First, use the formula \(\displaystyle A = 2\pi r (r+h)\) to find the surface area of the tank in feet.

\(\displaystyle A = 2\pi r (r+h) =A = 2\pi \cdot 25 \cdot (25+75) \approx 15,708\)

Now divide by 350, remembering to round up.

\(\displaystyle 15,708 \div 350 \approx 44.9\)

45 cans of paint need to be purchased.

The volume of a cylinder is given by \(\displaystyle \pi r^2\cdot height\)

Notice how the formula for the volume is defined as the area of a circle times the lateral height of the cylinder. It is as if we are taking little paper circles and stacking them one-by-one until we fill up the entire container.

Plugging in the numbers we get:

\(\displaystyle \pi (5)^2\cdot 10 = 250\pi\)\(\displaystyle cm^2\)

Because both the radius and height are unspecified, any real number could be it's height as long as an matching radius is also chosen.  There is no restriction on the height or radius being rational.

The general formula for the volume of a hollow cylinder is given by \(\displaystyle V=\pi (r_{o}^{2}-r_{i}^{2})l\) where \(\displaystyle r_{o}\) is the outer radius, \(\displaystyle r_{i}\) is the inner radius, and \(\displaystyle l\) is the length.

The question gives diameters and we need to convert them to radii by cutting the diameters in half.  Remember, \(\displaystyle d=2r\).  So the equation to solve becomes: 

\(\displaystyle V=\pi (4^{2}-1^{2})18\) or \(\displaystyle V=270\pi\; in^{3}\)

The correct answer is \(\displaystyle 160\pi \ in^{3}\). 

The formula for volume of a cylinder is

\(\displaystyle \pi r^{2} h\)

\(\displaystyle r=4\) and \(\displaystyle h=10\) 

The total surface area will be equal to the area of the two bases added to the area of the outer surface of the cylinder. If "unwrapped" the area of the outer surface is simply a rectangle with the height of the cylinder and a base equal to the circumference of the cylinder base. We can use these relationships to find a formula for the total area of the cylinder.

\(\displaystyle A=2A_{base}+A_{rectangle}\)

\(\displaystyle A=2(\pi r^2)+(2\pi r)(h)\)

Use the given radius and height to solve for the final area.

\(\displaystyle \small 2\pi(2)^{2} + 2\pi (2)(5)\)

\(\displaystyle \small 8\pi + 20\pi\)

\(\displaystyle \small 28\pi\)

The formula to find the volume of a cylinder is: \(\displaystyle V = \pi \cdot r^2 \cdot h\), where \(\displaystyle r\) is the radius of the cylinder and \(\displaystyle h\) is the height of the cylinder. 

A good point to start in this kind of a formula-based problem is to ask "What information do I have?" and "What information is missing that I need?"

In this case, the problem provides us with the height of the cylinder and its diameter. We have the \(\displaystyle h\) component of the equation, but we're missing the \(\displaystyle r\) component. Can we find out \(\displaystyle r\)? The answer is yes! Radius is half of diameter. So in this case, because the diameter is \(\displaystyle 2\:in\) , the radius must be \(\displaystyle 1\:in\). 

Now that we have \(\displaystyle r\) and \(\displaystyle h\), we are ready to solve for the volume after substituting in those values. 

\(\displaystyle V = \pi \cdot r^2 \cdot h\)

\(\displaystyle V = \pi \cdot (1)^2 \cdot 10\)

\(\displaystyle V = \pi \cdot \1 \cdot 10\)

\(\displaystyle V = 10 \pi\: in^3\)

To find the volume of a cylinder we need to find the area of the base and then multiply that by the height. The base is in the shape of a circle so the formula is given below: 

\(\displaystyle V=\pi\ r^{2}\ h\)

To find the radius of the base, we use what was found from the measurement for circumference of the base. We set the formula for circumference equal to the measured circumference given in the problem.

\(\displaystyle 2\pi\ (r)=10\pi\) 

From this we can solve to find that the radius =5. Which we can plug into the original Volume formula. With the given height of 20 ft. 

\(\displaystyle V=\pi\ 5^{2}\ (20)\)

Simplifying will give us the volume of the cylinder. 

\(\displaystyle V=500\pi\ ft^{3}\)

To find the surface area of a cylinder we need to find the area of the circular top and bottom and the area of the rectangular (but rounded) side, then add them together.  Lets start by finding the area of the circular ends to the cylinder, remember that we measured the diameter (d) and we need the radius (r) to find the area of a circle, the radius is half of the diameter:

\(\displaystyle A = r^2\pi = 2^2 \pi = 4\pi \; m^2\) (area of one circle)

Don't forget that we have two circles on the cylinder that have this area:

\(\displaystyle 4\pi \; m^2 + 4\pi \; m^2 = 8\pi \; m^2\)  (area for both circles)

Now we just need to find the area of rectangular side of the cylinder.  To do this we will take the hieght of the cylinder, 2.5 m, and multiply by the circumference (C) of the circle (the width of the rectangular side of the cylinder).

\(\displaystyle C = 2\pi r \; \text{or} \; C= d\pi = 4\pi \; cm\)

So now that we know the circumference we can multiply by the cylinder height to find the area of the rectangular side:

\(\displaystyle 4\pi \; cm \times 2.5 \; cm = 10\pi \; cm^2\) (area for rectangular side)

Now just add to find the total surface area:

\(\displaystyle 8\pi \; cm^2 + 10\pi \; cm^2 = 18\pi \; cm^2\)

and that is our total surface area for the cylinder!

To find the volume of a soup can, or a cylinder, we use the formula for volume:

\(\displaystyle \text {Base Area x Height = Volume}\)

The base of our cylinder is a circle and to find the area of the circle we use:

\(\displaystyle \text {Area =} \; \pi\times r^2 \;, \text {here our radius = }\frac{3.5}{2} = 1.75cm\)

Now we can use the formula for the area of our circular base:

\(\displaystyle \text {Area =} \; \pi\times (1.75)^2 = 3.0625\pi \; cm^2\)

All we need to do now is multiply the base by the hieght of our cylinder!

\(\displaystyle 3.0625 \pi \; cm^2 \times 8cm = 24.5\pi \; cm^3\)