. Armin is designing a bottle for a new
shampoo. The bottle should have a volume
of 450 cm3. Armin wants to minimize the
amount of plastic used in the bottle.
Determine the dimensions of the bottle
with the minimum surface area in each
case. Explain your method.
a) The bottle is a cylinder.
b) The bottle is a triangular prism with a
base that is an equilateral triangle. pls. help me am struggling to get it right. Thanks for you concern.
(a)
πr^2h = 450
a = 2πr^2+2πrh
= 2πr^2+2πr(450/πr^2)
= 2πr^2 + 900/r
Now just find r when da/dr=0
(b) is similar...
√3/4 s^2h = 450
a = 2(√3/4 s^2) + 3sh
= √3/2 s^2 + 3s(1800/√3 s^2)
= √3/2 s^2 + 1800√3/s
...
To determine the dimensions of the bottle with the minimum surface area in each case, we need to use calculus and optimization techniques. Let's start with each specific case:
a) The bottle is a cylinder:
In this case, we want to find the dimensions (radius and height) of the cylindrical bottle that minimize the surface area while maintaining a volume of 450 cm^3.
Step 1: Define the variables:
Let's denote the radius of the cylinder as "r" and the height as "h". We know that the volume (V) is 450 cm^3, and we need to minimize the surface area (A) of the cylinder.
Step 2: Set up the equations:
The formulas for volume (V) and surface area (A) of a cylinder are:
V = πr²h
A = 2πr² + 2πrh
Step 3: Substitute the given volume and simplify:
We can substitute the given volume V = 450 cm^3 into the volume equation:
450 = πr²h
Step 4: Solve the volume equation for h:
h = 450 / (πr²)
Step 5: Substitute the expression for h into the surface area equation:
A = 2πr² + 2πr(450 / (πr²))
Simplify:
A = 2πr² + (900 / r)
Step 6: Find the derivative of A with respect to r:
dA/dr = 4πr - (900 / r²)
Step 7: Set the derivative equal to zero and solve for r:
4πr - (900 / r²) = 0
4πr³ - 900 = 0
r³ = (900 / 4π)
r = ∛(225 / π)
Step 8: Substitute the value of r back into the equation for h:
h = 450 / (π(∛(225 / π))²)
The resulting values of r and h are the dimensions of the cylindrical bottle with the minimum surface area and a volume of 450 cm³.
b) The bottle is a triangular prism with a base that is an equilateral triangle:
In this case, we need to find the dimensions (base side length and height) of the triangular prism that minimize the surface area while maintaining a volume of 450 cm^3.
The steps to solve this case using calculus and optimization techniques are similar to those in case a) above. However, the formulas for volume and surface area will be different since the bottle is now a triangular prism. Here are the steps to follow:
Step 1: Define the variables:
Let's denote the base side length of the triangle as "s" and the height of the triangle as "h". We know that the volume (V) is 450 cm³, and we need to minimize the surface area (A) of the triangular prism.
Step 2: Set up the equations:
The formulas for volume (V) and surface area (A) of a triangular prism are:
V = (1/4) * sqrt(3) * s² * h
A = s² + 3s * h
Step 3: Substitute the given volume and simplify:
Substitute V = 450 cm³:
450 = (1/4) * sqrt(3) * s² * h
Step 4: Solve the volume equation for h:
h = 1800 / (sqrt(3) * s²)
Step 5: Substitute the expression for h into the surface area equation:
A = s² + 3s * (1800 / (sqrt(3) * s²))
Simplify:
A = s² + 5400 / (sqrt(3) * s)
Step 6: Find the derivative of A with respect to s:
dA/ds = 2s - (5400 / (sqrt(3) * s²))
Step 7: Set the derivative equal to zero and solve for s:
2s - (5400 / (sqrt(3) * s²)) = 0
2s³ - (5400 * sqrt(3)) = 0
s³ = (5400 * sqrt(3)) / 2
s = ∛[(5400 * sqrt(3)) / 2]
Step 8: Substitute the value of s back into the equation for h:
h = 1800 / (sqrt(3) * (∛[(5400 * sqrt(3)) / 2])²)
The resulting values of s and h are the dimensions of the triangular prism bottle with the minimum surface area and a volume of 450 cm³.
By following these steps, you can find the dimensions of the bottle in each case that minimize the surface area while maintaining the given volume.