. The length of the rectangle is 5. Write anequation that makes it easy to find the width of the rectangle if we knowthe area and the length.
Answers
1) Considering that the Area of a rectangle is given as:
[tex]A=lw[/tex]2) We can then write the following equation plugging into that the length= 5.
Say the area is "A", then we can find the width this way:
[tex]\begin{gathered} A=5ww \\ 5w=A \\ \frac{5w}{5}=\frac{A}{5} \\ w=\frac{A}{5} \end{gathered}[/tex]Note that we rewrote that to solve it for w (width).
All we need is to plug into the A the quantity of the area of this rectangle
Thus, the answer is w=A/5
The probability distribution for arandom variable x is given in the table.-105101520Probability.2015051.25115Find the probability that -5 < x < 5
Answers
Answer:
P = 0.30
Explanation:
From the table, we see that:
When: x = -5, Probability = 0.15
When: x = 0, Probability = 0.05
When: x = 5, Probability = 0.1
Therefore, the probability of -5 ≤ x ≤ 5 is obtained by the sum of the probabilities from -5 to 5, we have:
[tex]\begin{gathered} P=0.15+0.05+0.10 \\ P=0.30 \end{gathered}[/tex]Therefore, the probability is 0.30
28 * 81.5 can you help me
Answers
so the answer is 2282
Determine whether point (4, -3) lies on the line with equation y = -2x + 5 by using substitution and by graphing.
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The equation of the line is:
[tex]y=-2x+5[/tex]And we need to find if the point (4,-3) lies on the line.
To solve by using substitution, we need to remember that an ordered pair always has the form (x, y) --> the first number is the x-value, and the second number is the y-value.
In this case (4,-3):
x=4
and
y=-3
So now, we substitute the x value into the equation, and if the y-value we get in return is -3 --> the point lies on the line. If not, the point does not lie on the line.
[tex]\begin{gathered} y=-2x+5 \\ \text{Substituting x=4} \\ y=-2(4)+5 \\ y=-8+5 \\ y=-3 \end{gathered}[/tex]We do get -3 as the y-value which matches with the indicated y value of the point.
Thus, by substitution, we confirm that the point lies on the line.
To check the result by graphing, we need to graph the line. The graph of the line is shown in the following image:
In the image, we can see that the marked point on the line is the point we were looking for (4,-3). So we confirm by the graphing method that the point lies on the line.
Answer: It is confirmed by substitution and graphing that the point (4,-3) lies on the line.
A population of a certain species of bird is 22,000 animals and is decreasing by 700 birds per year..Write an equation for y, the population at time t (in years), representing the situation.y= How many birds are in the population after 7 years?
Answers
We can solve that problem using a linear function, we know that
[tex]y=mx+y_0[/tex]Where y0 is the initial population and m is the rate of decreasing, we know that for each "x" years we have -700 birds, therefore
[tex]y=-700x+22000[/tex]Let's use t instead of x
[tex]y=-700t+22000[/tex]That's the equation that represents the population at time t
[tex]\begin{gathered} y=-700t+22000\text{ \lparen t = 7\rparen} \\ \\ y=-700\cdot(7)+22000 \\ \\ y=-4900+22000 \\ \\ y=17100 \end{gathered}[/tex]Therefore after 7 years, the population will be 17100 birds.
Use dimensional analysis to determine which rate is greater. The pitcher for the Robins throws a baseball at 90.0 miles per hour. The pitcher on the Bluebirds throws a baseball 125.4 feet per second. Which pitcher throws a baseball faster? Complete the explanation:When I convert the Bluebirds pitcher's speed to the same units as the Robins pitcher's speed the speed is __ mi/h. Since the Bluebirds pitcher's speed is ____ the Robins pitcher's speed, the pitcher on the ____ throws a faster ball.
Answers
ANSWER and EXPLANATION
We want to solve the problem by using dimensional analysis.
To do this, let us convert the speed of the Bluebirds baseball to miles per hour.
We have that:
1 feet per second = 0.6818 miles per hour
125.4 feet per second = 85.50 miles per hour
As we can see the baseball of the Bluebirds is slower than the Robins (90 miles per hour)
Now, to complete the explanation:
When I convert the Bluebirds pitcher's speed to the same units as the Robins pitcher's speed, the speed is _85.50_ mi/h.
Since the Bluebirds pitcher's speed is _less than_ the Robins pitcher's speed, the pitcher on the __Robins_ throws a faster ball.
