what should be the dimensions of the cylinder if you wanted it to have the least amout of surface area as possible?
the cylinders volume is 100ft cubed
I assume this is calculus class.
Volume= PI*radius^2*h
h= volume/(PI*radius^2)
Surface=PI*radius^2*2 + PI*radius*height
Surface= PI*radius^2 * 2 + PI*radius*volume/(PI*radius^2)
Take the derivative of surface with respect to radius, set it to zero, and solve for radius.
i am only in 7th grade! Could you explain that a little more to me
Sorry, this question is a typical college level calculus question. For a seventh grader, it is a little harder.
Try this. Graph Surface area vs radius according to the equation I gave you...
Surface= 2*PI*radius^2 + 100/radius
You will see a graph that has a minimum. If you have a graphing calculator, it is easy to graph.
Once you have the minimum, read the radius point on the curve. Then,
Height= 100/(PI*radius^2)
This problem is a bit advanced for a seventh grader.
i have an answer but i don't know if it is right! will you tell me if i am right or wrong.
i got that the dimensions 31.83098862 ft by 2 ft would have the least surface area for the problem.
am i right or wrong?
can you help me????
I don't get that. I get the diameter to be 5.84 feet (twice the cube root of 25).
You can figure the height . I get less than four feet.
check me.
how did you figure that out?
this is what i did!
100 divided by pie and i got 31.83098862 ft and then that multiplied by 2 is 100!
please help me
One note. I included the ends in the surface area. If they are not to be included, then it is a different answer.
what ends?
please help! explain it to me in a 7th grade way! i am taking pre algebra
To determine the dimensions of the cylinder that would result in the least amount of surface area, we can use calculus and optimization principles.
Let's denote the radius of the cylinder as r and its height as h. The surface area (A) of a cylinder is given by the formula:
A = 2πr^2 + 2πrh
We are given that the volume (V) of the cylinder is 100 ft^3. The formula for the volume of a cylinder is:
V = πr^2h
To find the dimensions for the least amount of surface area, we need to minimize the surface area while keeping the volume constant.
1. Eliminate one variable:
Using the volume formula, we can rewrite one of the variables in terms of the other. Let's solve for h:
h = V / (πr^2)
2. Substitute the expression for h into the surface area formula:
Now, replace h with V / (πr^2) in the surface area formula:
A = 2πr^2 + 2πr(V / (πr^2))
Simplifying this equation gives:
A = 2πr^2 + 2V / r
3. Differentiate the surface area equation:
To optimize the surface area, we need to differentiate the equation with respect to r. Taking the derivative yields:
dA/dr = 4πr - 2V / r^2
4. Set the derivative equal to zero:
To find the critical points, we need to set the derivative equal to zero and solve for r:
4πr - 2V / r^2 = 0
Simplifying further gives:
4πr^3 = 2V
r^3 = V / (2π)
5. Solve for r:
We can now find the value of r by taking the cube root of (V / (2π)).
r = (V / (2π))^(1/3)
6. Calculate the height (h):
Substituting the value of r into the expression for h:
h = V / (πr^2)
h = V / (π((V / (2π))^(1/3))^2)
h = 2V^(2/3) / (π^(1/3))
7. Plug in the volume:
Since the volume was given as 100 ft^3, we can substitute V = 100 into the formulas for r and h:
r = (100 / (2π))^(1/3)
h = 2(100)^(2/3) / (π^(1/3))
Calculating these values will provide the dimensions (radius and height) for the cylinder with the least surface area for a given volume of 100 ft^3.