Two Identical Rectangular Prisms Each Have A Height Of 90 Centimeters. The Base Of Each Prism Is A Square, And The Surface Area Of Each Prism Is $K , \text{cm}^2$. If The Prisms Are Glued Together Along A Square Base, The Resulting Prism

Mar 7, 2025 by ADMIN 240 views

**Two Identical Rectangular Prisms: A Mathematical Discussion** ===========================================================

Introduction

In this article, we will explore the concept of two identical rectangular prisms, each with a height of 90 centimeters and a square base. We will examine the surface area of each prism and discuss the resulting prism when the two prisms are glued together along a square base.

The Surface Area of a Rectangular Prism

A rectangular prism has six faces: two rectangular faces with dimensions $l \times w$ , two rectangular faces with dimensions $l \times h$ , and two rectangular faces with dimensions $w \times h$ . The surface area of a rectangular prism is given by the formula:

SA = 2lw + 2lh + 2wh

where $l$ , $w$ , and $h$ are the length, width, and height of the prism, respectively.

The Surface Area of Each Prism

Let's assume that the base of each prism is a square with side length $s$ . Since the height of each prism is 90 centimeters, the surface area of each prism is:

SA = 2s^2 + 2s(90) + 2s(90)

Simplifying the expression, we get:

SA = 2s^2 + 360s

The Resulting Prism

When the two prisms are glued together along a square base, the resulting prism has a height of 90 centimeters and a square base with side length $2s$ . The surface area of the resulting prism is:

SA = 2(2s)^2 + 360(2s)

Simplifying the expression, we get:

SA = 8s^2 + 720s

Q&A

Q: What is the surface area of each prism?

A: The surface area of each prism is given by the formula $SA = 2s^2 + 360s$ , where $s$ is the side length of the square base.

Q: What is the surface area of the resulting prism?

A: The surface area of the resulting prism is given by the formula $SA = 8s^2 + 720s$ , where $s$ is the side length of the square base.

Q: How do the surface areas of the two prisms compare?

A: The surface area of the resulting prism is four times the surface area of each individual prism.

Q: What is the relationship between the side length of the square base and the surface area of the prism?

A: The surface area of the prism is directly proportional to the square of the side length of the square base.

Q: What is the effect of gluing the two prisms together on the surface area of the resulting prism?

A: Gluing the two prisms together increases the surface area of the resulting prism by a factor of four.

Q: What is the height of the resulting prism?

A: The height of the resulting prism is 90 centimeters, the same as the height of each individual prism.

Q: What is the shape of the base of the resulting prism?

A: The base of the resulting prism is a square with side length $2s$ , twice the side length of the base of each individual prism.

Q: What is the relationship between the side length of the square base and the height of the prism?

A: There is no direct relationship between the side length of the square base and the height of the prism.

Q: What is the effect of changing the side length of the square base on the surface area of the prism?

A: Increasing the side length of the square base increases the surface area of the prism, while decreasing the side length of the square base decreases the surface area of the prism.

Q: What is the effect of changing the height of the prism on the surface area of the prism?

A: Increasing the height of the prism increases the surface area of the prism, while decreasing the height of the prism decreases the surface area of the prism.

Q: What is the relationship between the surface area of the prism and the volume of the prism?

A: The surface area of the prism is not directly related to the volume of the prism.