the volume of an cylindrical can with a radius r cm and height h cm is 128000 c^3, show that the surface area of the can is A=2(22/7)r^2 + 246000/r. Find the value for r to minimize the surface area.
*i know what the quest. ask but i do not know how to apply it.
It looks like you are being asked to use 22/7 for pi. In that case,
2(22/7)r^2 is the combined area of the two circular ends.
If the volume is V = pi*r^2*h, the curved cylindrical area is
2 pi*r*h = 2V/r.
That is where your 246000/r term comes from.
To minimize the surface area, set the derivative of A(r) equal to zero.
dA/dr = (88/7) r -246,000/r^2 = 0
r = (246,000*7/88)^1/3 = 26.9 cm
If you have not yet studied differential calculus, I suggest you graph A vs r and see where it is a mininum.
To find the surface area of the cylindrical can, we first need to determine its height and radius using the volume formula.
Given that the volume (V) of the can is 128000 cm³ and the formula for the volume of a cylinder is V = πr²h, we can substitute the given value for V and solve for h in terms of r:
128000 = πr²h
Divide both sides of the equation by πr²:
h = 128000 / (πr²)
Now that we have the expression for the height in terms of the radius, we can proceed to find the surface area (A).
The formula for the surface area of a cylinder is:
A = 2πrh + 2πr²
Substituting the expression for h we derived earlier:
A = 2πr(128000 / (πr²)) + 2πr²
A = 256000 / r + 2πr²
Simplifying the equation, we can convert 2πr² to the equivalent form: (22/7)r²
A = 256000 / r + (22/7)r²
Now, to find the value of r that minimizes the surface area, we can take the derivative of A with respect to r, set it equal to 0, and solve for r.
dA/dr = -256000/r² + (44/7)r
Setting dA/dr = 0:
0 = -256000/r² + (44/7)r
Multiply both sides by r²:
0 = -256000 + (44/7)r³
Rearrange the equation:
256000 = (44/7)r³
Multiply both sides by 7/44:
r³ = 256000(7/44)
Simplify:
r³ = 4000
Take the cubic root of both sides:
r = ∛(4000)
r ≈ 15.92 cm
Therefore, to minimize the surface area, the value of r should be approximately 15.92 cm.
To find the surface area of the cylindrical can, we can use the volume and the formula for the surface area of a cylinder.
Step 1: Find the formula for the volume of a cylinder
The formula for the volume of a cylinder is:
V = πr^2h
Given that the volume of the cylindrical can is 128000 cm^3, we can set up the equation as:
128000 = πr^2h
Step 2: Solve for h in terms of r
To find the surface area, we need to express the height (h) in terms of r. Rearranging the equation from step 1, we get:
h = 128000 / (πr^2)
Step 3: Substitute the value of h in the formula for the surface area
The formula for the surface area of a cylinder is:
A = 2πr^2 + 2πrh
Substituting the value of h from step 2, we have:
A = 2πr^2 + 2πr(128000 / (πr^2))
= 2πr^2 + 2(128000 / r)
= 2πr^2 + 256000 / r
Simplifying further:
A = 2(22/7)r^2 + 246000 / r
This gives us the formula for the surface area of the cylindrical can.
To find the value for r that minimizes the surface area, we can take the derivative of the surface area formula with respect to r, set it equal to zero, and solve for r.
dA/dr = 4(22/7)r - 246000 / r^2
Setting dA/dr equal to zero:
4(22/7)r - 246000 / r^2 = 0
Multiply through by r^2 to remove the denominator:
4(22/7)r^3 - 246000 = 0
This is a cubic equation in terms of r. You can solve it by factoring, using the rational root theorem, or by using numerical methods like a graphing calculator or computer software.
Once you find the value(s) for r from solving the cubic equation, you can substitute it back into the surface area formula to get the corresponding minimum surface area.