(b) if the length, width, and height are increasing at a rate of 0.1 ft/sec, 0.2 ft/sec, and 0.5 ft/sec respectively, find the rate at which the surface area of the box is changing when x = 20 ft, y = 15 ft, and z = 25 ft.

Answers

Answer 1

Final answer:

By applying the derivative of the surface area of a rectangular prism formula with respect to time, it is found that the surface area of the box is increasing at a rate of 61 ft^2/sec when the dimensions are 20 ft by 15 ft by 25 ft.

Explanation:

The question involves calculating the rate of change of the surface area of a box with respect to time using derivatives, a concept from calculus. Given the rates at which the length (x), width (y), and height (z) are increasing, we can find the rate at which the surface area S is changing by using the surface area formula for a rectangular prism, S = 2(xy + xz + yz), and differentiating with respect to time.

At the given dimensions (x = 20 ft, y = 15 ft, z = 25 ft), and rate of increase (dx/dt = 0.1 ft/sec, dy/dt = 0.2 ft/sec, dz/dt = 0.5 ft/sec), the rate of change of surface area dS/dt is calculated by:

dS/dt = 2( (dy/dt)z + y(dz/dt) + (dx/dt)z + x(dz/dt) + (dx/dt)y + x(dy/dt) )

dS/dt = 2( (0.2)(25) + (15)(0.5) + (0.1)(25) + (20)(0.5) + (0.1)(15) + (20)(0.2) )

dS/dt = 2(5 + 7.5 + 2.5 + 10 + 1.5 + 4)

dS/dt = 2(30.5)

dS/dt = 61 ft2/sec

Therefore, the surface area of the box is increasing at a rate of 61 ft2/sec when the dimensions x, y, and z are 20 ft, 15 ft, and 25 ft respectively.

Related Questions

George spent 70 % of his saving to buy a camera. The camera cost $574 . How much did he originally have in savings?

Answers

574 : 70 * 100 =
820 $

Write an equation for "eight less than a number is seventy-five". What is the number? n - 8 < 75; -67 n - 8 = 75; 67 n - 8 = 75; 83 n - 8 < 75; 67

Answers

n-8=75
n is the number
so number minus 8 equals 75

n-8=75

n is the number

so number minus 8 equals 75

A six-sided rectangular box is to hold 1/2 cubic meter; what shape should the box be to minimize surface area?

Answers

A circle is an object which has the lowest surface area to volume ratio. So in this case, the shape should be like that of a circle but we cannot have a circle here because we want to form a 6 sided box. So the next thing closer to circle is a square. Hence the shape should be a square:

Answer:

square

Final answer:

To minimize surface area for a given volume of 1/2 cubic meter, the box should be shaped as a cube with each side measuring approximately 0.7937 meters. This cube shape offers the smallest surface area for the given volume.

Explanation:

To minimize the surface area of a six-sided rectangular box with a volume of 1/2 cubic meter, it is best to make the box in the shape of a cube. This shape has the smallest surface area for a given volume. To find the dimensions of the box, let's denote the side of the cube as 's'. We know that volume V of a cube is given by V = s³. So, if the volume is 1/2 cubic meter, we have:

s³ = 1/2

s = ∛(1/2) ≈ 0.7937 meters

Each side of the cube should be approximately 0.7937 meters to achieve the minimal surface area. The surface area SA of the cube can be calculated as:

SA = 6s²

SA = 6 × (0.7937)² ≈ 3.770 meters²

Thus, a cube with sides of about 0.7937 meters will have the minimal surface area for a volume of 1/2 cubic meter.

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which of the following steps were applied to abc obtain a'b'c'

Answers

d) shifted 2 units left and 4 units up

Answer:

Shifted 2 units left and 4 units up

Step-by-step explanation:

Given are two triangles

ABC is having vertices as (4,6) (7,6) and (5,9)

The transformed image A'B'C' has vertices as

(2,10) (5,10) (3,13)

On comparing the vertices corresponding we find a certain pattern.

X coordinate is reduced by 2 and y coordinate is increased by 4.

This implies the transformation is

Shifted 2 units left and 4 units up

the value of the 7 in 507,264 is 10 times the value of the 7 in which number?

