Question

a right triangular prism with a right isosceles triangular base is shown. the height of the prism is equal to 9 centimeters (cm), and the base of the right isosceles triangle is equal to 8 cm. what is the volume of the prism, rounded to the nearest cubic centimeter?

Explanation:

Step1: Find the area of the triangular base

The base is a right isosceles triangle with base \( b = 8 \, \text{cm} \). In a right isosceles triangle, the legs are equal, so the height of the triangle (let's call it \( h_{triangle} \)) is also \( 8 \, \text{cm} \) (wait, no—wait, in a right isosceles triangle, the legs are the two equal sides, and the base and height of the triangle (for area calculation) are the legs. Wait, actually, the area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). For a right isosceles triangle, if the base of the triangle (the side) is \( 8 \, \text{cm} \), then the height of the triangle (the other leg) is also \( 8 \, \text{cm} \)? Wait, no, wait—wait, the problem says "the base of the right isosceles triangle is equal to 8 cm". In a right isosceles triangle, the two legs are equal, and the hypotenuse is different. Wait, maybe I made a mistake. Wait, the area of a right triangle is \( \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 \). Since it's isosceles, \( \text{leg}_1 = \text{leg}_2 \). Wait, but the problem says "the base of the right isosceles triangle is equal to 8 cm". So maybe the base of the triangle (the side) is 8 cm, which is one of the legs. So the two legs are both 8 cm? Wait, no, maybe the base is 8 cm, and the height (the other leg) is also 8 cm? Wait, no, let's re-examine.

Wait, the formula for the volume of a prism is \( V = B \times h \), where \( B \) is the area of the base, and \( h \) is the height of the prism (the distance between the two bases).

The base is a right isosceles triangle. So the area of the triangular base \( B \) is \( \frac{1}{2} \times \text{base of triangle} \times \text{height of triangle} \). Since it's a right isosceles triangle, the base and height of the triangle (the two legs) are equal. Wait, the problem says "the base of the right isosceles triangle is equal to 8 cm". So the base of the triangle (one leg) is 8 cm, so the height of the triangle (the other leg) is also 8 cm? Wait, no, that can't be. Wait, maybe the base of the triangle is 8 cm, and the height of the triangle (the altitude to the base) is equal to the base? Wait, no, in a right isosceles triangle, the legs are the two equal sides, and the hypotenuse is \( \text{leg} \times \sqrt{2} \). So if the base of the triangle (one leg) is 8 cm, then the other leg (the height of the triangle) is also 8 cm. So the area of the triangle is \( \frac{1}{2} \times 8 \times 8 \)? Wait, no, that would be if both legs are 8. Wait, but let's check the problem again: "the base of the right isosceles triangle is equal to 8 cm". So the base (the side) is 8 cm, which is one leg. So the two legs are 8 cm each. So the area of the triangle is \( \frac{1}{2} \times 8 \times 8 = 32 \, \text{cm}^2 \)? Wait, no, that seems too big. Wait, maybe I'm misunderstanding. Wait, the height of the prism is 9 cm. Wait, let's re-express the volume formula for a triangular prism: \( V = \text{Area of base} \times \text{height of prism} \).

The base is a right isosceles triangle. Let's denote the legs of the right triangle as \( a \) and \( a \) (since it's isosceles), so the area of the base \( B = \frac{1}{2} \times a \times a = \frac{1}{2} a^2 \). The problem says "the base of the right isosceles triangle is equal to 8 cm". So the base of the triangle (the side) is 8 cm, which is one of the legs, so \( a = 8 \, \text{cm} \). Then the area of the base is \( \frac{1}{2} \times 8 \times 8 = 32 \, \text{cm}^2 \)? Wait, no, that can't be. Wait, maybe the base of the triangle is 8 cm, and the height of the triangle (the altitude) is \( \frac{8}{2} = 4 \, \text{cm} \)? Wait, no, that's for an isosceles triangle, not a right isosceles triangle. In a right isosceles triangle, the legs are the base and height, so they are equal. So if the base of the triangle (one leg) is 8 cm, the other leg (height of the triangle) is also 8 cm. Then the area is \( \frac{1}{2} \times 8 \times 8 = 32 \, \text{cm}^2 \). Then the volume of the prism is \( B \times \text{height of prism} = 32 \times 9 = 288 \, \text{cm}^3 \)? Wait, that seems high, but let's check again.

Wait, maybe I made a mistake. Wait, the problem says "a right triangular prism with a right isosceles triangular base". So the base is a right isosceles triangle, so the two legs are equal, and the right angle is between them. So the area of the base is \( \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 \). Since it's isosceles, \( \text{leg}_1 = \text{leg}_2 \). The problem says "the base of the right isosceles triangle is equal to 8 cm". So the base of the triangle (the side) is 8 cm, which is one leg, so \( \text{leg}_1 = 8 \, \text{cm} \), so \( \text{leg}_2 = 8 \, \text{cm} \). Then the area of the base is \( \frac{1}{2} \times 8 \times 8 = 32 \, \text{cm}^2 \). The height of the prism (the distance between the two triangular bases) is 9 cm. So the volume is \( 32 \times 9 = 288 \, \text{cm}^3 \). Wait, but that seems correct? Wait, no, wait—maybe the base of the triangle is 8 cm, and the height of the triangle (the altitude) is 4 cm? Wait, no, in a right isosceles triangle, the legs are the base and height, so they are equal. So if the base is 8, the height is 8. So the area is 32. Then volume is 32*9=288. So the answer is 288.

Wait, let's confirm the formula for the volume of a triangular prism: \( V = \text{Area of the triangular base} \times \text{height of the prism} \). The area of the triangular base (right isosceles triangle) is \( \frac{1}{2} \times \text{base} \times \text{height} \). Since it's right isosceles, base = height = 8 cm. So area is \( \frac{1}{2} \times 8 \times 8 = 32 \, \text{cm}^2 \). Then volume is \( 32 \times 9 = 288 \, \text{cm}^3 \).

Step2: Calculate the volume of the prism

Now that we have the area of the base \( B = 32 \, \text{cm}^2 \) and the height of the prism \( h = 9 \, \text{cm} \), we use the formula for the volume of a prism: \( V = B \times h \).

Substituting the values, we get:
\( V = 32 \times 9 = 288 \, \text{cm}^3 \)

Answer:

288