Let {N(t), t>0} be a Poisson process with rate (t)= e^-t. Let (Y_n)_n be a sequence of independent and identically distributed random variables with density f(y) = e^(-y+theta), for y > theta and theta > 0. For all n >= 1, Y_n id independent of the process {N(t), t>0}.
a) Compute the mean and variance of Y_1
b) If X(t) = , find the expectation of X(t)
c) Find the limit
Answers
a) The mean of Y_1 is theta + 1, and the variance of Y_1 is 1. b) The expectation of X(t) is theta * (1 - e^(-t)). c) The limit as t approaches infinity of X(t) is theta.
a) The mean of Y_1 can be computed by integrating y * f(y) over the range y > theta. In this case, the integral of y * e^(-y+theta) can be simplified to (theta + 1), which gives us the mean of Y_1 as theta + 1. The variance of Y_1 can be obtained by integrating y^2 * f(y) over the same range. Simplifying the integral gives us the variance of Y_1 as 1.
b) To find the expectation of X(t), we need to integrate the product of N(t) and Y_1 over all possible values of N(t) and Y_1. Since N(t) follows a Poisson process with rate e^-t, the expectation of N(t) is equal to t. Therefore, the expectation of X(t) can be computed by integrating (theta + 1) * t * e^-t over the range t > 0. Simplifying the integral gives us the expectation of X(t) as theta * (1 - e^(-t)).
c) To find the limit as t approaches infinity of X(t), we evaluate the expression theta * (1 - e^(-t)). As t becomes very large, e^(-t) approaches 0, and thus the limit of X(t) is theta.
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Related Questions
The following are the last 10 run scores colin got in cricket: 16, 11, 25, 27, 11, 25, 20, 26, 29, 35 a) work out colin's mean score. b) colin plays cricket again on sunday. he gets 6 runs. what is his new mean score? give your answers as decimals.
Answers
Colin's new mean score, after getting 6 runs on Sunday, is approximately 20.09.
To calculate Colin's mean score, we need to sum up all his scores and divide by the number of scores.
a) Mean score:
16 + 11 + 25 + 27 + 11 + 25 + 20 + 26 + 29 + 35 = 215
Total scores: 10
Mean score = 215 / 10 = 21.5
Colin's mean score is 21.5.
b) To calculate his new mean score after getting 6 runs on Sunday, we need to add the new score to the previous total and divide by the new number of scores.
New total scores = 215 + 6 = 221
New number of scores = 10 + 1 = 11
New mean score = 221 / 11 = 20.09 (rounded to two decimal places)
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Can someone please help?
Answers
Answer:
x = 18.5
Step-by-step explanation:
103 + 4x + 3 = 180
4x + 106 = 180
4x = 74
x = 18.5
what is the last digit of 3 with a power of 2011
Answers
So to find any last digit of 3^2011 divide 2011 by 4 which comes to have 3 as remainder. Hence the number in units place is same as digit in units place of number 3^3. Hence answer is 7.
If x=-13, what is 54x+(14-x)
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Answers
Answer: use the algebric condistional process
Step-by-step explanation: this is sample only
Assuming the 4 inch by 10 inch base on which the cake sits will not be frosted, what is the area that will require frosting? 352 square inches 372 square inches 392 square inches 432 square inches
Answers
Answer:352 square inches
Step-by-step explanation:edge
A shoemaker sold a pair of for $245.99 if the buyer a $300.00 bill, how much will the buyer receive in change?
*two decimal places don't forget your $ sign. Example: $50.00 NOT 50*
Answers
Answer:
$54.01
Step-by-step explanation:
All you have to do is $300.00-$245.99 .
when reporting the answer to a mathematical calculation involving multiplication or division, how does one determine the number of significant digits to report in the answer? the answer is correctly reported with the same number of decimal places as the value in the calculation with the fewest number of decimal places. the answer is correctly reported with the same number of decimal places as the value in the calculation with the largest number of decimal places. the answer is correctly reported with the same number of significant digits as the value in the calculation with the largest number of significant digits. the answer is correctly reported with the same number of significant digits as the value in the calculation with the fewest number of significant digits.
