/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 A triangular pyramid has a volum... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 3

A triangular pyramid has a volume of \(180 \mathrm{cm}^{3}\) and a height of \(12 \mathrm{cm} .\) Find the length of a side of the triangular base if the triangle's height from that side is \(6 \mathrm{cm} .\)

Short Answer

Expert verified
The length of the side of the triangular base is 15 cm.

Step by step solution

01

Understand the problem

We need to find the length of a side of the triangular base of a pyramid. Given values are: Volume of the pyramid = 180 cm鲁, height of the pyramid = 12 cm, and height of the triangle from that side = 6 cm.
02

Formula for the volume of a pyramid

The volume of a pyramid is given by \[ V = \frac{1}{3} A h \]where \( A \) is the area of the base and \( h \) is the height. Here, the height of the pyramid is 12 cm.
03

Substitute known values into the volume formula

Plug in the given values to find the area of the base triangle. We have: \[ 180 = \frac{1}{3} A \cdot 12 \]Solve for \( A \).
04

Solve for the base area

Rearrange the formula to solve for \( A \):\[ 180 = 4A \]Thus, \[ A = 45 \: \text{cm}^2 \]
05

Formula for the area of a triangle

The area of a triangle is given by \[ A = \frac{1}{2} b h_t \]where \( b \) is the length of the base and \( h_t \) is the height of the triangle from that base.
06

Substitute known values to find the base length

Using the known area and height of the base triangle, plug in the values to solve for the base length: \[ 45 = \frac{1}{2} b \cdot 6 \]Solve for \( b \).
07

Solve for the base length

Rearrange the formula to find \( b \):\[ 45 = 3b \]Thus, \[ b = 15 \: \text{cm} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangular Base
To fully grasp the problem at hand, it's essential to understand what a triangular base is. A triangular base means that the bottom of the pyramid is shaped like a triangle. This triangle is key to solving many aspects of the problem. Remember, a pyramid can have different types of bases such as square, rectangular, or in this case, triangular.
Working with a triangular base involves thinking about the properties of triangles, like their sides, angles, and height. Understanding these properties helps you use them in various formulas, as we did in this exercise.
Area of a Triangle
To solve for the volume of the pyramid, you first need to find the area of its triangular base. The area of a triangle can be calculated using the formula: \[ A = \frac{1}{2} b h_t \] where \( A \) is the area, \( b \) is the base length, and \( h_t \) is the triangle's height from that base.
In this exercise, we used a 6 cm height from the specified base. When you plugged in the values: \[ 45 = \frac{1}{2} b \times 6 \], you found that the base length \( b \) was 15 cm. This kind of substitution is a common way of isolating the unknown variable in math problems.
Also, remember that the height used in this formula is always perpendicular to the base. So, make sure you correctly identify the height from the right angle in your problems.
Height of a Pyramid
The height of a pyramid is a vertical line from the apex (top) down to the center of the base. It's a crucial part of our volume formula. In this problem, the height is given as 12 cm.
Always remember, the volume of a pyramid is given by: \[ V = \frac{1}{3} A h \], where \( V \) is volume, \( A \) is the area of the base, and \( h \) is the pyramid's height. Plugging in the given values, \[ 180 = \frac{1}{3} A \times 12 \], allowed us to find the base area \( A = 45 \text{ cm}^2 \).
Understanding how the height interacts with both the area of the base and the overall volume is vital to solving these kinds of geometric problems. This interaction shows the relationship between the dimensions in different shapes and how they contribute to the three-dimensional structure.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The volume of a cylinder is \(628 \mathrm{cm}^{3} .\) Find the radius of the base if the cylinder has a height of \(8 \mathrm{cm} .\) Round your answer to the nearest \(0.1 \mathrm{cm}\)

If you roll an 8.5 -by- 11 -inch piece of paper into a cylinder by bringing the two longer sides together, you get a tall, thin cylinder. If you roll an 8.5 -by- 11 -inch piece of paper into a cylinder by bringing the two shorter sides together, you get a short, fat cylinder. Which of the two cylinders has the greater volume?

Refer to the table. $$\begin{array}{|l|l|l|l|} \hline \text { Metal } & \text { Density } & \text { Metal } & \text { Density } \\\\\hline \text { Aluminum } & 2.81 \mathrm{g} / \mathrm{cm}^{3} & \text { Nickel } & 8.89 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Copper } & 8.97 \mathrm{g} / \mathrm{cm}^{3} & \text { Platinum } & 21.40 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Gold } & 19.30 \mathrm{g} / \mathrm{cm}^{3} & \text { Potassium } & 0.86 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Lead } & 11.30 \mathrm{g} / \mathrm{cm}^{3} & \text { Silver } & 10.50 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline \text { Lithium } & 0.54 \mathrm{g} / \mathrm{cm}^{3} & \text { Sodium } & 0.97 \mathrm{g} / \mathrm{cm}^{3} \\\\\hline\end{array}$$ When ice floats in water, one-eighth of its volume floats above the water level and seven-eighths floats beneath the water level. A block of ice placed into an ice chest causes the water in the chest to rise \(4 \mathrm{cm} .\) The right rectangular chest measures \(35 \mathrm{cm}\) by \(50 \mathrm{cm}\) by \(30 \mathrm{cm}\) high. What is the volume of the block of ice?

A sealed rectangular container \(6 \mathrm{cm}\) by \(12 \mathrm{cm}\) by \(15 \mathrm{cm}\) is sitting on its smallest face. It is filled with water up to \(5 \mathrm{cm}\) from the top. How many centimeters from the bottom will the water level reach if the container is placed on its largest face?

What is the volume of the largest hemisphere that you could carve out of a wooden block whose edges measure \(3 \mathrm{m}\) by \(7 \mathrm{m}\) by \(7 \mathrm{m} ?\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.