An energy drink container in the shape of a right circular cylinder must have a volume of 12 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches). The cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side. Find the dimensions that will minimize the cost. (Round your answers to two decimal places.)
v=pi*r^2*h=12(1.80469)=21.66
I got that r=1.20 but cant find h
Well, it seems like you're in quite a squeeze with this energy drink container. Let's see if I can help you find the height.
So, we know that the volume of the container is 12 fluid ounces, which translates to 21.66 cubic inches. We can use the formula for the volume of a cylinder, which is v = π * r^2 * h, to solve for h.
Since you already found that the radius (r) is approximately 1.20, we can substitute that into the equation:
21.66 = π * (1.20)^2 * h
Simplifying this equation, we get:
21.66 = 1.44π * h
Dividing both sides by 1.44π, we find:
h ≈ 21.66 / (1.44π)
Using a calculator, we can approximate this value:
h ≈ 21.66 / (1.44 * 3.14)
h ≈ 21.66 / 4.5232
h ≈ 4.78
Therefore, the height of the container (h) is approximately 4.78 inches.
Now, to minimize the cost, we need to determine the dimensions that will give us the lowest cost. With the given information, the cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side.
Unfortunately, you didn't provide the actual cost values, so I can't give you a precise answer. However, to minimize the cost, you would need to find the surface area of each component (top, bottom, and lateral side), multiply it by their respective cost per square inch, and compare the total costs for different dimensions.
I hope this helps, even if it's lacking in the humor department. Good luck finding that minimal cost!
To find the dimensions that will minimize the cost, we need to express the cost equation in terms of a single variable. Let's assume the radius of the cylinder is "r" and the height is "h".
We know that the volume of the cylinder is given by the formula:
V = πr^2h = 12
Also, we are given that the cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side. Let's denote the cost of constructing the top and bottom as "C_top_bottom" and the cost of constructing the lateral side as "C_lateral".
We can then express the cost equation as:
Cost = 2πr^2C_top_bottom + 2πrhC_lateral
To eliminate one variable, we can solve the volume equation for h:
h = 12 / (πr^2)
Substituting this expression for h in the cost equation, we get:
Cost = 2πr^2C_top_bottom + 2πr(12 / (πr^2))C_lateral
Simplifying further, we have:
Cost = 2C_top_bottomπr^2 + 24C_lateral / r
Now, we can differentiate the cost equation with respect to r and set it equal to zero to find the minimum cost:
d(Cost) / dr = 4C_top_bottomπr - 24C_lateral / r^2 = 0
4C_top_bottomπr = 24C_lateral / r^2
r^3 = 6C_lateral / (C_top_bottomπ)
Solving for r, we have:
r = (6C_lateral / (C_top_bottomπ))^(1/3)
To find the value of h, we can substitute this value of r back into the volume equation:
12 = πr^2h
h = 12 / (πr^2)
Now, we have the values of r and h that will minimize the cost.
To find the dimensions that will minimize the cost, we need to express the cost in terms of one variable and then find the minimum value of that variable. Let's break down the problem step by step:
1. Find the expression for the cost: To calculate the cost, we need to consider the top, bottom, and lateral side of the energy drink container separately. Let's assume the cost per square inch of the lateral side is C, then the cost per square inch of the top and bottom would be 2C (twice the cost of the lateral side).
2. Express the cost in terms of one variable: We need to find a relationship between the cost and the dimensions of the container. Using the formula for the volume of a cylinder (v = πr²h), we can express the height (h) in terms of the radius (r) and the volume (v). Substituting h in terms of r and v, we have:
v = πr²h
12 = πr²h
h = 12 / (πr²)
3. Express the cost function in terms of one variable: Now, we can express the cost in terms of r only. The cost of the lateral side would be the surface area of the side (A_lateral) multiplied by the cost per square inch (C). The cost of the top and bottom (A_top_bottom) would be the sum of the areas of both multiplied by the cost per square inch (2C). Therefore, the cost function (C_total) would be:
C_total = C * A_lateral + 2C * A_top_bottom
Substituting the formulas for the surface areas of the side and top/bottom:
C_total = C * (2πrh) + 2C * (2πr²)
C_total = 2Cπrh + 4Cπr²
4. Find the derivative of the cost function: To find the minimum cost, we need to find the derivative of the cost function with respect to r (C_total' with respect to r).
C_total' = 2Cπh + 8Cπr
5. Set the derivative equal to zero and solve for r: To find the critical points (where the derivative is zero), we need to solve the equation 2Cπh + 8Cπr = 0. However, since we are interested in the dimensions that minimize the cost, we can ignore the derivative with respect to h. Therefore, we have 8Cπr = 0.
Solving for r, we get r = 0.
6. Calculate the height (h): With r = 0, the height (h) will be undefined. However, since a container with zero radius isn't meaningful, we can disregard this case.
Therefore, we cannot find a value for h since the radius r must be greater than zero for a meaningful solution. In this scenario, it seems there might be an error in the problem statement or the cost function. Please recheck the problem requirements or conditions.
r^2 h = 21.66/pi = 6.89
now you still have to do the problem
cost = cost of top and bottom + cost of side
x is cost of side metal per in^2
C = 2x pi r^2 + x (2 pi r h)
call c = C/2pix what we minimize
c = r^2 + r h
but we already know h = 6.89/r^2
c = r^2 + 6.89/r
luckily you know calculus so take dc/dr and set to 0
dc/dr = 0 = 2 r - 6.89/r^2
0 = 2 r^3 - 6.89
r^3 = 3.44