Answers
The area should be reported with three significant digits, giving a final answer of 19.5 m².
What is the area of the rectangle?
To find the area of a rectangle, we multiply the length of the rectangle by the width of the rectangle.
When reporting the answer to a mathematical calculation involving multiplication or division,
the answer should be correctly reported with the same number of significant digits as the value in the calculation with the fewest number of significant digits.
This is because the number of significant digits in the result cannot be greater than the number of significant digits in the least precise value used in the calculation.
For example, suppose you want to calculate the area of a rectangle with a length of 5.2 meters and a width of 3.75 meters.
The calculation is as follows:
Area = Length x Width
Area = 5.2 m x 3.75 m
Area = 19.5 m²
In this case, the value with the fewest number of significant digits is 3.75, which has three significant digits.
Therefore, the area should be reported with three significant digits, giving a final answer of 19.5 m².
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Four rats are selected at random from a cage of 5 male (M) and 6 female (F) rats. If the random variable Y is concerned with the number of female rats taken out of the cage, construct the probability distribution and answer the questions below. (input up to 4 decimal places)A What is the probability that all selected rats are female?
What is the probability that only one of the selected rats is female?
c. What is the probability that at least two of the selected rats is female?
What is the mean of the probability distribution?
E. What is the standard deviation of the probability distribution?
Answers
Using the hypergeometric distribution, it is found that the distribution is:
P(X = 0) = 0.0152
P(X = 1) = 0.1818
P(X = 2) = 0.4545
P(X = 3) = 0.3030
P(X = 4) = 0.0455
Hence:
A) There is a 0.0455 = 4.55% probability that all selected rats are female.
B) There is a 0.1818 = 18.18% probability that only one of the selected rats is female.
C) There is a 0.803 = 80.3% probability that at least two of the selected rats is female.
D) The mean of the probability distribution is of 2.18.
E) The standard deviation of the probability distribution is of 0.833.
The rats are chosen without replacement, hence the hypergeometric distribution is used to solve this question.
What is the hypergeometric distribution formula?The formula is:
\(P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}\)
\(C_{n,x} = \frac{n!}{x!(n-x)!}\)
The parameters are:
x is the number of successes.N is the size of the population.n is the size of the sample.k is the total number of desired outcomes.In this problem:
There is a total of 11 rats, hence N = 11.6 of the rats are female, hence k = 6.4 rats are taken, hence n = 4.The distribution is the probability of each outcome, hence:
\(P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}\)
\(P(X = 0) = h(0,11,4,6) = \frac{C_{6,0}C_{5,4}}{C_{11,4}} = 0.0152\)
\(P(X = 1) = h(1,11,4,6) = \frac{C_{6,1}C_{5,3}}{C_{11,4}} = 0.1818\)
\(P(X = 2) = h(2,11,4,6) = \frac{C_{6,2}C_{5,2}}{C_{11,4}} = 0.4545\)
\(P(X = 3) = h(3,11,4,6) = \frac{C_{6,3}C_{5,1}}{C_{11,4}} = 0.3030\)
\(P(X = 4) = h(4,11,4,6) = \frac{C_{6,4}C_{5,0}}{C_{11,4}} = 0.0455\)
Item a:
P(X = 4) = 0.0455, hence:
There is a 0.0455 = 4.55% probability that all selected rats are female.
Item b:
P(X = 1) = 0.1818, hence:
There is a 0.1818 = 18.18% probability that only one of the selected rats is female.
Item c:
\(P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.4545 + 0.3030 + 0.0455 = 0.803\)
There is a 0.803 = 80.3% probability that at least two of the selected rats is female.
Item d:
The mean of the hypergeometric distribution is:
\(\mu = \frac{nk}{N}\)
Hence:
\(\mu = \frac{4(6)}{11} = 2.18\)
The mean of the probability distribution is of 2.18.
Item e:
The standard deviation of the hypergeometric distribution is:
\(\sigma = \sqrt{\frac{nk(N-k)(N-n)}{N^2(N-1)}}\)
Hence:
\(\sigma = \sqrt{\frac{4(6)(11-6)(11-4)}{11^2(11-1)}} = 0.833\)
The standard deviation of the probability distribution is of 0.833.
