PROBLEM 3. The radius of a sphere is measured to be 4 cm with an error of ±0.01 cm. Use differentials
to approximate the error in the surface area.
Answer: Now it’s time for the other differential formula. The formula for the surface area of a sphere is
S = 4πr^2The formula says that dS = S′dr, so first, we find the derivative of the surface area, (S′ = 8πr) and plug
away.
dS = 8π rdr = 8π (4)(±0.01) = ± 1.0053This looks like a big error, but given that the surface area of a sphere with radius 4 is approximately 201
cm^2 , the error is quite small.
PRACTICE PROBLEM SET 16
Use the differential formulas in this chapter to solve these problems. The answers are in Chapter 19.
1.Approximate .2.Approximate .3.Approximate tan 61°.4.The side of a cube is measured to be 6 in. with an error of ±0.02 in. Estimate the error in the volume
of the cube.5.When a spherical ball bearing is heated, its radius increases by 0.01 mm. Estimate the change in
volume of the ball bearing when the radius is 5 mm.6.A cylindrical tank is constructed to have a diameter of 5 meters and a height of 20 meters. Find the
error in the volume if(a)the diameter is exact, but the height is 20.1 meters; and(b)the height is exact, but the diameter is 5.1 meters.