Optimization problems

Answer Key - Grade 12 Math Worksheet - Challenge Me

Circle the best answer, or write your answer on the line.

1. 1.

   Esme fences a 300 m² rectangular plot. The front costs $5 per meter; the back and two sides cost $3 per meter. What front length x, in meters, costs least?

   15
2. 2.

   Kira fences a 800 m² rectangular plot. The front costs $3 per meter; the back and two sides cost $1 per meter. What front length x, in meters, costs least?

   20
3. 3.

   Luz fences a 300 m² rectangular plot. The front costs $5 per meter; the back and two sides cost $1 per meter. What front length x, in meters, costs least?

   10
4. 4.

   Milo picks two positive numbers x and y with x + y = 21 so that x · y² is as large as possible. What is y?

   14
5. 5.

   Ren picks two positive numbers x and y with x + y = 27 so that x · y² is as large as possible. What is y?

   18
6. 6.

   Zee picks two positive numbers x and y with x + y = 33 so that x · y² is as large as possible. What is y?

   22
7. 7.

   Tala's pen has area A(x) = x(20 − x), but the yard allows only 0 ≤ x ≤ 8. Which x gives the most area?

   • x = 0, the starting end of the range
   • x = 10, where the derivative A′(x) = 0
   • x = 8, the end of the allowed range
8. 8.

   A ball's height is h(t) = 40t − 5t² for 0 ≤ t ≤ 6, and Esme wants the highest point. At which t is it?

   • t = 0, the start of the range
   • t = 6, the end of the range
   • t = 4, where h′(t) = 0
9. 9.

   Jun's poster area is P(x) = x(30 − x), but the frame allows only 0 ≤ x ≤ 12. Which x gives the most area?

   • x = 0, the starting end of the range
   • x = 15, where the derivative P′(x) = 0
   • x = 12, the end of the allowed range
10. 10.

    Kira's cost is C(x) = x² − 20x + 150 for 0 ≤ x ≤ 6. Which x gives the least cost?

    • x = 10, where the derivative C′(x) = 0
    • x = 0, the starting end of the range
    • x = 6, the end of the allowed range