Optimization problems

Answer Key - Grade 12 Math Worksheet

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

1. 1.

   Esme uses 66 m of fence to make two pens side by side: three fences of length x and two of length y. What x gives the most total area?

   11
2. 2.

   Jun uses 72 m of fence to make two pens side by side: three fences of length x and two of length y. What x gives the most total area?

   12
3. 3.

   Milo cuts a square of side x from each corner of a 12 cm square card and folds up an open box. What x gives the largest volume?

   2
4. 4.

   Noah cuts a square of side x from each corner of a 18 cm square card and folds up an open box. What x gives the largest volume?

   3
5. 5.

   Ren cuts a square of side x from each corner of a 30 cm square card and folds up an open box. What x gives the largest volume?

   5
6. 6.

   Zee cuts a square of side x from each corner of a 24 cm square card and folds up an open box. What x gives the largest volume?

   4
7. 7.

   A(x) = 30x − x² has A′(15) = 0, and A″(x) = −2. What can Tala conclude about x = 15?

   • A″ < 0 there, so x = 15 gives a maximum
   • A″ < 0 there, so x = 15 gives a minimum
   • A′(15) = 0 is enough to call x = 15 a maximum
8. 8.

   C(x) = 2x + 50/x has C′(5) = 0, and C″(5) = 0.8. What can Esme conclude about x = 5?

   • C″ > 0 there, so x = 5 gives a maximum
   • C′(5) = 0 is enough to call x = 5 a minimum
   • C″ > 0 there, so x = 5 gives a minimum
9. 9.

   Jun's box volume V(x) = 4x³ − 48x² + 144x has V′(2) = 0, and V″(2) = −48. What does that show about x = 2?

   • V″ < 0 there, so x = 2 gives a maximum
   • V″ < 0 there, so x = 2 gives a minimum
   • V′(2) = 0 is enough to call x = 2 a maximum
10. 10.

    D(x) = x² − 8x + 20 has D′(4) = 0, and D″(x) = 2. What can Kira conclude about x = 4?

    • D′(4) = 0 is enough to call x = 4 a minimum
    • D″ > 0 there, so x = 4 gives a maximum
    • D″ > 0 there, so x = 4 gives a minimum