# calculus ( related rates )

a paper cup in the shape of a cone has a diameter of 10cm across the top, and is 8cm deep. if the cup is leaking out the bottom at 2pi cm^3/min, at what rate is thet area of the water surface ( only the top surface of the water) changing, when the cup has a the water depth of 5cm? exact solutions first; round to two decimal places

1. 👍 0
2. 👎 0
3. 👁 241
1. Let the radius of the water level be r cm
let the height of the water be h cm
by ratios of similar triangles (using a cross-section)
r/h = 5/8 ----> h = 8r/5

V= (1/3)πr^2h
= (1/3)πr^2(8r/5) = (8π/15)r^3
dV/dt = (8π/5)r^2 dr/dt
-2π = (8π/5)r^2 dr/dt
when r = 5
dr/dt = -1/20

A =πr^2
dA/dt = 2πr dr/dt
when r = 5
dA/dt = 2π(5)(-1/20) = -π/2 cm^2/min

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Related Rates

Gravel is being dumped from a conveyor belt at a rate of 10 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile

asked by Anonymous on March 1, 2017
2. ### Maths

A closed tin is in the shape of a cylinder of diameter 10cm and height 15cm. Use the value 3.14 for pier to find A.the total surface area B.the value of the tin to the r=nearest 10naira,if tin plate cost #4500 per m2

asked by Ste on April 18, 2017
3. ### math

A paper cone has a base diameter of 8cm and a height of 3cm a. Calculate the volume of the cone in term of pie b. Calculate the curve surface area of the cone

asked by Barry on October 9, 2016
4. ### Mathematics

a paper cone has a base diameter of 8cm and a height of 3cm.calculate the volume of the cone in terms of pie and make a sketch of the cone and hence use Pythagoras theorem to calculate its slant height and calculate the curve

asked by Love on July 22, 2020
1. ### Calc

A paper cup is to be designed in the shape of a right circular cone. It must have a capacity of 12 fluid ounces (1 fluid ounce = 1.80469 cubic inches) of soft drink but it must use a minimum amount of material in its construction.

asked by jo on July 30, 2012
2. ### mathematics

A closed tin is in the shape of a cylinder of diameter 10cm and height 28cm use 22\7 for ^ to find the value of the tin if tin plate cost #5000 per m2

asked by Mutiu on July 14, 2020
3. ### Calculus

Gravel is being dumped from a conveyor belt at a rate of 10 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal. How fast is the height of the pile

asked by Robin on July 5, 2013
4. ### Calculus Help Please Urgent!!!

A cone-shaped paper drinking cup is to be made to hold 30 cm^3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Round your answers to two decimal places.) Height = ? Radius = ? Show work

asked by Patrick on April 15, 2014
1. ### Geometry (check my work)

A scoop of ice cream shape like a sphere with a diameter of 6 cm is placed on an ice cream cone the has a diameter of 5 cm and a height of 9 cm, show that the cone is or is not big enough to hold all of the ice cream when it melts

asked by Sue on May 3, 2016
2. ### geometry

11. Infinitely many different sectors can be cut from a circular piece of paper with a 12-cm radius, and any such sector can be fashioned into a paper cone with a 12-cm slant height. (a) Show that the volume of the cone produced

asked by Sarah on April 15, 2010
3. ### math

The diameter and the slant height of a cone are both 24 cm. Find the radius of the largest sphere that can be placed inside the cone. (The sphere is therefore tangent to the base of the cone.) The sphere occupies a certain

asked by Riley on February 13, 2017
4. ### PreCalculus

Cone Problem Beginning with a circular piece of paper with a 4- inch radius, as shown in (a), cut out a sector with an arc of length x. Join the two radial edges of the remaining portion of the paper to form a cone with radius r

asked by Emma on January 1, 2011