(d) Differentiate with respect to time t to find an equation relating the rate of change of the volume of water in the cup to the rate of change of its height at any time t. (e) Find the rate at which the water is rising in the cup when its height in the cup is 3 cm. Round to two decimal places and include units. (f) Is the water rising most rapidly when it is 3, 4, or 5 cm high? Explain whether this makes sense in reality.
(d) Differentiate with respect to time t to find an equation relating the rate of change of the volume of water in the cup to the rate of change of its height at any time t. (e) Find the rate at which the water is rising in the cup when its height in the cup is 3 cm. Round to two decimal places and include units. (f) Is the water rising most rapidly when it is 3, 4, or 5 cm high? Explain whether this makes sense in reality.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you do d, e, and f please
Transcribed Image Text:**Outcomes:** Find a rate of change by relating it to other rates of change.
**MATH 2413: Calculus I – Activity 9**
1. A snow cone cup is 8 cm tall with a circular opening of radius 6 cm and is filling with water at 4 cm³/s.
(a) Draw the cup with water in it but not yet full. Label variable \( h \) for the water’s height and \( r \) for its radius.
(b) Find an equation relating the water’s height \( h \) to its radius \( r \) at any time \( t \). Hint: Use similar triangles.
(c) Find and simplify an equation relating the volume \( V \) of water in the cup to its height \( h \) at any time \( t \).
Hint: Use your equation from (b) to eliminate \( r \) from the volume formula for a cone: \( V = \frac{1}{3} \pi r^2 h \).
(d) Differentiate with respect to time \( t \) to find an equation relating the rate of change of the volume of water in the cup to the rate of change of its height at any time \( t \).
(e) Find the rate at which the water is rising in the cup when its height in the cup is 3 cm. Round to two decimal places and include units.
(f) Is the water rising most rapidly when it is 3, 4, or 5 cm high? Explain whether this makes sense in reality.
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