(b) What is the height of the arch at its center? (Round your answer to two decimal places.) 190.53 (c) At what points is the height 160 m? (Round your answers to two decimal places.) (x, y) = ( -47.02 ) (smaller x-value) (x, y) = (| 47.02 | (larger x-value) (d) What is the slope of the arch at the points in part (c)? (Round your answers to one decimal place.) (at the point with smaller x-value) ] (at the point with larger x-value)
(b) What is the height of the arch at its center? (Round your answer to two decimal places.) 190.53 (c) At what points is the height 160 m? (Round your answers to two decimal places.) (x, y) = ( -47.02 ) (smaller x-value) (x, y) = (| 47.02 | (larger x-value) (d) What is the slope of the arch at the points in part (c)? (Round your answers to one decimal place.) (at the point with smaller x-value) ] (at the point with larger x-value)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Arch Calculations
**(b) Height of the Arch at its Center**
*Question:* What is the height of the arch at its center? (Round your answer to two decimal places.)
*Answer:*
\[ \text{Height} = 190.53 \text{ m} \]
---
**(c) Points at Height 160 m**
*Question:* At what points is the height 160 m? (Round your answers to two decimal places.)
*Answer:*
\[ (x, y) = (-47.02, 160) \quad \text{(smaller x-value)} \quad \text{✘} \]
\[ (x, y) = (47.02, 160) \quad \text{(larger x-value)} \quad \text{✘} \]
---
**(d) Slope of the Arch at Specific Points**
*Question:* What is the slope of the arch at the points in part (c)? (Round your answers to one decimal place.)
\[
\text{Slope at the point with smaller x-value:} \quad \_\_\_\_\_\_
\]
\[
\text{Slope at the point with larger x-value:} \quad \_\_\_\_\_\_
\]
*Note:* Answers not provided.
---
**Need Help?**
[Read It]
---
### Explanation of Graphs/Diagrams:
In this section, there are two main parts to address: the height of the arch at its center, and the points at which the arch reaches a height of 160 meters. Although specific graphical elements are not displayed, the calculations likely relate to the geometry or algebraic representation of the arch. Users need to calculate specific values and the slope, implying a parabolic shape or another mathematical structure for the arch.
Additional support is available through a "Need Help?" prompt providing further resources.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabf3f71a-8f38-4387-af5c-ba72dda07b09%2F9d036c2a-6fbe-4861-a0a7-e9f8daefae70%2F16jog8i_processed.png&w=3840&q=75)
Transcribed Image Text:### Arch Calculations
**(b) Height of the Arch at its Center**
*Question:* What is the height of the arch at its center? (Round your answer to two decimal places.)
*Answer:*
\[ \text{Height} = 190.53 \text{ m} \]
---
**(c) Points at Height 160 m**
*Question:* At what points is the height 160 m? (Round your answers to two decimal places.)
*Answer:*
\[ (x, y) = (-47.02, 160) \quad \text{(smaller x-value)} \quad \text{✘} \]
\[ (x, y) = (47.02, 160) \quad \text{(larger x-value)} \quad \text{✘} \]
---
**(d) Slope of the Arch at Specific Points**
*Question:* What is the slope of the arch at the points in part (c)? (Round your answers to one decimal place.)
\[
\text{Slope at the point with smaller x-value:} \quad \_\_\_\_\_\_
\]
\[
\text{Slope at the point with larger x-value:} \quad \_\_\_\_\_\_
\]
*Note:* Answers not provided.
---
**Need Help?**
[Read It]
---
### Explanation of Graphs/Diagrams:
In this section, there are two main parts to address: the height of the arch at its center, and the points at which the arch reaches a height of 160 meters. Although specific graphical elements are not displayed, the calculations likely relate to the geometry or algebraic representation of the arch. Users need to calculate specific values and the slope, implying a parabolic shape or another mathematical structure for the arch.
Additional support is available through a "Need Help?" prompt providing further resources.
---
![### The Gateway Arch in St. Louis: Mathematical Representation
The Gateway Arch in St. Louis was designed by Eero Saarinen and constructed using the following equation to represent its central curve:
\[ y = 211.49 - 20.96 \cosh(0.03291765x) \]
Here, \(x\) and \(y\) are in meters and \(|x| \leq 91.20\).
#### (a) Graph the Central Curve
The following four graphs illustrate the central curve of the Gateway Arch, plotted according to the given equation. The graphs appear identical, each showing the characteristic symmetrical arch.
- The horizontal axis (x-axis) ranges from -100 to 100 meters.
- The vertical axis (y-axis) ranges from 0 to 200 meters.
The maximum height of the arch (approximately 192 meters) is observed at \(x = 0\), while the ends of the arch reach the ground where \(|x| \approx 91.20\).
Each plot of the central curve clearly demonstrates the iconic parabolic shape of the Gateway Arch:
1. **First Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Shows a symmetrical arch shape centered at the origin.
2. **Second Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Displays the same arch shape as the first graph.
3. **Third Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Identical to the first graph.
4. **Fourth Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Also matches the shape of the first graph.
These visual representations provide a clear understanding of the mathematical structure behind the graceful curve of the Gateway Arch.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabf3f71a-8f38-4387-af5c-ba72dda07b09%2F9d036c2a-6fbe-4861-a0a7-e9f8daefae70%2Fa92rje_processed.png&w=3840&q=75)
Transcribed Image Text:### The Gateway Arch in St. Louis: Mathematical Representation
The Gateway Arch in St. Louis was designed by Eero Saarinen and constructed using the following equation to represent its central curve:
\[ y = 211.49 - 20.96 \cosh(0.03291765x) \]
Here, \(x\) and \(y\) are in meters and \(|x| \leq 91.20\).
#### (a) Graph the Central Curve
The following four graphs illustrate the central curve of the Gateway Arch, plotted according to the given equation. The graphs appear identical, each showing the characteristic symmetrical arch.
- The horizontal axis (x-axis) ranges from -100 to 100 meters.
- The vertical axis (y-axis) ranges from 0 to 200 meters.
The maximum height of the arch (approximately 192 meters) is observed at \(x = 0\), while the ends of the arch reach the ground where \(|x| \approx 91.20\).
Each plot of the central curve clearly demonstrates the iconic parabolic shape of the Gateway Arch:
1. **First Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Shows a symmetrical arch shape centered at the origin.
2. **Second Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Displays the same arch shape as the first graph.
3. **Third Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Identical to the first graph.
4. **Fourth Graph:**
- Axes: x from -100 to 100, y from 0 to 200.
- Also matches the shape of the first graph.
These visual representations provide a clear understanding of the mathematical structure behind the graceful curve of the Gateway Arch.
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