if the radius of a sphere decreases by 0.1%,find the percentage decreases in (i)surface area
(ii)volume
To find the percentage decrease in the surface area of a sphere when the radius decreases by 0.1%, follow these steps:
Step 1: Calculate the initial surface area of the sphere using the formula: 4πr^2, where r is the initial radius.
Step 2: Calculate the new radius by subtracting 0.1% from the initial radius.
Step 3: Calculate the new surface area of the sphere using the formula: 4πr^2, where r is the new radius.
Step 4: Subtract the new surface area from the initial surface area.
Step 5: Divide the difference by the initial surface area.
Step 6: Multiply the result by 100 to get the percentage decrease.
To find the percentage decrease in the volume of a sphere when the radius decreases by 0.1%, follow these steps:
Step 1: Calculate the initial volume of the sphere using the formula: (4/3)πr^3, where r is the initial radius.
Step 2: Calculate the new radius by subtracting 0.1% from the initial radius.
Step 3: Calculate the new volume of the sphere using the formula: (4/3)πr^3, where r is the new radius.
Step 4: Subtract the new volume from the initial volume.
Step 5: Divide the difference by the initial volume.
Step 6: Multiply the result by 100 to get the percentage decrease.
To find the percentage decrease in surface area and volume of a sphere when the radius decreases by 0.1%, you'll need to know the formulas for surface area and volume of a sphere.
Let's denote the original radius of the sphere as "r" and the decrease in the radius as "Δr."
(i) Percentage decrease in surface area:
The surface area formula for a sphere is given by: A = 4πr^2
When the radius decreases by 0.1%, the new radius would be (r - Δr), where Δr = 0.001 * r.
To calculate the new surface area, substitute (r - Δr) into the surface area formula to get: A_new = 4π(r - Δr)^2
Now, let's calculate the percentage decrease using the formula: Percentage decrease = (A - A_new) / A * 100%
Substitute the values of A and A_new into the formula and simplify to get the final percentage decrease in surface area.
(ii) Percentage decrease in volume:
The volume formula for a sphere is given by: V = (4/3)πr^3
Using the same logic as above, substitute (r - Δr) into the volume formula to get: V_new = (4/3)π(r - Δr)^3
Calculate the percentage decrease using the formula: Percentage decrease = (V - V_new) / V * 100%
Substitute the values of V and V_new into the formula and simplify to get the final percentage decrease in volume.
By following these steps, you should be able to find the percentage decreases in surface area and volume when the radius of a sphere decreases by 0.1%.
for area, our new area is
4pi(.99r)^2
So, if we divide that by the original area, we have
4pi(.99r)^2 / 4pir^2 = .99^2
1-.99^2 = 0.0199
That is, a 1.99% decrease in area
Note that the area decreases by about twice the percentage as the radius.
I think you can expect the volume to decrease by about 3 times the %age of the radius.