If the radius of a sphere is increased by 125%, then by what percentage will its surface area increase (correct to two decimal places)?

The correct answer is 406.25%.

Let the original radius of the sphere be r.

Step 1: Find the new radius after a 125% increase.

An increase of 125% means the new radius is:

New radius = r + 125% of r = r + 1.25r = 2.25r

Step 2: Write the formula for surface area of a sphere.

Surface Area = 4\pi r^2

Step 3: Calculate the original and new surface areas.

Original Surface Area:

A_1=4\pi r^2

New Surface Area (using the new radius 2.25r):

A_2=4\pi {\left(2.25r\right)}^2=4\pi \times 5.0625r^2=5.0625\times 4\pi r^2

Note: {\left(2.25\right)}^2=5.0625

Step 4: Calculate the percentage increase in surface area.

\text{\% Increase}=\frac{A_2-A_1}{A_1}\times 100

=\frac{5.0625\times 4\pi r^2-4\pi r^2}{4\pi r^2}\times 100

=\frac{4\pi r^2(5.0625-1)}{4\pi r^2}\times 100

=(5.0625-1)\times 100

=4.0625\times 100=406.25%

Quick Tip: Since surface area depends on the square of the radius, if the radius is multiplied by a factor k, the surface area is multiplied by k². Here, the radius became 2.25 times the original (i.e., k = 2.25), so surface area became 2.25² = 5.0625 times the original, meaning a net increase of (5.0625 - 1) × 100 = 406.25%.