Answers

The value of seven is in the thousands place and is 10 time the number seven if it were at the hundreds place, which is where 2 is.

7000=10 x 700
7000=7000

Answer:

The value of the 7 in 507,264 is 10 times the value of the 7 in number 2

Step-by-step explanation:

Given Number : 507,264

To find : Number in which the value of the 7 in 507,264 is 10 times the value of the 7 .

Starting from the left ,

4 is in ones place

6 is in tens place

2 is in hundreds place

7 is in thousands place

0 is in ten thousands place

5 is in lakhs place

Here, 7 is in thousands place

7000 = 10 x 700

2 is in hundreds place

200 = 2 × 100

Therefore, the value of the 7 in 507,264 is 10 times the value of the 7 in number 2 .

after taking 1/4 pound of flour from a canister to make muffins, maria has 7/8 of the flour left. If each batch of muffins requires 2/5 pound of flour, find the greatest number of batches of muffins that Maria can make using the remaining flour/

Answers

You need a common denominator to calculate the amount of flour left, the easiest is 40 since 5*8 is 40 and 8*5 is 40. So you multiply the fractions to make them common by 40, you have 35/40 flour left and need 16/40 per batch. 16 goes into 35 2 times and you would have 3/40 of flour left.

How to determine if the function has x-axis sym, y-axis sym, and origin sym

y^2 -xy =2

Answers

To check for symmetry on the x axis, replace y with –y

-y^2 –x(-y) =2

Apply the product rule, since the equation is not identical tot eh original equation it is not symmetric about the x axis

Now do the same for y axis by replacing x with –x

Again using product rule the equations are not identical, so it is not symmetric about the y axis

To check the origin,

Replace both x & y with –x & -y

Again using product rule, the equations are not identical so it is not symmetric about the origin

Final answer:

To check for symmetry, replace x with -x and y with -y in the equation. Upon doing that for y^2 -xy =2, we find it lacks x-axis, y-axis, and origin symmetry.

Explanation:

To determine whether a function has x-axis symmetry, y-axis symmetry, or origin symmetry, we have to replace x with -x and y with -y and see whether we get the same function.

X-axis symmetry: Replace y with -y. The equation becomes (-y)^2-(-y)x=2 which simplifies to y^2+xy=2. This is different from the original function, so there is no x-axis symmetry.Y-axis symmetry: Replace x with -x. The equation becomes y^2-(-x)y=2 which simplifies to y^2+xy=2. This is different from the original function, so there is no y-axis symmetry.Origin symmetry: Replace x with -x and y with -y. The equation becomes (-y)^2-(-y x -x)=2, which simplifies to y^2+xy=2. This is different from the original function, so there is no origin symmetry.

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You walk north for 2.5 miles. then you walk east a distance of 3.0 miles. how many miles from the starting point are you?

Answers

All you do is add 3.0 and 2.5
This would give u 5.5

Correct me if i'm wrong, but i hope it helps

Prove the quotient of any two rational numbers is a rational number

Answers

That is not possible, a quotient or any equation for that fact can only be rational when it lacks repeating {x} values. Take for instance this 2x11x2^2+(1+10)=

In the equation the {x} values would be most likely described as {2}. Since 2 was the quotient that was being repeating hence that equation not being rational. Hope this helps!

The quotient of any two rational numbers is always a rational number. This can be proved by expressing the division as multiplication by the reciprocal and demonstrating that the result is a rational number. Therefore, the property of being rational is preserved under division.

Let's define rational numbers first. A rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero. Mathematically, if a and b are integers and b ≠ 0, then r = a/b is a rational number.

Consider two rational numbers r1 and r2.
Suppose r1 = a/b and r2 = c/d, where a, b, c, and d are integers, and b ≠ 0 and d ≠ 0.To find the quotient q of these two rational numbers, we compute r1 ÷ r2:
q = (a/b) ÷ (c/d)This division can be written as multiplication by the reciprocal:
q = (a/b) × (d/c) = (a × d) / (b × c)Since the product of two integers is also an integer, a × d and b × c are integers.We can write the quotient as:
q = (a × d) / (b × c), where (b × c) ≠ 0 because neither b nor c is zero.Therefore, q is the quotient of two integers where the denominator is not zero, which means q is a rational number.

