Question

1. 6.2 yd 15.7 yd 8.5 yd
2. hint: use pythagorean theorem to find diameter. 19.3 m 9.5 m
3. hint: use pythagorean theorem to find the height of triangular base. 17 mm 12 mm 12 mm

Explanation:

1. Problem 9:

• Since no specific question is asked about the triangular - prism in problem 9, we can't solve it. Let's assume we want to find the volume of the triangular - prism. The volume of a triangular prism \(V = Bh\), where \(B\) is the area of the base and \(h\) is the height of the prism. The base is a triangle with base \(b = 6.2\) yd and height \(h_{triangle}=8.5\) yd, and the height of the prism \(h_{prism}=15.7\) yd.
• First, find the area of the triangular base:
• # Explanation:

Step1: Calculate area of triangle

The formula for the area of a triangle is \(A=\frac{1}{2}bh\). Here, \(b = 6.2\) yd and \(h = 8.5\) yd. So \(A=\frac{1}{2}\times6.2\times8.5\).
\(A = 26.35\) square - yards.

• Then, find the volume of the prism:

Step2: Calculate volume of prism

The formula for the volume of a prism is \(V = Ah\), where \(A\) is the area of the base and \(h\) is the height of the prism. Here, \(A = 26.35\) square - yards and \(h = 15.7\) yd. So \(V=26.35\times15.7\).
\(V = 413.695\) cubic - yards.

1. Problem 10:

• We are given a cylinder with a diagonal of the cross - section \(d = 19.3\) m and height \(h = 9.5\) m. We need to find the diameter \(D\) of the base using the Pythagorean theorem. In a right - triangle formed by the height of the cylinder and the diameter of the base, if the diagonal of the cross - section is \(d\), the height is \(h\), and the diameter is \(D\), then \(d^{2}=h^{2}+D^{2}\) (by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = d\), \(a = h\), and \(b = D\)).
• # Explanation:

Step1: Rearrange Pythagorean formula for \(D\)

We have \(D^{2}=d^{2}-h^{2}\). Given \(d = 19.3\) m and \(h = 9.5\) m. So \(D^{2}=19.3^{2}-9.5^{2}\).
\(D^{2}=372.49 - 90.25=282.24\).

Step2: Find \(D\)

Take the square - root of both sides. \(D=\sqrt{282.24}=16.8\) m.

1. Problem 11:

• For the triangular prism, we are given the side lengths of the triangular base and the height of the prism. To find the height of the triangular base, we consider the right - triangle formed within the triangular base. Let the base of the right - triangle within the triangular base be \(a\) (half of the base of the isosceles triangle of the base of the prism), the height of the right - triangle be \(h_{triangle}\), and the hypotenuse be \(c\). Here, assume the base of the isosceles triangle of the base of the prism is \(12\) mm and the slant - side is \(17\) mm. The base of the right - triangle within the isosceles triangle base is \(a=\frac{12}{2}=6\) mm.
• # Explanation:

Step1: Apply Pythagorean theorem

Using the Pythagorean theorem \(c^{2}=a^{2}+h^{2}\), where \(c = 17\) mm and \(a = 6\) mm. We want to find \(h\). So \(h^{2}=c^{2}-a^{2}\).
\(h^{2}=17^{2}-6^{2}=289 - 36 = 253\).

Step2: Find the height of the triangular base

\(h=\sqrt{253}\approx15.9\) mm.

Answer:

Problem 9 volume: \(413.695\) cubic - yards
Problem 10 diameter: \(16.8\) m
Problem 11 height of triangular base: \(\sqrt{253}\approx15.9\) mm