### Problem 1: Birthday Cake Design A baker is creating a birthday cake. Let the region **R** be bounded by \( y = 20\sin\left(\frac{\pi x}{30}\right) \) and \( y = 0 \), as pictured in the graph below. Both \( x \) and \( y \) are measured in centimeters. #### Graph Description: The graph depicts a sine wave starting at the origin (0,0) and stretching to 30 centimeters along the x-axis. The wave's peak reaches up to 20 centimeters along the y-axis. This forms a single arch, as illustrated in region **R**. --- #### Tasks: **(a)** To plan the cake, the baker cuts region **R** out of a 30-centimeter by 20-centimeter rectangular piece of cardboard and then discards the remaining cardboard. Find the area of the discarded cardboard. **(b)** The baker is considering revolving region **R** about the line \( x = 40 \) to create a “bundt” style cake. Find the volume of his cake. **(c)** As a centerpiece of his “bundt” style cake in part (b), the baker would like to fill the “chimney” of the cake (the hole in the center) with meringue and light it on fire! Assuming the meringue is level with the top of the cake, find the volume of meringue necessary to construct the baker's centerpiece. --- ### Solution Approach: **Part (a):** - Calculate the area of region **R** using the integral: \[ \text{Area of R} = \int_0^{30} 20\sin\left(\frac{\pi x}{30}\right) \, dx \] - Determine the area of the rectangular cardboard: \[ \text{Area of rectangle} = 30 \times 20 = 600 \, \text{cm}^2 \] - Find the area of the discarded cardboard by subtracting the area of **R** from the area of the rectangle. **Part (b):** - To find the volume of the bundt cake, revolve region **R** around \( x = 40 \). Use the disk method to integrate: \[ \text{Volume} = \int_0^{30} \pi \left[
### Problem 1: Birthday Cake Design A baker is creating a birthday cake. Let the region **R** be bounded by \( y = 20\sin\left(\frac{\pi x}{30}\right) \) and \( y = 0 \), as pictured in the graph below. Both \( x \) and \( y \) are measured in centimeters. #### Graph Description: The graph depicts a sine wave starting at the origin (0,0) and stretching to 30 centimeters along the x-axis. The wave's peak reaches up to 20 centimeters along the y-axis. This forms a single arch, as illustrated in region **R**. --- #### Tasks: **(a)** To plan the cake, the baker cuts region **R** out of a 30-centimeter by 20-centimeter rectangular piece of cardboard and then discards the remaining cardboard. Find the area of the discarded cardboard. **(b)** The baker is considering revolving region **R** about the line \( x = 40 \) to create a “bundt” style cake. Find the volume of his cake. **(c)** As a centerpiece of his “bundt” style cake in part (b), the baker would like to fill the “chimney” of the cake (the hole in the center) with meringue and light it on fire! Assuming the meringue is level with the top of the cake, find the volume of meringue necessary to construct the baker's centerpiece. --- ### Solution Approach: **Part (a):** - Calculate the area of region **R** using the integral: \[ \text{Area of R} = \int_0^{30} 20\sin\left(\frac{\pi x}{30}\right) \, dx \] - Determine the area of the rectangular cardboard: \[ \text{Area of rectangle} = 30 \times 20 = 600 \, \text{cm}^2 \] - Find the area of the discarded cardboard by subtracting the area of **R** from the area of the rectangle. **Part (b):** - To find the volume of the bundt cake, revolve region **R** around \( x = 40 \). Use the disk method to integrate: \[ \text{Volume} = \int_0^{30} \pi \left[
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 44E
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Question
subpart c
![### Problem 1: Birthday Cake Design
A baker is creating a birthday cake. Let the region **R** be bounded by \( y = 20\sin\left(\frac{\pi x}{30}\right) \) and \( y = 0 \), as pictured in the graph below. Both \( x \) and \( y \) are measured in centimeters.
#### Graph Description:
The graph depicts a sine wave starting at the origin (0,0) and stretching to 30 centimeters along the x-axis. The wave's peak reaches up to 20 centimeters along the y-axis. This forms a single arch, as illustrated in region **R**.
---
#### Tasks:
**(a)** To plan the cake, the baker cuts region **R** out of a 30-centimeter by 20-centimeter rectangular piece of cardboard and then discards the remaining cardboard. Find the area of the discarded cardboard.
**(b)** The baker is considering revolving region **R** about the line \( x = 40 \) to create a “bundt” style cake. Find the volume of his cake.
**(c)** As a centerpiece of his “bundt” style cake in part (b), the baker would like to fill the “chimney” of the cake (the hole in the center) with meringue and light it on fire! Assuming the meringue is level with the top of the cake, find the volume of meringue necessary to construct the baker's centerpiece.
---
### Solution Approach:
**Part (a):**
- Calculate the area of region **R** using the integral:
\[
\text{Area of R} = \int_0^{30} 20\sin\left(\frac{\pi x}{30}\right) \, dx
\]
- Determine the area of the rectangular cardboard:
\[
\text{Area of rectangle} = 30 \times 20 = 600 \, \text{cm}^2
\]
- Find the area of the discarded cardboard by subtracting the area of **R** from the area of the rectangle.
**Part (b):**
- To find the volume of the bundt cake, revolve region **R** around \( x = 40 \). Use the disk method to integrate:
\[
\text{Volume} = \int_0^{30} \pi \left[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F715a84de-dd1a-49bc-bb04-088748385741%2F6f8a9be7-02c2-4584-b064-53a09ada4fb7%2F95rhztm_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 1: Birthday Cake Design
A baker is creating a birthday cake. Let the region **R** be bounded by \( y = 20\sin\left(\frac{\pi x}{30}\right) \) and \( y = 0 \), as pictured in the graph below. Both \( x \) and \( y \) are measured in centimeters.
#### Graph Description:
The graph depicts a sine wave starting at the origin (0,0) and stretching to 30 centimeters along the x-axis. The wave's peak reaches up to 20 centimeters along the y-axis. This forms a single arch, as illustrated in region **R**.
---
#### Tasks:
**(a)** To plan the cake, the baker cuts region **R** out of a 30-centimeter by 20-centimeter rectangular piece of cardboard and then discards the remaining cardboard. Find the area of the discarded cardboard.
**(b)** The baker is considering revolving region **R** about the line \( x = 40 \) to create a “bundt” style cake. Find the volume of his cake.
**(c)** As a centerpiece of his “bundt” style cake in part (b), the baker would like to fill the “chimney” of the cake (the hole in the center) with meringue and light it on fire! Assuming the meringue is level with the top of the cake, find the volume of meringue necessary to construct the baker's centerpiece.
---
### Solution Approach:
**Part (a):**
- Calculate the area of region **R** using the integral:
\[
\text{Area of R} = \int_0^{30} 20\sin\left(\frac{\pi x}{30}\right) \, dx
\]
- Determine the area of the rectangular cardboard:
\[
\text{Area of rectangle} = 30 \times 20 = 600 \, \text{cm}^2
\]
- Find the area of the discarded cardboard by subtracting the area of **R** from the area of the rectangle.
**Part (b):**
- To find the volume of the bundt cake, revolve region **R** around \( x = 40 \). Use the disk method to integrate:
\[
\text{Volume} = \int_0^{30} \pi \left[
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