the following rational function gives the number of days it would take two construction crews, working together, to frame a house that crew 1 (working alone)could complete in t days and crew 2 (working alone) could complete in t+3 days. f(t)=(t^2+3t)/(2t+3). (a) if crew 1 could frame a certain house in 24 days, how long would it take both crews working together? (b) if crew 2 could frame a certain house in 32 days, how long would it take both crews working together?

It's easier to work with

1/f(t) = 1/t + 1/(t+3)

For crew 1, 1/t = 1/24, so t=24
Thus 1/f(t) = 1/24 + 1/27 = 17/216
So, both crews can do it in 216/17 days

For crew 2, 1/(t+3) = 1/32, so t=29
Now 1/f(t) = 1/29 + 1/32 = 61/928
So, both crews can do it in 928/61 days

To answer both parts of the question, we need to substitute the given values into the rational function.

(a) If crew 1 could frame a certain house in 24 days, we substitute t = 24 into the function f(t) = (t^2 + 3t)/(2t + 3).

f(24) = (24^2 + 3 * 24)/(2 * 24 + 3)
= (576 + 72)/(48 + 3)
= 648/51
= 12

Therefore, crew 1 and crew 2 working together would frame the house in 12 days.

(b) If crew 2 could frame a certain house in 32 days, we substitute t = 32 into the function f(t) = (t^2 + 3t)/(2t + 3).

f(32) = (32^2 + 3 * 32)/(2 * 32 + 3)
= (1024 + 96)/(64 + 3)
= 1120/67

Therefore, crew 1 and crew 2 working together would frame the house in approximately 16.7164 days (rounded to 4 decimal places).

To find how long it would take both crews working together, we need to evaluate the rational function f(t) at the given values of t.

(a) If crew 1 could frame a certain house in 24 days, we substitute t = 24 into the function f(t):

f(t) = (t^2 + 3t) / (2t + 3)
f(24) = (24^2 + 3*24) / (2*24 + 3)
= (576 + 72) / (48 + 3)
= 648 / 51
= 12

Therefore, it would take both crews working together 12 days to frame the house.

(b) If crew 2 could frame a certain house in 32 days, we substitute t = 32 into the function f(t):

f(t) = (t^2 + 3t) / (2t + 3)
f(32) = (32^2 + 3*32) / (2*32 + 3)
= (1024 + 96) / (64 + 3)
= 1120 / 67

Therefore, it would take both crews working together approximately 16.7164 days (rounded to four decimal places) to frame the house.

Thank you Steve, that was a big help.