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A cardboard box has a square base, with each edge of the base having length x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in. (a) Express the volume V of the box as a function of x. V(x) = (b) What is the domain of V? (Use the fact that length and volume must be positive. Enter your answer using interval notation.)

A cardboard box has a square base, with each edge of the base having length x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in. (a) Express the volume V of the box as a function of x. V(x) = (b) What is the domain of V? (Use the fact that length and volume must be positive. Enter your answer using interval notation.)

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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The image consists of four graphs, each depicting a curve on a coordinate plane with axes labeled "V" (volume) on the vertical axis and "x" on the horizontal axis. Here’s a detailed description of each graph: 1. **Top Left Graph:** - The curve starts at (0,0) and initially rises steeply. - The peak of the curve, indicating the maximum volume, occurs around x = 10, with the peak value of V slightly over 1000. - The curve then declines sharply as x approaches 20. 2. **Top Right Graph:** - The curve starts near (0,0) and increases gradually. - It reaches a maximum volume at approximately x = 7, with a peak value of V around 500. - After peaking, the curve decreases continuously and levels off as it nears the end at x = 20. 3. **Bottom Left Graph:** - The curve begins near (0,0) and shows a gentle rise. - It achieves a peak volume at about x = 10, with a maximum V slightly over 500. - The volume decreases steadily, leveling out towards x = 20. 4. **Bottom Right Graph:** - The curve starts at (0,0) and rises sharply. - It reaches its maximum near x = 15, with the peak volume approaching 1500. - After reaching the peak, the curve declines sharply towards x = 20. **Task:** - Use the appropriate graph to estimate the maximum volume for a box. Round your answer to the nearest whole number and enter it in the provided box labeled with "in³" (cubic inches). This educational exercise involves analyzing the graphs to determine the point at which the maximum volume occurs for each scenario.
Transcribed Image Text:The image consists of four graphs, each depicting a curve on a coordinate plane with axes labeled "V" (volume) on the vertical axis and "x" on the horizontal axis. Here’s a detailed description of each graph: 1. **Top Left Graph:** - The curve starts at (0,0) and initially rises steeply. - The peak of the curve, indicating the maximum volume, occurs around x = 10, with the peak value of V slightly over 1000. - The curve then declines sharply as x approaches 20. 2. **Top Right Graph:** - The curve starts near (0,0) and increases gradually. - It reaches a maximum volume at approximately x = 7, with a peak value of V around 500. - After peaking, the curve decreases continuously and levels off as it nears the end at x = 20. 3. **Bottom Left Graph:** - The curve begins near (0,0) and shows a gentle rise. - It achieves a peak volume at about x = 10, with a maximum V slightly over 500. - The volume decreases steadily, leveling out towards x = 20. 4. **Bottom Right Graph:** - The curve starts at (0,0) and rises sharply. - It reaches its maximum near x = 15, with the peak volume approaching 1500. - After reaching the peak, the curve declines sharply towards x = 20. **Task:** - Use the appropriate graph to estimate the maximum volume for a box. Round your answer to the nearest whole number and enter it in the provided box labeled with "in³" (cubic inches). This educational exercise involves analyzing the graphs to determine the point at which the maximum volume occurs for each scenario.
A cardboard box has a square base, with each edge of the base having length \(x\) inches, as shown in the figure. The total length of all 12 edges of the box is 144 inches. (a) Express the volume \(V\) of the box as a function of \(x\). \(V(x) =\) [ ] (b) What is the domain of \(V\)? (Use the fact that length and volume must be positive. Enter your answer using interval notation.) [ ] **Diagram Explanation:** The image shows a three-dimensional rectangular prism (a box) with a pink hue. The base of the box is square with sides labeled as \(x\). The height of the box is not explicitly labeled. To solve for the volume, we observe that the total length of all 12 edges is given as 144 inches. Each base has 4 edges of length \(x\), and there are 4 vertical edges. Therefore, the equation for the total edge length is: \[4x + 4x + 4h = 144\] Here, \(h\) is the height. Solving for \(h\) will enable the expression of the volume \(V = x^2h\) as a function of \(x\).
Transcribed Image Text:A cardboard box has a square base, with each edge of the base having length \(x\) inches, as shown in the figure. The total length of all 12 edges of the box is 144 inches. (a) Express the volume \(V\) of the box as a function of \(x\). \(V(x) =\) [ ] (b) What is the domain of \(V\)? (Use the fact that length and volume must be positive. Enter your answer using interval notation.) [ ] **Diagram Explanation:** The image shows a three-dimensional rectangular prism (a box) with a pink hue. The base of the box is square with sides labeled as \(x\). The height of the box is not explicitly labeled. To solve for the volume, we observe that the total length of all 12 edges is given as 144 inches. Each base has 4 edges of length \(x\), and there are 4 vertical edges. Therefore, the equation for the total edge length is: \[4x + 4x + 4h = 144\] Here, \(h\) is the height. Solving for \(h\) will enable the expression of the volume \(V = x^2h\) as a function of \(x\).
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