Question

Any help with these 3 calc questions will be appreciated!
The link to the chart is below:



media. cheggcdn.com /media/a86/ a86f85b4-821e-4c06-863d-126b863efdae/ phpwYmomy





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.



C. The cost to maintain inventory for the weather balloons is given by C(x)=21,000/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

Answer 1

media.cheggcdn.com/media/a86/a86f85b4-821e-4c06-863d-126b863efdae/phpwYmomy

Answer 2

A. for a sphere, v = 4/3 πr^3
∆v ≈ dv/dt ∆t
so, ∆t = 0.5, making
∆v ≈ 4πr^2 ∆t = 4π*4*0.5 = 8π
If the graph is concave down, then the tangent line lies above the graph, meaning the approximation is too large.

B. using the chain rule,
dv/dt = 4πr^2 dr/dt
dv/dt = 4π*2^2(-2/π) = -32 ft^3/s

C. The marginal cost is dC/dx = -21000/x^2 + 2.4
So now just find C'(100)