Question

Michael is paid $9 per hour to work at the movie theater and $7 per hour when he helps his aunt at her bakery. Michael cannot work more than 32 hours in a week, but he wishes to earn at least $251 each week. Which work schedule is within Michael’s constraints?
A) 8 hours at the movie theater and 25 hours at the bakery
B) 15 hours at the movie theater and 20 hours at the bakery
C) 19 hours at the movie theater and 12 hours at the bakery
D) 21 hours at the movie theater and 8 hours at the bakery

Answer 1

The answer is C.) 19 hours at the movie theater and 12 hours at the bakery. 19 times 9 + 12 times 7 = $255 each week. Also, 19 + 12 hours = 31, which is one hour less than the maximum Michael can work.

Answer 2

Let

x-------> the number of hours working to the movie theater

y------->  the number of hours working at the bakery

we know that

$x+y \leq 32\ hours$ ------> inequality A

$9x+7y \geq \ 251$ -----> inequality B

we will proceed to verify each case to determine the solution of the problem

 If the work schedule is within Michael's limitations, then he must meet both of these inequalities.

case A)  $8$ hours at the movie theater and $25$ hours at the bakery

$x=8\ hours\\y=25\ hours$

Verify inequality A

$8+25 \leq 32\ hours$

$33 \leq 32\ hours$ ------> is not true

therefore

The work schedule of case A) is not within Michael's limitations

case B)  $15$ hours at the movie theater and $20$ hours at the bakery

$x=15\ hours\\y=20\ hours$

Verify inequality A

$15+20 \leq 32\ hours$

$35 \leq 32\ hours$ ------> is not true

therefore

The work schedule of case B) is not within Michael's limitations

case C)  $19$ hours at the movie theater and $12$ hours at the bakery

$x=19\ hours\\y=12\ hours$

Verify inequality A

$19+12 \leq 32\ hours$

$31 \leq 32\ hours$ ------> is  true

Verify inequality B

$9*19+7*12 \geq \ 251$

$\ 255 \geq \ 251$ ------> is true

therefore

The work schedule of case C) is within Michael's limitations

case D)  $21$ hours at the movie theater and $8$ hours at the bakery

$x=21\ hours\\y=8\ hours$

Verify inequality A

$21+8 \leq 32\ hours$

$29 \leq 32\ hours$ ------> is  true

Verify inequality B

$9*21+7*8 \geq \ 251$

$\ 245 \geq \ 251$ ------> is not true

therefore

The work schedule of case D) is not within Michael's limitations

the answer is the option C

$19$ hours at the movie theater and $12$ hours at the bakery