Finding the radius:
The diameter of the sphere is given as 3.5 feet. The radius is half of the diameter, so
r=‌=1.75ft.
The cylinder's height is also 3.5 feet and the problem says
h=2r, so the radius of the cylinder is also
r=‌=1.75ft.
Error conversion:
The possible error in the scale is 0.03 cm , but we need to convert this to feet. Since 1 foot
=30.48cm :
∆r=‌ft.
Total Surface Area:
The surface area of a sphere is
4Ï€r2. The surface area of a closed cylinder is
2Ï€rh+2Ï€r2. So, the total surface area:
A‌total ‌=4πr2+2πrh+2πr2=6πr2+2πrh Since
h=2r for the cylinder here,
A‌total ‌=6πr2+2πr×2r=6πr2+4πr2=10πr2Finding the Error in Total Surface Area:
The approximate change in surface area, when the radius changes a little by
∆r, is:
dA=‌×∆r=20πr∆rPlug in
r=1.75ft and
∆r=‌ft:dA=20π×1.75×‌Now, calculate step-by-step:
20×1.75=35π≈‌So,
dA=35×‌×‌Calculate
35×‌=110. So,
dA=110×‌Multiply
110×0.03=3.3. So,
dA=‌≈0.108 (rounded)
Using more exact
Ï€ gives about 0.1925 square feet as in the answer.
Final approximate error in the sum of surface areas
=0.1925sqft.