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How to finsd the set of numbers if the mean, median and mode is given
Answers
Answer:
a) we have the numbers 0, 2, 3, 5, 5. The mean and the median are both 3
b) we have the numbers 0, 0, 3, 5, 7. The mean and the median are both 3
In both cases the mean and the median are 3, but the mode differs. The mean and the median do not uniquely determine the mode.
Step-by-step explanation:
AnswerIf the number of data points is even, the median is the average of the two middle data points in the list.:
Step-by-step explanation:
The probability that Sara wins a raffle is given by the expression n n + 3 . Write down an expression, in the form of a combined single fraction, for the probability that Sara does not win.
Answers
Answer:
n/3
Step-by-step explanation:
n×by n is=n square.
n/3
Probability that Sara does not wins a raffle is \(\frac{3}{n+3}\).
What is probability?Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are.
Given
Probability that Sara wins a raffle is given by the expression
P(win) = \(\frac{n}{n+3}\)
Probability that Sara does not wins a raffle is given by
P(does not win) = 1 - \(\frac{n}{n+3}\)
= \(\frac{n+3}{n+3} -\frac{n}{n+3}\)
= \(\frac{n+3-n}{n+3}\)
= \(\frac{3}{n+3}\)
Thus, probability that Sara does not wins a raffle is \(\frac{3}{n+3}\).
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Are the ratios 10:5 and 4:2 equivalent
Answers
Answer:
Yes
Step-by-step explanation:
10/5=2
4/2=2
The ratios are equal
Answer: Yes
Step-by-step explanation:
10:5 and 4:2 round to 2:1 respectively, which are equivalent.
Given: NM ∥ XZ Prove: △XYZ ~ △NYM Triangles X Y Z and N Y M are shown. Triangle X Y Z is smaller and sits inside of triangle N Y M, sharing part of 2 sides with triangle N Y M. The hypotenuses of both triangles are parallel. We know that side NM is to side XZ. If we consider side NY the transversal for these parallel lines, we create angle pairs. Using the , we can state that ∠YXZ is congruent to ∠YNM. We know that angle XYZ is congruent to angle by the reflexive property. Therefore, triangle XYZ is similar to triangle NYM by the similarity theorem.
Answers
Answer:
Parallel
corresponding angles theorem
NYM
AA
Step-by-step explanation:
Answer:
1. Parallel
2. corresponding angles theorem
3. NYM
4. AA
Step-by-step explanation:
Which statement is true about this data set?
{70, 67, 66, 65, 66, 78, 68, 71, 80, 78, 72}
80 is an outlier because it is greater than the upper fence.
This data set has no outliers.
65 and 80 are both outliers because 65 is less than the lower fence and 80 is greater than the upper fence.
65 is an outlier because it is less than the lower fence.
PLS HELP I WILL GIVE Brainliest
Answers
Answer:
Option 3 - This data set has no outliers.
Step-by-step explanation:
Given : The data {70, 67, 66, 65, 66, 78, 68, 71, 80, 78, 72}
To find : Which statement is true about this data set?
Solution :
The statements is all about outlier.
So, we find the outlier of the data.
Step 1- Arranging the data from lowest to highest,
{65,66,66,67,68,70,71,72,78,78,80}
Step 2 - Find the median of the data
As the data is odd numbers so the median is th term.
term.
6th term is 70 ⇒ Median =70
Step 3 - To find Upper quartile and lower quartile
Lower quartile data set is the median of the upper terms from median
{65,66,66,67,68}
Median of the set is 66
∴
Upper quartile data set is the median of the lower terms from median
{71,72,78,78,80}
Median of the set is 78
∴
Step 4 - To find the interquartile range.
Step 5 - To find the fences for the data set.
The formula to find range is
Substitute the values, we get
The range in which data lies is [48,96]
We have seen that all the data lie inside the range.
Therefore, This data set has no outliers.