Thus, we have proved that the quotient of any two rational numbers is a rational number.

Richard estimates that he can apply fertilizer to 4,485 square feet of grass in 3/4 of an hour. How many square feet of grass can Richard fertilize per hour?

Answers

We know that Richard can apply fertilizer to 4485 square feet = 3/4 hour

To solve this problem, one approach is to find 1/4 then multiply by 4 to get the whole 4/4 or 1 hour.

So the solution would be:

4485/3=1495
Then multiply this by 4.

1495 multiplied by 4 is equal to 5980 square feet.

So Richard can apply fertilizer to 5980 square feet in 1 hour.

Find the product of 4×7×25tell which strategy you used

Answers

A product is simply the outcome of multiplying two or more numbers together, so just type 4x7x25 into a calulator = 700

A protective cover for a wireless tablet has a height of 8 inches and base of 7 inches. What is the area of the cover?

Answers

So to find the answer of the area, you have to multiply 7 and 8 which would equal to 56.

Answer: The area of the cover = 56 square inches.

Step-by-step explanation:

We know that the shape of a tablet is rectangle.

Thus to calculate the area of the protective cover we apply the formula to calculate area of rectangle.

The area of rectangle is given by :-

[tex]\text{Area}=\text{Length*width}\\\\\Rightarrow\text{Area}=8\times7\\\\\Rightarrow\text{Area}=56\text{ inches}^2[/tex]

Therefore, the area of the protective cover is 56 square inches.

Choose the answer. 3. (1 pt) Suzanne cut a ribbon into 9 equal pieces. She sewed 5 pieces onto her quilt. Which number line shows a dot representing the fraction of the ribbon that was sewn onto the quilt? A. B. C. D.

Answers

Given that Sussan sew 5 out of 9 cut out ribbins unto her quilt, the number line that shows a dot representing the fraction of the ribbon that was sewn onto the quilt is given by:

A number line with 9 partitions with a dot marked on the 5th partition.

Find two consecutive odd integers such that their product is 71 more than 3 times their sum.

Answers

Let's put this in terms of a system of equations. Let's call the numbers x and y, with x being the larger integer and y the smaller one. Since they are consecutive odd integers, we know that:
x = y + 2
The constraint is:
xy= 3(x+y) + 71 Simplify
xy = 3x + 3y + 71 Plug in (y+2) for x:
(y + 2) y = 3(y + 2) + 3y + 71 Simplify
y² + 2y = 3y + 6 + 3y + 71 Subtract 6y from both sides, add 71+6
y² -4y = 77
y² - 4y - 77 = 0
Factor:
(y - 11) (y+7) = 0
y = 11 or y=-7

When y = 11, x = 13, since x = y+2
Plug in to make sure:
xy = 3(x+y) + 71
11*13 = 3 (11 + 13) + 71
143 = 3 (24) + 71
143 = 72 + 71
143 = 143
Our values work!
Answer: 11, 13

A car is traveling at a rate of 40 miles per hour. What is the car's rate in kilometers per hour? How many kilometers will the car travel in 4 hours? In your computations, assume that 1 mile is equal to 1.6 kilometers. Do not round your answers.

Answers

1 mile = 1.6 kilometers

multiply 40 mph by 1.6 = 64

car's rate = 64 kilometers per hour

multiply kilometers per hour by 4 = 256

car will travel 256 kilometers in 4 hours

What he said. Multiply 40 by the numbers

The basic recipe for split pea soup serves 8 people. Find the amount of each ingredient needed to serve 32 people.

Answers

[tex]\bf \begin{array}{ccll} \stackrel{basic}{recipe}&people\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 1&8\\ r&32 \end{array}\implies \cfrac{1}{r}=\cfrac{8}{32}\implies \cfrac{1\cdot 32}{8}=r[/tex]

so, 4 time as much the recipe, or 4 times as much of each ingredient.

Time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value 8 min and standard deviation 3 min. if five individuals fill out a form on one day and six on another, what is the probability that the sample average amount of time taken on each day is at most 11 min? (round your answer to four decimal places.)

Answers

Calculate the z-score of the given by the equation,

z-score = (x - m)/s

where x is the data point, m is the mean, and s is the standard deviation. Substituting the known values,

z-score = (11 - 8)/ 3 = 1

This translates to 0.84.

