a circular has a radius between 6.5cm and 7cm and a volume of 315cm. find the smallest height and the largest height of the cylinder with that volue. round to the nearest 100th
try radius of 6.5, then
πr^2h = 315
π(6.5)^2h = 315
h = 2.37
find the height for r = 7, then decide
just for fun, try a radius of 6.8 and see if it falls between your two values.
volume=h*PI*r^2
smallest height will be at 7cm
315/PI*7^2=2.05cm
largest height will be at 6.5cm
315/(PI* 6.5^2)=2.37cm
v = πr^2h, so h = v/(πr^2)
a larger r means a smaller h, so
315/(π*7^2) <= h <= 315/(π*6.5^2)
2.05 <= h <= 2.37
To find the smallest and largest height of the cylinder with a given volume, we can use the formula for the volume of a cylinder:
Volume = π * radius^2 * height
Given:
- Radius is between 6.5 cm and 7 cm
- Volume is 315 cm^3
Let's find the smallest height first:
1. Calculate the minimum and maximum volume using the given range of the radius:
Minimum volume = π * (6.5 cm)^2 * height
Maximum volume = π * (7 cm)^2 * height
2. Equate the minimum volume to the given volume and solve for the height:
π * (6.5 cm)^2 * height = 315 cm^3
3. Divide both sides of the equation by π * (6.5 cm)^2 to isolate the height:
height = 315 cm^3 / (π * (6.5 cm)^2)
4. Use a calculator to evaluate the expression and round the result to the nearest hundredth to find the smallest height.
Now, let's find the largest height:
1. Repeat steps 2 and 3 using the maximum volume instead:
π * (7 cm)^2 * height = 315 cm^3
2. Divide both sides of the equation by π * (7 cm)^2 to isolate the height:
height = 315 cm^3 / (π * (7 cm)^2)
3. Use a calculator to evaluate the expression and round the result to the nearest hundredth to find the largest height.
By following these steps, you'll be able to find the smallest and largest heights of the cylinder with the given volume.