Answer:
out estimation is underestimate when we increase the cost.
Step-by-step explanation:
Given diameter of the racquetballs (x)= 4 cm
New diameter = 4.05 cm
Therefore, change in diameter ( Δ x) = 0.05.
Cost Function of x diameter of 30,000 racquetballs at the rate 0.005 per square centimeter
C(x) = 30000 × 0.005 × S/4
C(x) = 150 × S/4 = 37.5 × S.
We are given S = π r^2
Therefore,
C(x)= 37.5 π r^2.
C(4) = 37.5 *π*(4)^2 = 37.5 π*16
C (4+0.05) = 37.5 *π*(4.05)^2 = 37.5 * π * 16.4025.
Finding first derivative of cost function, we get
C'(x) = 2 * 37.5 * π r = 75 π r
C(4) = 75 * π * 4 = 300 π
We know,
Δ C(x)/ Δ x = C'(x).
Plugging value of Δ x and x
Δ C(4) = Δ x * C '(4) = 0.05 * 300 π = 15 π = 15 * 3.141 = 47.12.
We also know
Δ C(x) = C(x+Δx) - C(x) = (37.5 * π * 16.4025) - (37.5 π*16) = 37.5 π (16.4025 -16)
= 37.5 π(0.4025)
= 37.5 * 3.14 * 0.4025
= 47.42.
We can see that 47.42 is greater than 47.12.
Therefore, out estimation is underestimate when we increase the cost.