ANSWER: 0.84.

A data firm records a large amount of data. historically, 1.0% of the pages of data recorded by the firm contain errors. if 200 pages of data are randomly selected?
a. what is the probability that six or more pages contain errors?
b. what is the probability that more than 10 pages contain errors?
c. what is the probability that none of the pages contain errors?
d. what is the probability that fewer than five pages contain errors?

Answers

a) P(X >= 6) = 0.0160

b) P(X > 10) ≈ 6.88 x 10^-6

c) P(X = 0) ≈ 0.1339

d) P(X < 5) ≈ 0.9482

a. Probability of six or more pages with errors:

This is the sum of probabilities for 6, 7, 8, ..., 200 errors:

P(X >= 6) = 1 - [P(X < 6)] = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)]

Using a calculator or statistical software, we can find these individual probabilities and sum them up. Alternatively, we can use binomial distribution tables or functions available in programming languages.

P(X >= 6) ≈ 0.0160

b. Probability of more than 10 pages with errors:

Similar to part (a), but calculating from 11 errors onwards:

P(X > 10) = 1 - [P(X=0) + P(X=1) + ... + P(X=10)]

P(X > 10) ≈ 6.88 x 10^-6 (very small probability)

c. Probability of none of the pages containing errors:

P(X = 0) = (0.99)^200 ≈ 0.1339

d. Probability of fewer than five pages containing errors:

P(X < 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

P(X < 5) ≈ 0.9482

Use place value to find the product 3x600=3x

Answers

x = 600 because 3x600 = 1800 meaning x needs to be 1800 and 3 x 600 is 1800

Mrs blanda truck holds 19.7 gallons of gas she used 4 full tanks of gas last month and paid $3 for each gallon how much did mrs blanka spend on gas last month

Answers

multiply 19.7 x 4 for total gallons

19.7 * 4 = 78.8 total gallons

multiply that by price:

78.8 * $3 = $236.40 total price

Final answer:

Mrs. Blanka spent $236.4 on gas last month by filling up her truck's 19.7-gallon tank 4 times at a price of $3 per gallon.

Explanation:

To calculate how much Mrs. Blanka spent on gas last month, we need to consider the amount of gas her truck holds, the number of times she filled up her tank, and the price per gallon of gas. Mrs. Blanka's truck holds 19.7 gallons of gas, and she used 4 full tanks of gas last month. Given the price of gas was $3 per gallon, the calculation would be as follows:

First, find the total gallons of gas used: 19.7 gallons/tank × 4 tanks = 78.8 gallons.Next, calculate the total cost: 78.8 gallons × $3/gallon = $236.4.

Therefore, Mrs. Blanka spent $236.4 on gas last month.

Four friends split equally a lunch bill of 36.96 plus 20% tip. How much did each person pay

Answers

Let P = how much each person paid.

P = [(36.96)(0.20) + 36.96]/4

Solve for P to find your answer.

Find the simple interest. Principal $2300 rate 20% time 27 months The interest is $ ?

Answers

[tex]\bf \qquad \textit{Simple Interest Earned}\\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\to& \$2300\\ r=rate\to 20\%\to \frac{20}{100}\to &0.20\\ t=years\to \frac{27}{12}\to &\frac{9}{4} \end{cases} \\\\\\ I=2300\cdot 0.2\cdot \cfrac{9}{4}[/tex]

the difference of twice a number and 3

Answers

2n/3
the difference means divide 2 times(multiply)a number

Based on the graph of the line below, which describes the relationship between the value of the y-intercept and the value of the slope?

Answers

y intercept x = 0, y = 3

(-9,0)(0,3)
slope = (3-0)/(0+9) = 1/3

answer
A. y intercept is 3 and slope is 1/3

Jack is mowing a lawn that has a shed. Jack wants to know the area of the lawn he has to mow. here are the dimensions of the yard and the shed. 13x-4 length 5x withd shed is 2x+4length withd is 2x
find a polynomial that describes the area of the lawn he has to mow. i.e. find the area of the area of the shed.

