Problem solving involving polynomial function
- 2. Study these examples. Example 1. A square base wooden food tray with a volume of 450 cubic inches is to be constructed from a square sheet of wood by cutting 2-inch squares from the corners and bending up sides. What must be the size of the sheet of wood to be used? Find the dimensions of the wooden tray?
- 3. Draw the indicated figure in the problem.
- 4. Steps Solution I. Identify the given information. Let h = 2 inches be the height of the wooden tray l = length and width of the square l – 2 length and width of the wooden tray V= 450 cubic inches be the volume of the wooden tray 2. Represent the volume of the wooden tray V = lwh = lwh = (l – 2)(l - 2) (2) = (l2 – 4l + 4)(2) = 2l2 – 8l + 8
- 5. Steps Solution 3. Find the quadratic equation using the result in no. 3 Since the volume is 450 we have, = 2l2 - 8l + 8 = 450 = l2 –4l + 4 = 225 = l2 – 4l + 4 – 225 = 0 = l2 – 4l -221 = 0 4. Determine the size of the wood by computing the roots 𝑥 = −𝑏± 𝑏2−4𝑎𝑐 2𝑎 l= −(−4)± (−4)2−4(1)(−221) 2(1) l = 4 ± 16+884 2 = 4 ± 900 2 = 4 ±30 2 l = 4+30 2 = 17 and l = 4 −30 2 = -13
- 6. Determine the dimension l = 17 – 2 w = 17 – 2 h = 2 17 – 2 = 15 Thus, the sheet of wood is 17 in by 17 in. Dimensions of the wooden tray: The height is 2 inches, the length is 15 inches and the width is also 15 1nches.
- 7. Example 2. The area of a right triangle is 240 cm2. Find the length of the legs of the right triangle if the length of one of the legs is 2 cm less than the twice the length of the smaller leg. Draw the indicated figure in the problem.
- 8. Steps Solution 1. Identify the given information Let x = the smaller leg 2x – 2 be the length of the leg 2. Represent the area of the triangle A = 1 2 bh = 1 2 x 2x − 2 A = 2𝑥2 −2𝑥 2 = x2 - x A = 240cm2 x2 – x - 240= 0
- 9. 𝑥 = −𝑏 ± 𝑏2 − 4𝑎𝑐 2𝑎 𝑥 = −(−1) ± (−1)2−4(1)(−240) 2(1) 𝑥 = 1± 1+960 2 = 1± 961 2 = 1±31 2 x = 1+31 2 = 16 and x = 1 −31 2 = -15
- 10. x = the smaller leg = 16cm 2x – 2 be the length of the longer leg = 2(16) – 2 = 30 cm Thus the smaller leg of the right triangle is 16 cm and the longer is 30 cm.