Final answer:
The ratio of the surface area to volume for a cylinder is obtained by dividing the surface area formula (2πr(h + r)) by the volume formula (πr²h). For a cylinder with a 7-inch radius and a 28-inch height, the ratio simplifies to 35/98.
Step-by-step explanation:
Finding the Ratio of Surface Area to Volume for a Cylinder
To find a rational expression for the ratio of the surface area of a cylinder to its volume, we use the formulas for surface area (SA) and volume (V) of a cylinder. The surface area of a cylinder is calculated by SA = 2πr(h + r) where r is the radius and h is the height of the cylinder. The volume is found using the formula V = πr²h. To obtain the ratio, we simply divide SA by V:
Ratio = πr²h / (2πr(h + r))
Now, plugging in the values for a cylinder with a radius of 7 inches and a height of 28 inches, we get:
SA = 2π(7 inches)(28 inches + 7 inches) = 2π(7 inches)(35 inches) = 490π square inches
V = π(7 inches)²(28 inches) = 1372π cubic inches
Therefore, the ratio of the surface area to volume for this specific cylinder is:
Ratio = 490π / 1372π = 35/98, after simplifying the π's out and reducing the fraction.