A paper cup is to be designed in the shape of a right circular cone. It must have a capacity of 12 fluid ounces (1 fluid ounce = 1.80469 cubic inches) of soft drink but it must use a minimum amount of material in its construction. What should the dimensions of this paper cup be and how much material is needed for its construction?
a) One will need to compute derivatives and find and test critical values to obtain the answer.
a) is correct. Did you do that?
v = 1/3 pi r^2 h
12*1.80469 = pi/3 r^2 h
h = 64.9688/(pi*r^2)
surface area of cone is pi r √(r^2+h^2)
a = pi *r √(r^2+(64.9688/(pi*r^2))^2)
= pi √(r^6 + 427.671)/r
Now find da/dr
da/dr = (6.283r^6 - 1343.57)/[r^2 √(r^6 + 427.671)]
The denominator is never zero, so we just need to have
6.283r^6 = 1343.57
r = 2.445 in
h = 3.459 in
Better check my math. That's a strange-shaped paper cup!
To design the paper cup with a minimum amount of material, we need to find the dimensions of the cone that will have a capacity of 12 fluid ounces.
Let's break down the problem into steps:
Step 1: Find the formula for the volume of a cone
The volume of a cone is given by the formula V = (1/3)πr^2h, where V is the volume, r is the radius of the base, and h is the height.
Step 2: Convert the capacity to cubic inches
Since 1 fluid ounce is equal to 1.80469 cubic inches, the capacity of 12 fluid ounces is equal to 12 * 1.80469 = 21.65628 cubic inches.
Step 3: Set up the equation for the volume of the cone
We know that the volume of the cone should be equal to the desired capacity, so we have:
(1/3)πr^2h = 21.65628
Step 4: Minimize the amount of material
To minimize the amount of material, we need to minimize the surface area of the cone.
The surface area of a cone is given by the formula A = πr(r + √(r^2 + h^2)), where A is the surface area.
Step 5: Use calculus to find the critical values
Differentiate the surface area formula, set it equal to zero, and solve for r and h to find the critical values.
Step 6: Test the critical values
Evaluate the surface area using the critical values to determine which one gives the minimum surface area.
By following these steps, we can find the dimensions of the cone and determine the amount of material needed for its construction.
To design the paper cup in the shape of a right circular cone, we need to find the dimensions that will maximize its capacity while minimizing the amount of material used.
Let's start by defining the variables:
- r = radius of the base of the cone
- h = height of the cone
To find the dimensions, we need to set up an objective function and a constraint.
Objective function: We want to maximize the capacity of the cup. The capacity of a cone can be calculated using the formula: V = (1/3) * π * r^2 * h
Constraint: The capacity of the cup should be 12 fluid ounces. We need to convert this into cubic inches as given: 12 fluid ounces = 12 * 1.80469 cubic inches = 21.65628 cubic inches.
Now, let's set up the equations and solve for the dimensions.
Objective function: V = (1/3) * π * r^2 * h
Capacity constraint: V = 21.65628 cubic inches
Substituting the value of V in the objective function, we get:
21.65628 = (1/3) * π * r^2 * h
To minimize the amount of material, we can minimize the surface area of the cone. The surface area of a cone can be calculated using the formula: A = π * r * l, where l is the slant height of the cone.
To find the slant height, we can use the Pythagorean theorem: l^2 = r^2 + h^2
Since r and h are positive, we can solve for l as l = √(r^2 + h^2)
Now we have two equations:
1. 21.65628 = (1/3) * π * r^2 * h
2. A = π * r * √(r^2 + h^2)
To find the dimensions, we can solve this system of equations using calculus.
Step 1: Take the derivative of A with respect to r to find the critical points.
Step 2: Set the derivative equal to zero and solve for r.
Step 3: Take the second derivative of A to determine if the critical point found in step 2 is a maximum or minimum.
Step 4: Substitute the value of r obtained in step 2 back into the equation of A to find the corresponding value of h.
Step 5: Calculate the amount of material used by finding the surface area using the values of r and h.
Once you have the dimensions (r and h), you can calculate the amount of material needed by finding the surface area and multiplying it by the thickness of the cup material.
Note: The above explanation describes the general process, but it may require additional mathematical calculations to find the exact dimensions and the amount of material needed for the paper cup.