How do you slove this promblem 207.4÷61
Answers
we have
207.4÷61
[tex]207.4\div61=\frac{207.4}{61}=\frac{2,074}{610}=\frac{1,830}{610}+\frac{244}{610}=3+\frac{244}{610}=3\frac{244}{610}[/tex]simplify
244/610=122/305=4/10=2/5
therefore
the answer is 3 2/5Use the formula for the probability of the complement of an event.A single card is drawn from a deck. What is the probability of not drawing a 7?
Answers
occur
the answer is 12/13 or 0.932
Explanation
when you have an event A, the complement of A, denoted by.
[tex]A^{-1}[/tex]consists of all the outcomes in wich the event A does NOT ocurr
it is given by:
[tex]P(A^{-1})=1-P(A)[/tex]Step 1
find the probability of event A :(P(A)
The probability of an event is the number of favorable outcomes divided by the total number of outcomes possible
[tex]P=\frac{favorable\text{ outcomes}}{\text{total outcomes}}[/tex]so
let
favorable outcome = 4 (there are four 7 in the deck)
total outcomes=52
hence,replacing
[tex]\begin{gathered} P=\frac{4}{52}=\frac{1}{13} \\ P(A)=\frac{1}{13} \end{gathered}[/tex]Step 2
now, to find the probability that the event does NOT ocurrs ( not drawing a 7)
let's apply the formula
[tex]P(A^{-1})=1-P(A)[/tex]replace
[tex]\begin{gathered} P(A^{-1})=1-\frac{1}{13} \\ P(A^{-1})=\frac{13-1}{13}=\frac{12}{13} \\ P(A^{-1})=0.923 \end{gathered}[/tex]therefore, the answer is 12/13 or 0.932
I hope this helps you
Which value of x makes the equation true 3x-6/3= 7x-3/6
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The value of x that makes the equation true is - 3 / 8.
How to solve equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Therefore, the value of x that makes the equation true is the value that makes the two sides of the equation equal.
Hence,
3x - 6 / 3 = 7x - 3 / 6
3x - 2 = 7x - 1 / 2
add 2 to both sides of the equation
3x - 2 = 7x - 1 / 2
3x - 2 + 2 = 7x - 1 / 2 + 2
3x = 7x - 1 / 2 + 2
3x = 7x + 3 / 2
subtract 7x from both side of the equation
3x - 7x = 7x - 7x + 3 / 2
- 4x = 3 / 2
cross multiply
- 8x = 3
x = - 3 / 8
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The value of x which makes the equation true is - 3 / 8.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given equation is 3x-6/3= 7x-3/6
Three x minus six divided by three equal to seven times of x minus three divided by six
3x-6/3= 7x-3/6
(9x-6)/3=(42x-3)/6
Apply cross multiplication
6(9x-6)=3(42x-3)
Apply distributive property
54x-36=126x-9
add 36 on both sides
54x=126x-9+36
54x=126x+27
-27=126x-54x
-27=72x
x=-27/72
x=-9/24=-3/8
Hence value of x is -3/8 for equation 3x-6/3= 7x-3/6.
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after a 30% discount, Kay's sneakers cost $36, if she would have waited another week to buy them, they would have been on sale for 40% off the original price, how much more money could she have saved
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Let the original price of sneakers be x.
Determine the original price of the sneakers.
208 x 26 using long multiplication
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Answer:
2 0 8
× 2 6
+ 1 2 4 8
+ 4 1 6
= 5 4 0 8
The Answer of 208 × 26 Is 5.408
Explanation.= 208 × 26
= (208 × 6) + (208 × 20)
= 1.248 + 4.160
= 5.408
__________________
Class: Elementary School
Lesson: Multiplication
[tex]\boxed{ \colorbox{lightblue}{ \sf{ \color{blue}{ Answer By\:CyberPresents}}}}[/tex]
Write 5^-15 with a positive exponent
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Given:
[tex]5^{-15}[/tex]To change a negative exponent to a positive exponent, the variable will change from numerator to denominator and vice versa.