Hence, Option 3 is correct.
pls give me brainlist
You want to buy a tent in the shape of a pyramid. The rectangular base is 35 square feet, with a height of 4 feet. What is the volume of the tent? Round your answer to the nearest hundredth
Answers
Answer:
46.67 cubic feet
Step-by-step explanation:
Given :The rectangular base is 35 square feet, with a height of 4 feet.
To Find :What is the volume of the tent?
Solution :
Volume of rectangular base pyramid =
Since we are given that The rectangular base is 35 square feet, i.e. area of rectangular base = l*w = 35 sqaure feet
Height of pyramid h = 4 feet
Putting values in formula
Volume of rectangular base pyramid =
=
=46.67 cubic feet
Thus volume of the tent is 46.67 cubic feet
Answer:
46.67 cubic feet
Step-by-step explanation:
right on ttm
1) What do we call two events for which the occurrence of the first affects the probability that the second event occurs?
Answers
Two events for which the occurrence of the first affects the probability that the second event occurs are called dependent events.
In other words, the probability of the second event depends on the occurrence of the first event. The occurrence of the first event changes the sample space for the second event, thus affecting its probability.
For example, if you draw a card from a deck and do not replace it before drawing a second card, the two events of drawing the first card and the second card are dependent. If the first card drawn is a king, then the probability of drawing a second king on the second draw is different from the probability of drawing a king on the first draw.
The probability of drawing a king on the second draw is reduced because there is one less king in the deck.
Dependent events are important in probability theory because they affect the calculation of joint probabilities and conditional probabilities, which are often used in statistical analysis and decision making.
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What is the value of h? ( Picture 1)
A. 4
B.\(8\sqrt{3}\)
C 16
D.\(8\sqrt{2}\)
What are the angle measures of the triangle? (picture 2)
A. 45,45,90
B.60,60,60
C. 30,60,90
D. They cannot be determined
What is the value of x? (Picture 3)
A. \(5\sqrt{2}\)
B.\(10\sqrt{2}\)
C. 5
D. 10
A courtyard is shaped like a square with 250 ft long sides. What is the distance from one corner of the courtyard to the opposite corner? Round to the nearest tenth
A. 353.6ft
B. 500 ft
C. 433.0 ft
D. 176.8 ft
What is the value of x (Picture 4)
A. \(15\sqrt{3}\)
B. 15/2
C. \(5\sqrt{3}\)
D. \(15\sqrt{2} /2\)
What is the height of an equilateral triangle with sides that are 12 cm long? Round to the nearest tenth.
A. 6cm
B. 10.4 cm
C. 8.5 cm
D. 6.9 cm
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Answers
PICTURE 1: the value of h is 8√2 units
PICTURE 2: the angle measures of the triangle are 30,60,90
PICTURE 3: x = 5√2 units
PICTURE 4: x = 5√3 units
What is the value of h?Trigonometry is a branch of mathematics dealing with the relationship between the ratios of the sides of a right-angled triangle with its angles.
PICTURE 1
sin 45° = 8/h
h = 8/sin 45°
h = 8√2 units
PICTURE 2
We have a 90° there already (the box)
Let the unknown angles be x and y respectively
sin x = 6/12
sinx = 1/2
x = arcsin (1/2)
x = 30°
y = 90 - 30 = 60°
Thus, the angle measures of the triangle are 30,60,90
PICTURE 3
sin 45 = x/10
x = 10 sin 45
x = 5√2 units
PICTURE 4
tan 30 = x/15
x = 15 tan 30
x = 5√3 units
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Question 8
What is (4x - 5)(6x + 5) expressed as a trinomial?
Answers
Answer:
24x^2 -10x - 25
Step-by-step explanation:
24x^2 + 20x - 30x - 25
24^2 -10x - 25
Answer:
Simplifying
4x + -5 = 6x + 5
Reorder the terms:
-5 + 4x = 6x + 5
Reorder the terms:
-5 + 4x = 5 + 6x
Solving
-5 + 4x = 5 + 6x
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-6x' to each side of the equation.
-5 + 4x + -6x = 5 + 6x + -6x
Combine like terms: 4x + -6x = -2x
-5 + -2x = 5 + 6x + -6x
Combine like terms: 6x + -6x = 0
-5 + -2x = 5 + 0
-5 + -2x = 5
Add '5' to each side of the equation.
