A contractor charged a client $240 to paint a fence, estimating that the work would take a certain number of hours to complete. However, the contractor completed the work 2 hours earlier than estimated, resulting in a $6 increase in his per hour rate. How many hours did the contractor originally estimate the work would take?
When I factorize the quadratic equation, where t is time, I get (t-12)(t+10), i.e., t=12 and t=-10. yet, the answer here is 10. Some help please? Thanks!
Nevermind, messed up in the calculations, got the answer. Not able to delete this post for some reason, so here lies a mark of eternal embarrassment :’)
t * r = 240
(t - 2)(r + 6) = 240
(t - 2)(\frac{240}{t} + 6) = 240
240 + 6t - \frac{480}{t} - 12 = 240
6t - \frac{480}{t} - 12 = 0
6t^2 - 480 - 12t = 0
t^2 - 80 - 2t = 0
t^2 -10t + 8t - 80 = 0
t(t - 10) + 8(t - 10) = 0
(t + 8) (t - 10)
t = -8, 10
T1 = estimated time to complete the job (in hours)
T2 = actual time taken (in hours)
We know:
Step-by-step:
From equation (1):
T2 = T1 − 2
Substitute into equation (2):
(240 / (T1 − 2)) − (240 / T1) = 6
Multiply both sides by T1(T1 − 2) to eliminate denominators:
240 * T1 − 240 * (T1 − 2) = 6 * T1 * (T1 − 2)
Simplify left-hand side:
240T1 − 240T1 + 480 = 6T1² − 12T1
480 = 6T1² − 12T1
Divide both sides by 6:
80 = T1² − 2T1
Rearrange:
T1² − 2T1 − 80 = 0
Solve the quadratic equation:
T1 = 10 (only the positive solution is valid)
