Question

The area of a rectangle is given by the function A(x)=x(x−2). What is the value of x if the area is doubled?

A vehicle's distance d in miles is given by the function d(r,t)=r•t, in which r is the rate in miles per hour and t is the time in hours. What happens to the value of d if the rate is tripled and the time increases by 3 hours?

Answer 1

1. Let x(x-a) = a. Then x²-2x-a =0 so that x = [2±√(4+4a)]/2 =[1±√(1+a)] = 1+√(1+a) as a is positive. When x(x-2) = 2a, then x²-2x-2a =0 so that x = [2±√(4+8a)]/2 =[1±√(1+2a)] = 1+√(1+2a) as a is positive. Hence the value of x now is [1+√(1+2a)]/[ 1+√(1+a)] times the earlier value of x, where a is the earlier area of the rectangle.

2. We have d(r,t) = r*t where r is the rate in miles per hour and t is the time in hours. When r is tripled and t increases by 3 hours, then the new value of d is 3r(t+3). Thus, the new value of d is [3r(t+3)/rt] = 3(t+3)/t times the earlier value of d .