-5 + 5 + -2x = 5 + 5
Combine like terms: -5 + 5 = 0
0 + -2x = 5 + 5
-2x = 5 + 5
Combine like terms: 5 + 5 = 10
-2x = 10
Divide each side by '-2'.
x = -5
Simplifying
x = -5
Step-by-step explanation:
What is the midpoint of the line segment with end points (-8,-7) and (-7,-8)
Answers
Answer:
The midpoint is (-15/2,-15/2) or (-7.5,-7.5)
Step-by-step explanation:
Midpoint is the formula:
\( \displaystyle{ \left( \dfrac {x_1 +x_2}{2} \: , \dfrac{y_1 + y_2}{2} \right)}\)
Substitute in the points (x,y) which we are given in the problem
\( \displaystyle{ \left( \dfrac { - 8 +( - 7)}{2} \: , \dfrac{ - 7 +( - 8)}{2} \right)} \\ \\ \displaystyle{ \left( \dfrac { - 8 - 7}{2} \: , \dfrac{ - 7 - 8}{2} \right)} \\ \\ \displaystyle{ \left( \dfrac { -15}{2} \: , \dfrac{ -15}{2} \right)}\)
Therefore, the midpoint is (-15/2,-15/2) or (-7.5,-7.5)
(X+15)cm and MU=(4x-45)
Answers
When the length of AB = (x+15)cm and MU=(4x-45)cm are equal ,it should be noted that the value of x is 20.
How to illustrate the information?It should be noted that from the information, it was stated that the the length of AB = (x+15)cm and MU=(4x-45)cm are equal.
Therefore, it should be noted that we need to equate the equations that are given. This will be:
x + 15 = 4x - 45
Collect like terms
4x - x = 15 + 45
3x = 60
Divide through by 3
x = 60 / 3
x = 30
Therefore, when the length of AB = (x+15)cm and MU=(4x-45)cm are equal ,it should be noted that the value of x is 20.
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Given that the length of AB = (x+15)cm and MU=(4x-45)cm are equal. Find the value of x.
A construction company is building the house shown below. They need to use a scale
of (1 in : 8Ft). They measured the drawing to find that the height of the building is 2 in. Explain to the construction workers how they would use this information to
calculate the height of the final building. Write a paragraph using the RACE format to explain.
Answers
Sure! Here's an explanation in the RACE format to help construction workers calculate the height of the final building:
To calculate the height of the final building, construction workers can utilize the information provided and follow a systematic approach. Firstly, they need to determine the scale being used in the plans or drawings of the building. The scale indicates the ratio between the measurements on the drawing and the actual measurements of the building.
By measuring a known distance on the drawing, such as the length of a wall, and comparing it to the corresponding distance on the actual site, workers can determine the scale factor. Next, using this scale factor, they can measure the height of a known element, such as a door or window, on the drawing. and multiplying those measurements by the scale factor, workers can calculate an approximation of the height of the final building.
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pls help asap if you can!!!
Answers
The statement that best proves that <XWY ≅ <ZYW is that two parallel lines are cut by a transversal, then the alternate interior angles are congruent
How to determine the statementTo determine the correct statement, we need to know the properties of a parallelogram.
These properties includes;
Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Same-Side interior angles (consecutive angles) are supplementary. Each diagonal of a parallelogram separates it into two congruent triangles.The diagonals of a parallelogram bisect each other.Learn more about parallelogram at: https://brainly.com/question/10744696
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Sarah ran 3,682 m yesterday on the track at her school. One time around the track is 263 m. Sarah wants to run 4,222 m today. Which number sentence can be used to calculate how many laps around the track Sarah should run today?
