Using Calculus for the Weather Balloon Problem
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Weather Bots is a manufacturer of neoprene weather balloons. As the 350-gram spherical balloon is being inflated with hydrogen, the radius of the balloon, in feet, is modeled by a twice-differentiable function r, of time t, where t is measured in minutes. The table below gives selected values of the rate of change, r ' (t) for the radius of Balloon A over the time interval 0 < t < 10.
t (minutes) 0 2 4 7 9 10
r ' (t) (feet per minute) 3.7 2..6 1.3 0.9 0.4 0.25
A. The radius of Balloon A is 5 feet when t = 4 minutes. Estimate the radius of the balloon when t = 4.5, using the tangent line approximation at t = 4. It is known that the graph of r is concave down for the time 0 < t < 10. Is your approximation greater than or less than the true value?
B. Balloon B has been removed from service and the radius of the balloon is decreasing at a rate of 2/pi feet per second. Find the rate at which the volume is decreasing when the radius of Balloon B is 2 feet. Given: V = 4/3 pi r^3
C. The cost to maintain inventory for the weather balloons is given by
C(x) = 21,500/x + 2.4x, where x is the number of balloons in inventory. Find the marginal cost for adding the 101st balloon to the inventory. Explain the meaning of this extra balloon in context to the scenario.
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Solution Summary
Step-by-step calculations are shown for each part of the calculus problem.
Solution Preview
A)
Use the slope intercept form of the line, y = mx + b.
Substitute m = 1.3, x = 4 and y = 5 into the slope intercept form to get
5 = 1.3 * 4 + b
Solve for b.
5 = 5.2 + b
b = 5.2 - 5
b = -0.2
Substitute the values for b and ...
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- MBA, University of California, Riverside
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- BSc, California State Polytechnic University, Pomona
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