MATHEMATICS COLLEGE

A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 141 inches. Find the dimensions of the package of maximum volume that can be sent. (The cross section is circular.)

Answers

Answer 1

Answer:

Answer:

Length = 47 in

Radius = 47/π in

Step-by-step explanation:

Let 'h' be the length of the package, and 'r' be the radius of the cross section.

The length and girth combined are:

$L+G=141=h+2\pi r\nh=141-2\pi r$

The volume of the cylindrical package is:

$V=A_b*h\nV=\pi r^2*h$

Rewriting the volume as a function of 'r':

$V=\pi r^2*h\nV=\pi r^2*(141-2\pi r)\nV=141\pi r^2-2\pi^2 r^3$

The value of 'r' for which the derivate of the volume function is zero yields the maximum volume:

$V=141\pi r^2-2\pi^2 r^3\n(dV)/(dr)=282\pi r-6\pi^2r^2=0\n 6\pi r=282\nr=(47)/(\pi) \ in$

The length is:

$h=141-2\pi r=141-2\pi*(47)/(\pi)\nh=47\ in$

The dimensions that yield the maximum volume are:

Length = 47 in

Radius = 47/π in

Related Questions

28Here are two rectangles.РeBDSRQR = 20 cmBC = PQThe perimeter of ABCD is 52 cmThe area of PQRS is 90 cm?Find the length of AB.
Point C is the center of the circle. angle ACB measures 49. What is the of arc ADB
José tiene 30 años menos que su padre y 27 más que su hijo. entre los 3 suman 135 años ¿ cuántos años tiene cada uno?
Which equation represents a line that passes through -2,4 and has a slope of 2/5 need help Now
The quadratic equation x^2 - 8x = -4 is rewritten in the form (x + p)^2 = q. when q is a constant. What is the value of p ?

How do I write 4.18 as a mixed number

Answers

Answer:

4 18/100

Step-by-step explanation

Which function, g or h, is the inverse function for function f?

Answers

Answer:

first option

imagine the line y = x as a mirror

Answer:

The answer is A. "the function h because the graphs of ƒ and h are symmetrical about the line y = x"

Find the midpoint of the segment with these endpoint (4,6),(9,3)

Answers

Answer:

The midpoint of the segment is (6, 4.5).

Step-by-step explanation:

Midpoint of a segment:

The midpoint of a segment is given by the mean of the coordinates of their endpoints.

Segment with coordinates (3,6) and (9,3).

Mean x-coordinate: (3+9)/2 = 12/2 = 6

Mean y-coordinate: (6+3)/2 = 9/2 = 4.5

The midpoint of the segment is (6, 4.5).

Four friends want to share two largepizzas equally. Each pizza is cut into
ten slices. How many slices will each
person get?

Answers

Answer: 5

Step-by-step explanation:

2 pizzas ten slices eatch, so 20 slices. 20 slices devided by 4 is five. :)

What is the square footage of 9.5 feet x 12 feet

Answers

9.5 x 12 = 114 sq ft

The surface of a mountain is modeled by the equation h(x, y) = 5000 - 0.001x² - 0.004y². A mountain climber is at the point (500, 300, 4390). In what direction should the climber move in order to ascend at the greatest rate?

Answers

Answer:

i-2.4j

Step-by-step explanation:

Given that,

The surface of a mountain is modeled by the equation as follows :

$h(x,y)=5000 -0.001x^2-0.004y^2$

A mountain climber is at the point (500, 300, 4390).

We need to find the direction in which he should move in order to ascend at the greatest rate.

To find direction, first finding the gradient of h as follows :

$\nabla h(x,y)=-0.002xi-0.008yj$

Now put x = 500 and y = 300

So,

$\nabla h(x,y)=-0.002(500)i-0.008(300)j\n\n\nabla h(x,y)=i-2.4j$

The direction of the climber is i-2.4j

Other Questions

You are making a 2 ingredient trail mix with almonds and raisins. There is a 5:7 ratio of almonds to raisins. How many ounces of almonds would be in a 3-pound bag of this trail mix? (There are 16 ounces in 1 pound.) *

Please help ASAP! Thank you!

An ice cream factory makes 310 quarts of ice cream in 5 hours. How many quarts can they make per hour?

Jake buys a gaming system and equipment for $2200.00 with a credit card that has a 28% APR. He makes a $100.00 monthly payment. What is the total amount of interest based on the APR? Enter your answer as a dollar amountsuch as: $324.