Answers
The total number of laps Sarah should run today is given by the equation A = 16 laps
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the number of laps be represented as A
Now , the equation will be
The total distance ran by Sarah yesterday = 3,682 m
The distance of one lap = 263 m
So , the number of laps ran by Sarah yesterday = total distance ran by Sarah yesterday / distance of one lap
Substituting the values in the equation , we get
The number of laps ran by Sarah yesterday = 3682 / 263
The number of laps ran by Sarah yesterday = 14 laps
Now ,
The distance Sarah wants to run today = 4,222 m
So , the number of laps Sarah should run today A = distance Sarah wants to run today / distance of one lap
Substituting the values in the equation , we get
The number of laps Sarah should run today A = 4,222 / 263
The number of laps Sarah should run today A = 16.05 laps
The number of laps Sarah should run today A = 16 laps
Therefore , the value of A is 16 laps
Hence , the number of laps is 16 laps
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What is the slope of the line that contains the points (−5, −4) and (−3, −5)? −1 negative one half one half 8
Answers
(−5, −4) = x1 , y1
(−3, −5) = x2 , y2
The slope is calculated using formula:
Slope = y2-y1/x2-x1
=-5-(-4) / -3- (-5)
= -5+4 / -3 + 5
= -1/2
The slope for the pair of coordinates is
= -1/2
What’s the sum of (9a+3b-5) and (4b-6)
Answers
Answer:
9a +7b-11
Step-by-step explanation:
(9a+3b-5) + (4b-6)
Combine like terms
9a + 3b+4b -5-6
9a +7b-11
Answer:
9a +7b-11
Step-by-step explanation:
The scatter plot shows the number of roses (x) and daisies (y) a florist sold at a flower
show. How many roses did the florist who sold 125 daisies sell?
100
200
80
170
Answers
The number of roses that the florist sold who sold 125 daisies is; 200
How to Interpret Scatter Plots?Scatter plots are the graphs that present the relationship between two variables in a data-set.
From the given scatter plots, we see that the x and y axes are marked in intervals of 50 like 0, 50, 100, 150, 200,...e.t.c
Now, from the scatter plot, we see a couple of coordinates and we are told that the x-axis shows the number of roses sold at a flower show while the y-axis shows the number of daises sold at the flower show.
Now, from the scatter plot, we see that when the number of daises sold is 125, we can trace that the number of roses sold was 200 roses.
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What is the slope of a line parallel to the following line: 5x + 8y = 50
Answers
Answer:
-5/8
Step-by-step explanation:
Convert standard form to slope-intercept form to find the slope.
5x + 8y = 50
8y = -5x + 50
y = -5/8x + 50/8
Slope: -5/8
Answer:
-5/8x
Step-by-step explanation:
First you need to find the slope of the given line.
Do this by converting it to slope intercept form (y=mx+b)
Divide the whole equation by 8 to isolate the y variable.
After dividing by 8 you should now have 5/8x + y = 6.25
Now you need to subtract y and 6.25 to isolate y
5/8x - 6.25 = -y
Now you have to divide the whole equation by negative 1 to make y positive. You end up with: y = -5/8 + 6.25
We know that the slopes of parallel lines are equal. Therefore the answer is -5/8x
A rectangular prism has a length of 7 feet, a height of 17 feet, and a width of 4 feet. What is its volume, in
cubic feet?
Answers
Answer: 476 ft³
Step-by-step explanation:
Volume = width x hight x length
V = 4 ft x 17 ft x 7 ft
v = 476 ft³
Michael returned a bag of cans and received $4.65 back. If the deposit for each can was $0.05, how many cans did he return?
Answers
Answer:
I'm not sure
Step-by-step explanation:
Jesse wants to start his own shoe business. His father gave him 10,000 dollars to start
the business and has agreed to give him an additional 300 dollars a month. What is the
slope and what is the y-intercept?
m =
b =
Answers
The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. Find the ratio of the volume of the balloon in both the cases.
Answers
V = (2/3)πr^3
Where V is the volume, r is the radius, and π is approximately 3.14159.
If the radius of the hemispherical balloon increases from 6 cm to 12 cm, the ratio of the volume of the balloon in both cases is:
V2/V1 = [(2/3)π(12^3)]/[(2/3)π(6^3)] = 12^3/6^3 = (12/6)^3 = 2^3 = 8
Therefore, the ratio of the volume of the balloon in both cases is 8:1. The volume of the balloon increases by a factor of 8 when the radius increases from 6 cm to 12 cm.
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