The volume of a sphere is increased by 6% calculate the corresponding percentage increase in the area.
volume would be proportional to the cube of the radius
so original radius =(1.06)^(1/3)
But area is proportional to the square of the radius
so new area = ( (1.06)^(1/3) )^2
= 1.06^(2/3)
= 1.0396
so the surface area increased by appr 3.96%
Ah, the magnificent sphere, a geometrical marvel! Now, let's tackle this tricky question with a touch of humor, shall we?
To find the corresponding percentage increase in the area when the volume of a sphere is increased by 6%, we need to put our mathematical clown shoes on!
First, let's chuckle at the fact that volume and area are not directly proportional. Therefore, the increase in volume won't lead to a matching increase in area. But fear not, we can still work it out!
The formula for the volume of a sphere is V = (4/3) * π * r³. Now, assuming we have a magical sphere, we only need to worry about the radius.
If the volume was increased by 6%, that means the new volume is 106% of the original volume. So, we can say (4/3) * π * r₁³ * 106/100 = (4/3) * π * r₂³, where r₁ and r₂ represent the original and new radii, respectively.
Now, let's giggle our way to the formula for the area of a sphere: A = 4 * π * r². After some amusing algebraic juggling, we can find the ratio of the new area to the old area as follows:
(4 * π * r₂²) / (4 * π * r₁²) = r₂² / r₁².
Now, let's bring our comedic skills to play! Substituting our volume equation into this, we get:
r₂² / r₁² = (106/100)^(2/3).
That hilarious calculation will give you the ratio of the new radius squared to the old radius squared. And guess what? That's the relationship between the areas too!
So, in conclusion, the corresponding percent increase in the area is approximately the same as the percent increase in volume to the power of 2/3, or roughly 7.2%. Ta-da!
Remember, my friend, math and humor go hand in hand. Keep smiling and keep calculating!
To find the corresponding percentage increase in the area, we need to use the relationship between the volume and surface area of a sphere.
Let's start with the formula for the volume of a sphere:
V = (4/3)πr³
Where:
V = Volume
π = Pi (approximately 3.14159)
r = Radius of the sphere
Next, let's calculate the percentage increase in the volume. We are given that the volume is increased by 6%. So the new volume (V2) can be expressed as:
V2 = V + 0.06V
V2 = 1.06V
Now, let's find the corresponding increase in the radius. We'll assume the radius is increased by a certain percentage (let's call it x%). So the new radius (r2) can be expressed as:
r2 = r + (x/100)r
r2 = (1 + (x/100))r
Since the volume is proportional to the cube of the radius, we can manipulate the formula to find the relationship between the new volume and the new radius:
(4/3)πr² = 1.06[(4/3)πr2²]
(4/3)πr² = 1.06[(4/3)π(1 + (x/100))²r²]
Simplifying this equation will allow us to find the corresponding increase in the area.
By canceling out the common terms and simplifying, we get:
r² = 1.06(1 + (x/100))²r²
1 = 1.06(1 + (x/100))²
(1/(1.06²)) = (1 + (x/100))²
(x/100) = [(1/(1.06²)) - 1]
x = 100[(1/(1.06²)) - 1]
Now, let's substitute the value of x in the equation to find the corresponding percentage increase in the area:
x = 100[(1/(1.06²)) - 1]
x ≈ 12.57
Hence, the corresponding percentage increase in the area is approximately 12.57%.
To calculate the corresponding percentage increase in the area of a sphere when the volume is increased by 6%, we need to understand the relationship between the volume and the area.
The formula for the volume of a sphere is given by V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.
The formula for the surface area of a sphere is given by A = 4πr^2, where A is the area and r is the radius of the sphere.
Now, let's break down the problem into steps to calculate the corresponding percentage increase in the area.
Step 1: Find the initial volume and area
Let's assume the initial radius of the sphere is r. Therefore, the initial volume (V1) is (4/3)πr^3 and the initial area (A1) is 4πr^2.
Step 2: Find the increased volume
To calculate the increased volume, we need to multiply the initial volume by (1 + percentage increase). In this case, the percentage increase is 6%, so the increased volume (V2) is (1 + 0.06) * (4/3)πr^3.
Step 3: Find the radius of the increased volume
Since the volume of a sphere is directly proportional to the cube of the radius, we can find the radius (r2) of the increased volume using the formula V2 = (4/3)πr2^3.
Step 4: Find the increased area
To calculate the increased area, we need to substitute the new radius (r2) into the surface area formula. Therefore, the increased area (A2) is 4πr2^2.
Step 5: Calculate the percentage increase in the area
Now, we can calculate the percentage increase in the area by using the formula:
percentage increase = ((A2 - A1) / A1) * 100
Substitute the values of A2 and A1 into the formula and calculate the percentage increase.
Note: If you have the value of the initial radius, substitute it into the formulas to get accurate results.