Questions and Answers
What is the left Riemann sum approximation for the volume of the tank?
176.3 cubic feet
Does the approximation in Part A overestimate or underestimate the volume of the tank?
overestimate
What is the volume of the tank based on the model f(h) = 50.3/[e^(0.2h) + h]?
101.325 cubic feet
What is the rate at which the volume of water is changing when the height is 5 feet?
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What is the area of region R bounded by the curve r=f(θ) = 1 + sinθ cos(2θ) from 0 to π/2?
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Write an equation that would determine the value of k which divides the area of region S into two equal parts.
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What is the expression for w(θ), the distance between the points with polar coordinates (f(θ), θ) and (g(θ), θ)?
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What is the average value of w(θ) over the interval 0 ≤ θ ≤ π/2?
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Study Notes
Tank Volume and Area Calculations
- A tank with a height of 10 feet has a decreasing function A(h) for horizontal cross-section area at height h.
- Volume of the tank can be approximated using a left Riemann sum:
- Volume = ∫{0,10} A(h) dh
- Approximation: 2 * 50.3 + 3 * 14.4 + 5 * 6.5 = 176.3 cubic feet.
Overestimation of Volume
- Using a left Riemann sum results in an overestimate when A(h) is decreasing, as it captures the area at the left endpoint of each subinterval.
Exact Volume Using Model
- The area function is modeled as f(h) = 50.3/[e^(0.2h) + h].
- Volume calculated as ∫{0,10} f(h) dh = 101.325 cubic feet, demonstrating that the integral of area gives volume.
Rate of Volume Change
- When water is at a height of 5 feet, it rises at 0.26 feet per minute.
- Rate of volume change: dV/dt = (dV/dh) * (dh/dt) = f(5) * 0.26 = 1.694 cubic feet per minute.
Polar Area Calculations
- For polar curves r = f(θ) = 1 + sin(θ)cos(2θ) and r = g(θ) = 2cos(θ) over the interval [0, π/2]:
- Area of region R is found using the formula:
- Area = (1/2) ∫{0, π/2} (f(θ))² dθ = 0.648.
- Area of region R is found using the formula:
Equation for Equal Areas
- The ray θ = k divides region S into equal areas, leading to the equation:
- ∫{0,k} ((g(θ))² - (f(θ))²) dθ = (1/2) ∫{0, π/2} ((g(θ))² - (f(θ))²) dθ.
Distance Between Points in Polar Coordinates
- The distance w(θ) between points (f(θ), θ) and (g(θ), θ) is defined as:
- w(θ) = g(θ) - f(θ).
- Average value of w(θ) over [0, π/2]:
- w_A = [1/(π/2 - 0)] [∫{0, π/2} w(θ) dθ] = 0.485.
Finding Specific Value of θ
- Determining θ for which w(θ) = w_A requires solving the equation with respect to the average distance w_A found above.
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Description
Prepare for the AP BC Calculus exam with this quiz focused on the 2017 free response questions. This specific problem involves calculating the volume of a tank using a left Riemann sum, emphasizing integration and approximation techniques. Test your understanding of calculus concepts with this targeted review.