For example:
[tex]\begin{gathered} P^{-1}\text{ = }\frac{1}{P} \\ \\ We\text{ know that:} \\ 5^{-15}=5^{(15)-1} \\ \\ 5^{(15)-1}\text{ = }\frac{1}{5^{15}} \end{gathered}[/tex]Therefore, we have:
[tex]5^{-15}\text{ = }\frac{1}{5^{15}}[/tex]ANSWER:
[tex]\frac{1}{5^{15}}[/tex]5. The number of hours spent in an airplane on a single flight is recordedon a dot plot. The mean is 5 hours. The median is 4 hours. The IQR is 3hours. The value 26 hours is an outlier that should not have been includedin the data. When 26 is removed from the data set, calculate the following(some values may not be used):*H0 2 4 6 8 10 12 14 16 18 20 22 24 26 28number of hours spent in an airplane1.4 hours1.5 hours3 hours3.5 hoursWhat is themean?OWhat is themedian?оOOWhat is the IQR?OOOO
Answers
Solution
Since the outlier that is 26 has been removed
We will work with the remaining
Where X denotes the number of hours, and f represent the frequency corresponding to eaxh hours
We find the mean
The mean (X bar) is given by
[tex]\begin{gathered} mean=\frac{\Sigma fx}{\Sigma f} \\ mean=\frac{1(2)+2(2)+3(3)+4(3)+5(2)+6(2)}{2+2+3+3+2+2} \\ mean=\frac{2+4+9+12+10+12}{2+2+3+3+2+2} \\ mean=\frac{49}{14} \\ mean=\frac{7}{2} \\ mean=3.5 \end{gathered}[/tex]We now find the median
Median is the middle number
Since the total frequency is 14
The median will be on the 7th and 8th term in ascending order
[tex]\begin{gathered} median=\frac{7th+8th}{2} \\ median=\frac{3+4}{2} \\ median=\frac{7}{2} \\ median=3.5 \end{gathered}[/tex]Lastly, we will find the interquartile range
The formula is given by
[tex]IQR=Q_3-Q_1[/tex]Where
[tex]\begin{gathered} Q_3=\frac{3}{4}(n+1)th\text{ term} \\ Q_1=\frac{1}{4}(n+1)th\text{ term} \end{gathered}[/tex]We calculate for Q1 and Q3
[tex]\begin{gathered} Q_1=\frac{1}{4}(n+1)th\text{ term} \\ \text{n is the total frequency} \\ n=14 \\ Q_1=\frac{1}{4}(14+1)th\text{ term} \\ Q_1=\frac{1}{4}(15)th\text{ term} \\ Q_1=3.75th\text{ term} \\ Q_1\text{ falls betwe}en\text{ the frequency 3 and 4 in ascending order} \\ \text{From the table above} \\ Q_1=2 \end{gathered}[/tex][tex]\begin{gathered} Q_3=\frac{3}{4}(n+1)th\text{ term} \\ Q_3=\frac{3}{4}(14+1)th\text{ term} \\ Q_3=\frac{3}{4}(15)th\text{ term} \\ Q_3=11.25th\text{ term} \\ \text{From the table above} \\ Q_3=5 \end{gathered}[/tex]Therefore, the IQR is
[tex]\begin{gathered} IQR=Q_3-Q_1 \\ IQR=5-2 \\ IQR=3 \end{gathered}[/tex]A rainstorm in Portland, Oregon, wiped out the electricity in 10% of the households in the city. Suppose that a random sample of 50 Portland households is taken after the rainstorm.Answer the following.(a)Estimate the number of households in the sample that lost electricity by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response.(b)Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.
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Solution
Question A:
[tex]\begin{gathered} Mean=np \\ where, \\ n=\text{ Number of sample values} \\ p=\text{ Probability of mean} \\ \\ n=50 \\ p=10\%=\frac{10}{100} \\ \\ \therefore Mean=50\times\frac{10}{100}=5 \end{gathered}[/tex]- The number of households in the sample that lost electricity is 5
Question B:
[tex]\begin{gathered} \sigma=\sqrt{npq} \\ where, \\ \sigma=\text{ Standard deviation} \\ n=\text{ Number of data points in the sample} \\ p=\text{ Probability of obtaining the mean} \\ q=\text{ Probability of NOT}obtaining\text{ the mean}=1-p \\ \\ n=50 \\ p=10\%=0.1 \\ q=1-0.1=0.9 \\ \\ \sigma=\sqrt{50\times0.1\times0.9} \\ \sigma=2.121320343...\approx2.121 \end{gathered}[/tex]- The standard deviation is 2.121
Final answers
- The number of households in the sample that lost electricity is 5
- The standard deviation is 2.121
The figure below shows a rectangular court. 74 ft (a) Use the calculator to find the area and perimeter of the court. Make sure to include the correct units. Area: 93 ft Perimeter: (b) The court will have a wood floor. Which measure would be used in finding the amount of wood needed? Perimeter O Area (c) A strip of tape will be placed around the court. Which measure would be used in finding the amount of tape needed? Perimeter O Area ft X ft² Ś ft³ ?
