### 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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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.