Optimization worksheet with answers

Free printable Optimization worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
riverxxy

A farmer has 320 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the length y of the side parallel to the river for the largest area.

320 m2x + y = 320
y = m
2.Easy
xxyx

A rectangular pen is fenced with 360 m of fencing and divided into two parts by one extra fence parallel to the sides of length x (three fences of length x and two of length y in total). Find the largest total area A that can be enclosed.

360 m3x + 2y = 360
A = m²
3.Medium
rh

A closed cylindrical can (with a top and a bottom) must hold 330 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

V = 330 cm³
A = cm²
4.Medium
rh

A closed cylindrical can (with a top and a bottom) must hold 330 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

V = 330 cm³
r = cm
5.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 180 − 0.25q euros per unit, and the cost is C(q) = 9600 + 25q euros. Find the maximum weekly profit P.

p = 180 − 0.25qC = 9600 + 25q
P = €
6.Hard
qP(q)maximum

A company sells q units per week. The price at which it can sell them is p = 90 − 2q euros per unit, and the cost is C(q) = 200 + 15q euros. Find the maximum weekly profit P.

p = 90 − 2qC = 200 + 15q
P = €

Answer key

  1. 1. y = 160 m
  2. 2. A = 5400 m²
  3. 3. A = 264.4 cm²
  4. 4. r = 3.745 cm
  5. 5. P = 14425 €
  6. 6. P = 503.1 €

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