Answer:
Part A: 12 cm
Part B: 840 cm^2
Step-by-step explanation:
Part A:
Draw an isosceles triangle with a horizontal base 10 cm long and two sides 13 cm long. Now draw a segment from the vertex where the two 13-cm sides meet to the midpoint of the 10-cm base. This segment is perpendicular to the base and is the height of the triangle. We need to find its length. The large triangle with 13-cm, 13-cm, and 10-cm sides is now divided into two smaller right triangles. Each of the right triangles has a leg 5 cm long and a 13-cm hypotenuse. The height of the large triangle is the other leg of the smaller triangles.
We now use the Pythagorean theorem:
a^2 + b^2 = c^2
5^2 + b^2 = 13^2
25^2 + b^2 = 169
b^2 = 144
b = 12
The height of the triangular base of the prism is 12 cm long.
Part B:
The amount of cardboard used to make the box is the total surface area of the prism.
total surface area = area of the bases + lateral area
The area of the bases is the sum of the areas of the two bases. Since the two bases are congruent triangle, we multiply the area of one base by 2.
The lateral area is the sum of the area of the three rectangular sides. Two of the rectangles have dimensions 13 cm by 20 cm. One rectangle has dimensions 10 cm by 20 cm.
total surface area = 2 * bh/2 + 2(13 cm * 20 cm) + (10 cm * 20 cm)
total surface area = 2 * (10 cm * 12 cm)/2 + 2(13 cm * 20 cm) + (10 cm * 20 cm)
total surface area = 10 cm * 12 cm + 2(260 cm^2) + 200 cm^2
total surface area = 840 cm^2
