Question

A candy bar box is in the shape of a triangular prism. The volume of the box is 2,400 cubic centimeters. A triangular prism is shown with base of triangle labeled 16 cm, sides of triangles labeled 17 cm, and length of the box equal to 20 cm. Part A: What is the height of the base? Show your work. (5 points) Part B: What is the approximate amount of cardboard used to make the candy box? Explain how you got your answer. (5 points)

Answer 1

Answer:

  A) 15 cm

  B) 1240 cm²

Step-by-step explanation:

Part A: The height of the triangular base can be found a couple of ways. One is to use the Pythagorean theorem.

The triangular base is isosceles, so the height, half its base (8 cm) and the long edge (17 cm) form a right triangle. The height is then found from ...

  17² = 8² + height²

  289 -64 = height² . . . . . subtract 64

  √225 = 15 = height . . . . take the square root

The height of the triangular base is 15 cm.

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Part B: The volume of a prism is given by ...

  V = Bh

where B is the area of the base and h is the length ("heigh") of the prism. We can use this formula to find B, the area of each of the triangular bases of the prism.

  2400 cm³ = B·(20 cm)

  2400 cm³/(20 cm) = B = 120 cm² . . . . . area of one end of the prism

Now the lateral area of the prism is the product of its length (20 cm) and the perimeter of its base (17 cm + 17 cm + 16 cm). That area is ...

  lateral area = (20 cm)(50 cm) = 1000 cm²

Together with the areas of the two ends, we find the total area of the cardboard box to be ...

  total area = lateral area + 2×base area = 1000 cm² + 2×120 cm²

  total area = 1240 cm²

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Note that you can also find the height of the base triangle from the base area.

  A = (1/2)bh

  120 cm² = (1/2)(16 cm)h

  120 cm²/(8 cm) = h = 15 cm