Sonya can paint at a rate of

*v*(*t*) = 150 – 4*t*

square feet per hour, where *t* is the number of hours since she started painting. Can Sonya paint the walls of a 12 foot by 12 foot office with an 8 foot high ceiling in 3 hours?

What about the walls of a 14 foot by 14 foot office with an 8 foot high ceiling?

Answer

We need to figure out how many square feet Sonya can paint in 3 hours. Then for each office we need to determine how many square feet there are to paint, and see if this is greater or smaller than the number of square feet Sonya can paint in 3 hours.

To find out how many square feet Sonya can paint in 3 hours, we integrate her rate of painting from time *t* = 0 hours to time *t* = 3 hours:

The first office is 12 by 12 feet with a height of 8 feet. Since each wall has width 12 and height 8, and there are 4 walls, the office has

12 × 8 × 4 = 384

square feet of wall that need to be painted. Since 384 is less than 432, Sonya can paint the walls of this office in 3 hours.

The second office is 14 x 14 feet with a height of 8 feet. Since each wall has width 14 and height 8, and there are 4 walls, the office has

14 × 8 × 4 = 448

square feet of wall that need to be painted. Since 448 is greater than 432, Sonya cannot paint the walls of this office in 3 hours.