Answers
Given a rectangle with sides "a" and "b":
The area of the rectangle is:
[tex]A=ab[/tex]The perimeter of the rectangle is:
[tex]P=2a+2b[/tex]Given the sides of the rectangle:
a = 74 ft
b = 93 ft
(a)
The area of the rectangle is:
[tex]\begin{gathered} A=74ft*93ft \\ A=6882ft^2 \end{gathered}[/tex]The perimeter of the rectangle is:
[tex]\begin{gathered} P=2*74ft+2*93ft \\ P=148ft+186 \\ P=334ft \end{gathered}[/tex](b) The wood will cover all the area of the court, then the area must the used.
(c) The tape will be placed around the court, then the perimeter must be used.
Answer:
(a)
(b)
(c) Perimeter
Solve the following system of linear equations using elimination.
x – y - 3z = 4
2x + 3y – 3z = -2
x + 3y – 2z = -4
Answers
By applying the elimination method, the solutions to this system of three linear equations include the following:
x = 2.y = -2.z = 0.How to solve these system of linear equations?In order to determine the solutions to a system of three linear equations, we would have to evaluate and eliminate each of the variables one after the other, especially by selecting a pair of linear equations at each step and then applying the elimination method.
Given the following system of linear equations:
x – y - 3z = 4 .........equation 1.
2x + 3y – 3z = -2 .........equation 2.
x + 3y – 2z = -4 .........equation 3.
From equation 1 and equation 3, we would eliminate x as follows:
x – y - 3z = 4
x + 3y – 2z = -4
-4y - z = 8 .........equation 4.
Next, we would pick a different pair of linear equations to eliminate x:
(x – y - 3z = 4) × 2 ⇒ 2x - 2y - 6z = 8
2x - 2y - 6z = 8
2x + 3y - 3z = -2
-5y - 3z = 10 ........equation 5.
From equation 4 and equation 5, we would eliminate z to get the value of y:
(-4y - z = 8) × 3 ⇒ -12y - 3z = 24
-12y - 3z = 24
-5y - 3z = 10
-7y = 14
y = 14/7
y = -2.
For the value of z, we have:
-4y - z = 8
z = -4y - 8
z = -4(-2) - 8
z = 8 - 8
z = 0
For the value of x, we have:
x – y - 3z = 4
x = 4 + y + 3z
x = 4 - 2 + 3(0)
x = 2
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Divide 8 1/8 by 7 1/12 simplify the answer and write as a mixed number
Answers
The division of 8 1/8 by 7 1/12 is 91/136.
What is division?Division simply has to do with reduction of a number into different parts. On the other hand, a mixed number is the number that's made up of whole number and fraction.
Dividing 8 1/8 by 7 1/12 will go thus:
8 1/8 ÷ 7 1/12
Change to improper fraction
65/8 ÷ 85/7
= 65/8 × 7/85
= 91/136
The division will give a value of 91/136.
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The volume of a right circular cylinder with a radius of 4 in. and a height of 12 in. is ___ π in^3.
Answers
For the given right cylinder:
Radius = r = 4 in
Height = h = 12 in
The volume of the cylinder =
[tex]\pi\cdot r^2\cdot h=\pi\cdot4^2\cdot12=192\pi[/tex]So, the answer will be the volume is 192π in^3
he center of the circle below is at P. If arc AB measures 86 °, then what is the measure of the angle < APB ?
Answers
Answer:
D. 86°
Explanation:
Given:
• The center of the circle = P
,• The measure of arc AB = 86°
We want to find the measure of the angle APB.
By Circle's theorem: The measure of an arc is equal to the measure of the central angle subtended by the same arc.
Applying this theorem, we have that:
[tex]\begin{gathered} m\angle APB=m\widehat{AB} \\ \implies m\angle APB=86\degree \end{gathered}[/tex]The measure of the angle APB is 86 degrees.
Option D is correct.
In the figure, m < 1= (x-6)º and m2 2= (5x).
Answers
We have the measure of angles 1 and angle 2, as we can see from the diagram in the image, angles 1 and 2 added form the right angle (90°) in the figure.
