A rectangular box is 12 in. * 14 in. * 18 in. What is the surface area of the largest size sphere that could fit in this box? leave you answer in terms of pi
To find the surface area of the largest sphere that can fit in a rectangular box, we need to determine the sphere's diameter. Since the sphere needs to fit inside the box, its diameter must be equal to the box's smallest dimension.
In this case, the smallest dimension of the box is 12 inches. Therefore, the diameter of the sphere is also 12 inches.
The surface area of a sphere can be calculated using the formula: A = 4πr^2, where A represents the surface area and r is the radius of the sphere.
To find the radius, we need to divide the diameter by 2: r = 12 / 2 = 6 inches.
Now, we can calculate the surface area of the sphere: A = 4π(6^2) = 4π(36) = 144π square inches.
Hence, the surface area of the largest sphere that can fit in the given rectangular box is 144π square inches.
To fit a sphere inside a rectangular box, the diameter of the sphere must be less than or equal to the smallest dimension of the box (because a sphere's diameter is the same in every direction). In this case, the dimensions of the box are 12 inches, 14 inches, and 18 inches. The smallest dimension is 12 inches.
Therefore, the diameter of the largest sphere that could fit into this box is 12 inches, which means the radius (r) is half that, or 6 inches.
The surface area (A) of a sphere is given by the formula:
\[A = 4\pi r^2\]
Substitute the radius (r = 6 inches) into the formula:
\[A = 4\pi (6\text{ in})^2\]
\[A = 4\pi (36\text{ in}^2)\]
\[A = 144\pi\text{ in}^2\]
So, the surface area of the largest sphere that could fit into this box, in terms of \(\pi\), is \(144\pi\text{ in}^2\).
just give me a number
The value of π (pi) is approximately 3.14159. To get a numerical value for the surface area of the largest sphere that can fit inside the box, we'll multiply the surface area in terms of π by this approximate value.
\[ A \approx 144\pi \text{ in}^2 \times 3.14159 \]
\[ A \approx 452.38976 \text{ in}^2 \]
Rounded to the nearest whole number, the surface area of the largest sphere that could fit into this box is approximately 452 square inches.