Answers

We know that the formula for area is simply:

area = l w

where l is length and w is width

yard dimensions: l = 13x – 4, w = 5x

Ayard = (13x – 4) 5x = 65x^2 – 20x

shed dimensions: l = 2x + 4, w = 2x

Ashed = (2x + 4) 2x = 4x^2 + 8x

Therefore the area to mow is:

Amow = Ayard – Ashed

Amow = 65x^2 – 20x – (4x^2 + 8x)

Amow = 65x^2 – 20x – 4x^2 – 8x

Amow = 61x^2 – 28x

Round 354,738 to the nearest thousand?

Answers

your answer is 400,000.

Hope this helps!

354,738 rounded to the nearest thousand is 355,000.

What is a place value?

In Mathematics and Euclidean Geometry, a place value is a numerical value (number) which denotes a digit based on its position in a given number and it includes the following:

TenthsHundredthsThousandthsUnitTensHundredsThousands.

The number in the thousand place in the number 354,738 is 7 and it is greater than or equal to 5. This ultimately implies that, we would round up the number 4 to 5.

354,738 ≈ 355,000.

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Max has a monthly salary of $1,100 and also earns 7.5% commission on his sales. If Max had $43,000 in sales last month, what were his total earnings for the month? a. $3,225 b. $4,050 c. $4,325 d. $5,150

Answers

If he makes a base salary of 1100$, then Max's total salary is 1100+<some number>. That <some number> is the amount he makes in commissions, which is 7.5% (or 0.075 by moving the decimal 2 spots to the left). Multiplying 43000 by 0.075 to get his commision money, we get 3225. Remember that we still have to add 1100 to it (as that's his base salary), so we end up with
1100+3225=4325

An equation is formed when two equal expressions. Max's total earnings for the given month are $4,325. The correct option is C.

What is an equation?

An equation is formed when two equal expressions are equated together with the help of an equal sign '='.

Given that Max has a monthly salary of $1,100 and also earns a 7.5% commission on his sales. Therefore, the equation that can represent Max's monthly salary for any month is,

y = $1,100 + $(0.075x)

where y is the month's salary and x is the number of sales he made.

Given in the last month Max had $43,000 in sales last month. Therefore, Max's total salary for the given month is,

Max Salary = $1,100 + $(0.075x)

= $1,100 + $(0.075 × 43,000)

= $1,100 + $3,225

= $4,325

Hence, Max's total earnings for the given month are $4,325.

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Ahmad is fertilizing his garden. The garden is in the shape of a rectangle. Its length is 12 feet and its width is 10 feet. Suppose each bag of fertilizer covers 30 square feet. How many bags will he need to cover the garden?

Answers

You will need 4 bags.

Hope it helps!

The length of a rectangle is given by 5t+4sqrt(t), and its height is sqrt(t), where t is time in seconds and the dimensions are in centimeters. Fine the rate of change of the area of the rectangle with respect to time when the length of the rectangle is 9 centimeters.

Answers

Answer:

11.5 cm²/s

Step-by-step explanation:

We must solve two problems to answer this question:

find the time at which the length is 9 cmfind the rate of change of area with respect to time (at that time)

For the first problem, we have ...

... 9 = 5t +4√t

... 9 -5t = 4√t . . . . . subtract 5t

... 81 -90t +25t² = 16t . . . . square both sides

... 25t² -106t +81 = 0 . . . subtract 16t

... (t -1)(25t -81) = 0 . . . . factor

... t = 1 or 3.24 . . . . . . t=3.24 is an extraneous solution

_____

For the second problem, we have area (a(t)) is ...

... a(t) = length×height = (5t +4√t)(√t) = 5t^(3/2) +4t

Then the derivative is ...

... a'(t) = (3/2)(5√t) +4

and

... a'(1) = (3/2)(5√1) +4 = 11.5 . . . . . cm²/s

_____

Comment on the attachment

A graphing calculator can help solve both problems. It can help find the time at which length is 9 cm by solving ... length-9 = 0. (The nice thing here is that there is no extraneous solution.) It can compute the derivative of a function, too.

Joshua earns $8.20 per hour when he works at the college library. How much money does he earn by working at the library for $4.5 hours.

PLEASE HELP!!!

Answers

8.20 x 4.5 36.9

he earns $36.90

36.9 is the answer that should be it

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