The radius of a solid sphere is measured as 11.24 cm. What is the surface area of the sphere to appropriate significant figures?

The correct option is 1588 cm².

Step 1: Identify the given value and the number of significant figures.
The measured radius of the solid sphere is:

$$r=11.24\text{cm}$$

The measurement 11.24 cm contains four significant figures.

Step 2: State the formula for the surface area of a sphere.
The surface area $A$ of a sphere is given by the formula:

$$A=4\pi r^2$$

Here, 4 is an exact number, and the constant $\pi$ (approximately 3.14159) can be taken to as many significant figures as needed so that it does not limit the precision of the final calculation.

Step 3: Calculate the surface area.
Substitute the value of the radius $r$ and $\pi$ into the formula:

$$A=4\times 3.14159\times {\left(11.24\right)}^2$$

First, calculate the square of the radius:

$${\left(11.24\right)}^2=126.3376{\text{cm}}^2$$

Now, substitute this value back to find the surface area:

$$A\approx 4\times 3.14159\times 126.3376$$

$$A\approx 1588.12{\text{cm}}^2$$

Step 4: Round to the appropriate number of significant figures.
According to the rules of significant figures for multiplication, the final result must contain the same number of significant figures as the measurement with the least number of significant figures. Since the measured radius has four significant figures, we round our calculated area to four significant figures:

$$A\approx 1588{\text{cm}}^2$$