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
media.cheggcdn.com/media/a86/a86f85b4-821e-4c06-863d-126b863efdae/phpwYmomy
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)
A. To estimate the radius of the balloon when t = 4.5 using the tangent line approximation at t = 4, we need to find the equation of the tangent line at t = 4.
1. First, locate the point on the graph where t = 4 and find the corresponding radius. Let's call this point (4, r1).
2. Next, find the slope of the tangent line at t = 4. This can be done by finding the derivative of the radius function with respect to time (t) and evaluating it at t = 4. Let's call this slope m.
3. Once you have the slope, use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is the point on the line.
Now, it's important to note that the given graph of r is concave down for 0 < t < 10. This means that the graph is curving downwards or decreasing. Since we are finding the tangent line at t = 4, and the graph is concave down, the tangent line will be below the actual graph at that point.
To answer the question, the approximation using the tangent line will be lower (less than) the true value.
B. To find the rate at which the volume is decreasing when the radius of Balloon B is 2 feet, we need to use the relationship between volume (V) and radius (r) of a balloon.
1. The volume of a balloon is given by the formula V = (4/3)*pi*r^3, where pi is a constant.
2. We're told that the radius is decreasing at a rate of 2/pi feet per second, so the rate of change of radius with respect to time (dr/dt) is -2/pi ft/s (negative sign indicates decreasing).
3. To find the rate at which the volume is decreasing (dV/dt), we can use the chain rule of differentiation. The chain rule states that dV/dt = dV/dr * dr/dt, where dV/dr is the derivative of volume with respect to radius, and dr/dt is the rate of change of radius with respect to time.
4. Taking the derivative of V with respect to r gives us dV/dr = 4*pi*r^2.
5. Substitute the given value for dr/dt (-2/pi) and the given radius (2 ft) into the equation dV/dt = dV/dr * dr/dt, and calculate the result.
This will give you the rate at which the volume is decreasing when the radius of Balloon B is 2 feet.
C. To find the marginal cost for adding the 101st balloon to the inventory, we need to calculate the derivative of the cost function C(x) with respect to the number of balloons x.
1. Start with the given cost function C(x) = 21,000/x + 2.4x.
2. Take the derivative of C(x) with respect to x (dC/dx) using the rules of derivative.
3. Simplify the derivative expression to find the marginal cost for adding one more balloon to the inventory.
4. Once you have the derivative, substitute x = 101 into the expression to find the marginal cost for adding the 101st balloon to the inventory.
The meaning of this extra balloon in the context of the scenario is that it represents the additional cost required to add one more balloon to the existing inventory of 100 balloons. The marginal cost measures how much the total cost will increase by adding one more unit (in this case, one more balloon) to the inventory.