At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Sure, let's break down the given problem step-by-step.
### Given Information:
1. The length of the rectangular prism (box) is given as 24 inches.
2. The height of the box is represented as [tex]\( x \)[/tex] inches.
3. The width of the box is 7 inches less than the height, which makes the width equal to [tex]\( x - 7 \)[/tex] inches.
### Volume of a Rectangular Prism:
The volume [tex]\( V \)[/tex] of a rectangular prism is given by the formula:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Plugging in the given values:
[tex]\[ V = 24 \times (x - 7) \times x \][/tex]
### Simplifying the Volume Expression:
First, distribute [tex]\( x \)[/tex] across the terms inside the parenthesis:
[tex]\[ V = 24 \times (x^2 - 7x) \][/tex]
Then, distribute 24 inside the parenthesis:
[tex]\[ V = 24x^2 - 168x \][/tex]
So, the equation that models the volume of the box in terms of its height [tex]\( x \)[/tex] is:
[tex]\[ V = 24x^2 - 168x \][/tex]
### Can the Height of the Box be 15 inches?
To determine whether the height of the box can be 15 inches, we need to plug [tex]\( x = 15 \)[/tex] into the simplified volume expression and verify a few things:
1. Calculate the width when [tex]\( x = 15 \)[/tex]:
[tex]\[ \text{Width} = 15 - 7 = 8 \text{ inches} \][/tex]
2. Calculate the volume when [tex]\( x = 15 \)[/tex]:
[tex]\[ V = 24 \times 15^2 - 168 \times 15 \][/tex]
3. Ensure that the volume is non-negative (since volume cannot be negative):
Let's simplify this step further:
[tex]\[ 24 \times 15^2 - 168 \times 15 = 24 \times 225 - 2520 \][/tex]
[tex]\[ = 5400 - 2520 = 2880 \][/tex]
Since [tex]\( 2880 \)[/tex] is a positive value, the calculated volume is non-negative. This verifies that having a height of 15 inches results in a valid volume.
Thus, it is viable for the height of the box to be 15 inches.
### Summary:
1. Volume Formula Modeling the Box in terms of Height [tex]\( x \)[/tex]: [tex]\( 24x^2 - 168x \)[/tex]
2. Width when Height is 15 Inches: 8 inches
3. Viable for Height to be 15 Inches: Yes
Therefore:
- The equation modeling the volume of the box in terms of its height [tex]\( x \)[/tex] is [tex]\( V = 24x^2 - 168x \)[/tex].
- The width is 8 inches when the height is 15 inches.
- The height can be 15 inches since it results in a non-negative volume value.
### Given Information:
1. The length of the rectangular prism (box) is given as 24 inches.
2. The height of the box is represented as [tex]\( x \)[/tex] inches.
3. The width of the box is 7 inches less than the height, which makes the width equal to [tex]\( x - 7 \)[/tex] inches.
### Volume of a Rectangular Prism:
The volume [tex]\( V \)[/tex] of a rectangular prism is given by the formula:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Plugging in the given values:
[tex]\[ V = 24 \times (x - 7) \times x \][/tex]
### Simplifying the Volume Expression:
First, distribute [tex]\( x \)[/tex] across the terms inside the parenthesis:
[tex]\[ V = 24 \times (x^2 - 7x) \][/tex]
Then, distribute 24 inside the parenthesis:
[tex]\[ V = 24x^2 - 168x \][/tex]
So, the equation that models the volume of the box in terms of its height [tex]\( x \)[/tex] is:
[tex]\[ V = 24x^2 - 168x \][/tex]
### Can the Height of the Box be 15 inches?
To determine whether the height of the box can be 15 inches, we need to plug [tex]\( x = 15 \)[/tex] into the simplified volume expression and verify a few things:
1. Calculate the width when [tex]\( x = 15 \)[/tex]:
[tex]\[ \text{Width} = 15 - 7 = 8 \text{ inches} \][/tex]
2. Calculate the volume when [tex]\( x = 15 \)[/tex]:
[tex]\[ V = 24 \times 15^2 - 168 \times 15 \][/tex]
3. Ensure that the volume is non-negative (since volume cannot be negative):
Let's simplify this step further:
[tex]\[ 24 \times 15^2 - 168 \times 15 = 24 \times 225 - 2520 \][/tex]
[tex]\[ = 5400 - 2520 = 2880 \][/tex]
Since [tex]\( 2880 \)[/tex] is a positive value, the calculated volume is non-negative. This verifies that having a height of 15 inches results in a valid volume.
Thus, it is viable for the height of the box to be 15 inches.
### Summary:
1. Volume Formula Modeling the Box in terms of Height [tex]\( x \)[/tex]: [tex]\( 24x^2 - 168x \)[/tex]
2. Width when Height is 15 Inches: 8 inches
3. Viable for Height to be 15 Inches: Yes
Therefore:
- The equation modeling the volume of the box in terms of its height [tex]\( x \)[/tex] is [tex]\( V = 24x^2 - 168x \)[/tex].
- The width is 8 inches when the height is 15 inches.
- The height can be 15 inches since it results in a non-negative volume value.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