Thus, the sum of x-6 and 5x, must be equal to 90°.
(a) Write an equation:
[tex]x-6+5x=90[/tex](b) To find the degree measure of each angle, first we need to solve for the value of x in the equation.
Combining like terms:
[tex]6x-6=90[/tex]Adding 6 from both sides:
[tex]\begin{gathered} 6x-6+6=90+6 \\ 6x=96 \\ \end{gathered}[/tex]Divide both sides by 6:
[tex]\begin{gathered} \frac{6x}{6}=\frac{96}{6} \\ x=16 \end{gathered}[/tex]Now that we have x, we find angle 1:
[tex]m\angle1=x-6=16-6=10[/tex]And the measure of angle 2:
[tex]m\angle2=5x=5(16)=80[/tex]The point (4, 16) is on the graph of f(x) = 2^x. Determine the coordinates of this point under the following transformations.
f(x) = 2^4x: ____________
Answers
The coordinate of the image after the transformation is (4, 65536)
How to determine the coordinate of the image?From the question, the coordinate of the point is given as
(4, 16)
From the question, the equation of the function is given as
f(x) = 2^x
When the function is transformed. we have the equation of the transformed function to be given as
f(x) = 2^4x65536
So, we substitute 4 for in the equation f(x) = 2^4x
So, we have
f(4) = 2^(4 x 4)
Evaluate the products
f(4) = 2^16
Evaluate the exponent
f(4) = 65536
So, we have (4, 65536)
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there are about 6*10^24 molecules in a litre of water. it is estimated that a person drinks about 2.2 *10^3 litres of water a year. how many molecules of water does a person drink in a year?
Answers
As per the concept of multiplication, the amount of molecules of water does a person drink in a year is 13.2 x ²⁷ or 1.32 x 10²⁸.
Molecules:
Molecules are referred as the smallest particle of a substance that has all of the physical and chemical properties of that substance. It is made up of one or more atoms.
Given,
There are about 6 x 10²⁴ molecules in a liter of water. it is estimated that a person drinks about 2.2 x 10³ liters of water a year.
Here we need to find the amount of molecules of water does a person drink in a year.
To calculate the total amount of molecules consumption for the year we have to use the following formula,
That is,
Total molecules per year = amount of water per year x molecules in water.
Here we know that,
the amount of water consumption per year = 2.2 x 10³ liters
And the amount of molecules of one liter water = 6 x 10²⁴
When we apply the values on the formula, then we get,
=> total amount of molecules consumption = (2.2 x 10³) x (6 x 10²⁴)
=> (2.2 x 6) x (10³ x 10²⁴)
=> 13.2 x 10³⁺²⁴
=> 13.2 x ²⁷
Therefore, the amount of molecules of water does a person drink in a year is 13.2 x ²⁷ or 1.32 x 10²⁸.
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help meeeeeeeeee pleaseee !!!!!
Answers
The solution to the composite function is as follows;
(f + g)(x) = x² + 3x + 5(f - g)(x) = x² - 3x + 5(f. g)(x) = 3x³ + 15x(f / g)(x) = x² + 5 / 3xHow to solve composite function?The composition of a function is an operation where two functions say f and g generate a new function say h in such a way that h(x) = g(f(x)).
If we are given two functions, it is possible to create or generate a “new” function by composing one into the other.
Composite functions are when the output of one function is used as the input of another.
In other words, a composite function is generally a function that is written inside another function.
Therefore,
f(x) = x² + 5
g(x) = 3x
Hence, the composite function can be solved as follows:
(f + g)(x) = f(x) + g(x) = x² + 5 + 3x = x² + 3x + 5
(f - g)(x) = f(x) - g(x) = x² + 5 - 3x = x² - 3x + 5
(f. g)(x) = f(x) . g(x) = (x² + 5)(3x) = 3x³ + 15x
(f / g)(x) = f(x) / g(x) = x² + 5 / 3x
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hi, i need help finding the mean and standard deviation. the reeses pieces are just a filler for the sample subject.
Answers
Solution
For this case we can calculate the mean in the following way:
[tex]E(X)=np=30\cdot0.52=15.6[/tex]And the standard deviation would be:
[tex]Sd(x)=\sqrt[=]{30\cdot0.52\cdot(1-0.52)}=2.736[/tex]The data in the table show how long (in minutes, t) it takes several commuters to drive to work. Find the correlation coefficient and the equation of the best fit for the data. Treat the commute distance d as the independent variable.
