(d) Over the time interval 0 ≤ t ≤ 6, at what time t is the amount of sand on the beach least? What is this minimum value? Explain and justify your answers fully.
(d) Over the time interval 0 ≤ t ≤ 6, at what time t is the amount of sand on the beach least? What is this minimum value? Explain and justify your answers fully.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I believe I have answered 7 a-c correctly. ? But I am still needing help on answering 7d. Any help would be appreciated.
![**Math 152 - Net Change Theorem: Applications of FTC**
**7. The tide removes sand from the beach at a small ocean park at a rate modeled by the function:**
\[ R(t) = 2 + 5 \sin \left( \frac{4\pi t}{25} \right) \]
A pumping station adds sand to the beach at a rate modeled by the function:
\[ S(t) = \frac{15t}{1 + 3t} \]
Both \( R(t) \) and \( S(t) \) are measured in cubic yards of sand per hour. \( t \) is measured in hours, and the valid times are \( 0 \leq t \leq 6 \). At time \( t = 0 \), the beach holds 2500 cubic yards of sand.
**(a) What definite integral measures how much sand the tide will remove during the time period \( 0 \leq t \leq 6 \)? Why?**
Solution steps involve setting up and evaluating the integral:
\[ \int_0^6 R(t) \, dt = \int_0^6 \left( 2 + 5 \sin \left( \frac{4\pi t}{25} \right) \right) \, dt \]
Partial solutions involve calculations like finding antiderivatives and evaluating from 0 to 6, which results in approximately 31.815 cubic yards.
**(b) Write an expression for \( Y(x) \), the total number of cubic yards of sand on the beach at time \( x \). Carefully explain your thinking and reasoning.**
The solution involves using:
\[ Y(x) = 2500 + \int_0^x S(t) \, dt - \int_0^x R(x) \, dx \]
And calculating individual integrals for \( S(t) \) and \( R(t) \).
**(c) At time \( t = 4 \), is the total amount of sand on the beach increasing or decreasing? At what instantaneous rate is the total number of cubic yards of sand on the beach at time \( t = 4 \) changing?**
Express the rate of change as:
\[ Y'(t) = S(t) - R(t) \]
Calculations using specific values show:
\[ Y'(4) = \frac{15(4)}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf909276-6ad3-4144-b98c-235c5a32e436%2F688adbbb-f68f-476c-8402-314e83c4adba%2Fzpj8j4w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Math 152 - Net Change Theorem: Applications of FTC**
**7. The tide removes sand from the beach at a small ocean park at a rate modeled by the function:**
\[ R(t) = 2 + 5 \sin \left( \frac{4\pi t}{25} \right) \]
A pumping station adds sand to the beach at a rate modeled by the function:
\[ S(t) = \frac{15t}{1 + 3t} \]
Both \( R(t) \) and \( S(t) \) are measured in cubic yards of sand per hour. \( t \) is measured in hours, and the valid times are \( 0 \leq t \leq 6 \). At time \( t = 0 \), the beach holds 2500 cubic yards of sand.
**(a) What definite integral measures how much sand the tide will remove during the time period \( 0 \leq t \leq 6 \)? Why?**
Solution steps involve setting up and evaluating the integral:
\[ \int_0^6 R(t) \, dt = \int_0^6 \left( 2 + 5 \sin \left( \frac{4\pi t}{25} \right) \right) \, dt \]
Partial solutions involve calculations like finding antiderivatives and evaluating from 0 to 6, which results in approximately 31.815 cubic yards.
**(b) Write an expression for \( Y(x) \), the total number of cubic yards of sand on the beach at time \( x \). Carefully explain your thinking and reasoning.**
The solution involves using:
\[ Y(x) = 2500 + \int_0^x S(t) \, dt - \int_0^x R(x) \, dx \]
And calculating individual integrals for \( S(t) \) and \( R(t) \).
**(c) At time \( t = 4 \), is the total amount of sand on the beach increasing or decreasing? At what instantaneous rate is the total number of cubic yards of sand on the beach at time \( t = 4 \) changing?**
Express the rate of change as:
\[ Y'(t) = S(t) - R(t) \]
Calculations using specific values show:
\[ Y'(4) = \frac{15(4)}{
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