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1. A snow cone cup is 8 cm tall with a circluar 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.

1. A snow cone cup is 8 cm tall with a circluar 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.

Advanced Engineering Mathematics
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 a, b, and c please

## MATH 2413: Calculus I – Activity 9 ### Outcomes: Find a rate of change by relating it to other rates of change. --- ### Problem Statement: 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. **Tasks:** (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. --- ### Further Explanations: - **Similar Triangles:** Use the concept of similar triangles to find a relationship between the height and radius of the water in the cone. - **Volume Formula:** The volume of a cone is given by the formula \( V = \frac{1}{3} \pi r^2 h \). Use algebraic manipulation to express \( V \) in terms of \( h \) only, utilizing the relationship from part (b). - **Differentiation:** Apply differentiation techniques to relate rates of change, showing the interdependence of the varying parameters over time. - **Rate Calculation
Transcribed Image Text:## MATH 2413: Calculus I – Activity 9 ### Outcomes: Find a rate of change by relating it to other rates of change. --- ### Problem Statement: 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. **Tasks:** (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. --- ### Further Explanations: - **Similar Triangles:** Use the concept of similar triangles to find a relationship between the height and radius of the water in the cone. - **Volume Formula:** The volume of a cone is given by the formula \( V = \frac{1}{3} \pi r^2 h \). Use algebraic manipulation to express \( V \) in terms of \( h \) only, utilizing the relationship from part (b). - **Differentiation:** Apply differentiation techniques to relate rates of change, showing the interdependence of the varying parameters over time. - **Rate Calculation
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