Answers
Given the set of data
sort
Commute data (x)
24,25,27,30, 35,35,46,50,52
Commute distance (y)
20,20,29,20,34,39,29,34,50
The line of best fit is given by
with
[tex]R^2=0.5592[/tex][tex]R=\sqrt{0.5592}[/tex][tex]R=0.747[/tex]R= 0.75
with function
[tex]t=0.7+5.5[/tex]Correct answer
option D
Answer: r ≈ 0.75
t ≈ 0.8d + 11.5
Step-by-step explanation:
You have to use a graphing calculator to solve this problem.
This is the correct answer (I just took the test).
Which equation is true when the value of x is - 12 ?F: 1/2x+ 22 = 20G: 15 - 1/2x = 21H: 11 - 2x = 17 J: 3x - 19 = -17
Answers
Substitute x = - 12 in each of the given equation, if the equation satisfy then tha x = -1 2
F) 1/2x + 22 = 20
1/2 ( -12) + 22 = 20
(-6) + 22 = 20
16 is not equal to 22
G) 15 -1/2x = 21
Substitute x = -12 in the expression :
15 - 1/2( -12) = 21
15 + 1/2(12) =21
15 + ( 6) = 21
21 = 21
Thus, The equation 15 - 1/2x = 21 is true for x = -12
H) 11 - 2x = 17
Susbstitute x = ( -12) in the equation :
11 - 2x = 17
11 - 2( -12) = 17
11 + 24 = 17
35 = 17
Since, 35 is not equal to 17
D) 3x - 19 = -17
SUsbtitute x = ( -12)
3( -12) - 19 = -17
-36 - 19 = -17
-36 = -17 + 19
-36 = 2
Since - 36 is not equal 2
Answer : G) 15 - 1/2x = 21
I need to know the initial size of the culture Find the doubling period Find the population after 65 min When will the population reach 10000
Answers
Given:
The population was 100 after 10 mins.
The population was 1500 after 30 mins.
To fill the blanks:
Explanation:
According to the problem, we write,
[tex]\begin{gathered} P=P_0e^{kt} \\ 100=P_0e^{10k}.........(1) \\ 1500=P_0e^{30k}............(2) \end{gathered}[/tex]Dividing equation (2) by equation (1), we get
[tex]\begin{gathered} \frac{1500}{100}=\frac{P_0e^{30k}}{P_0e^{10k}} \\ 15=e^{20k} \\ ln15=20k \\ 2.708=20k \\ k=\frac{2.708}{20} \\ k=0.1354 \end{gathered}[/tex]So, the equation becomes,
[tex]P=P_0e^{0.1354t}....................(3)[/tex]a) To find: The initial population
When P = 100 and t = 10, then the initial population would be,
[tex]\begin{gathered} 100=P_0e^{0.1354(10)} \\ 100=P_0e^{1.354} \\ 100=P_0(3.873) \\ P_0=\frac{100}{3.873} \\ P_0\approx25.82 \end{gathered}[/tex]Therefore, the initial population is 25.82.
b) To find: The doubling time
Using the formula,
[tex]\begin{gathered} t=\frac{\ln2}{k} \\ t=\frac{\ln2}{0.1354} \\ t=5.1192 \\ t\approx5.12mins \end{gathered}[/tex]The doubling time is 5.12 mins.
c) To find: The population after 65 mins
Substituting t = 65 and the initial population is 25.82 in equation (3) we get,
[tex]\begin{gathered} P=25.82e^{0.1354(65)} \\ P\approx171467.56 \end{gathered}[/tex]Therefore, the population after 65 mins is 171467.56.
d) To find: The time taken for the population to reach 10000
Substituting P = 10000 and the initial population is25.82 in equation (3) we get,
[tex]\begin{gathered} 10000=25.82e^{0.1354t} \\ e^{0.1354t}=\frac{10000}{25.82} \\ e^{0.1354t}=387.297 \\ 0.1354t=\ln(387.297) \\ 0.1354t=5.959 \\ t=\frac{5.959}{0.1354} \\ t\approx44.01 \end{gathered}[/tex]Therefore, the time taken for the population to reach 10000 is 44.01 mins.
Final answer:
• The initial population is 25.82.
,• The doubling time is 5.12 mins.
,• The population after 65 mins is 171467.56.
,• The time taken for the population to reach 10000 is 44.01 mins.