The solution to this is simple, though just take it step-by-step.  First find the radius of the circular cross-section.  This will give us the radius of the sphere (since this cross-section is at the center of the sphere).  If the cross-section has an area of 225π, we know its area is defined by:

A = 225π = πr²

Solving for r, we get r = 15.

From here, we merely need to use our formula for the volume of a sphere:

V = (4 / 3)πr³

For our data this is: (4 / 3)π * 15³ = 4π * 15² * 5 = 4500π

We need to ascertain two values: The center point and the point of intersection with the surface.  Let's do that first:

The center is defined by the x-intercept. To find that, set the line equation equal to 0 (y = 0 at the x-intercept):

0 = 4x + 12; 4x = –12; x = –3; Therefore, the center is at (–3,0)

Next, we need to find the point at which the line intersects with the sphere's surface. To do this, solve for the point with x-coordinate at 3:

y = 4 * 3 + 12; y = 12 + 12; y = 24; therefore, the point of intersection is at (3,24)

Reviewing our data so far, this means that the radius of the sphere runs from the center, (–3,0), to the edge, (3,24).  If we find the distance between these two points, we can ascertain the length of the radius.  From that, we will be able to calculate the volume of the sphere.

The distance between these two points is defined by the distance formula:

d = √( (x₁ – x₀)² + (y₁ – y₀)² )

For our data, that is:

√( (3 + 3)² + (24 – 0)² ) = √( 6² + 24² )  = √(36 + 576) = √612 = √(2 * 2 * 3 * 3 * 17) = 6√(17)

Now, the volume of a sphere is defined by: V = (4/3)πr³

For our data, that would be:  (4/3)π * (6√(17))³ = (4/3) * 6³ * 17√(17) * π = 4 * 2 * 6² * 17√(17) * π = 4896π√(17)

To begin, we must solve for the radius of our sphere.  To do this, recall the equation for the surface area of a sphere: A = 4πr²

For our data, that is: 676π = 4πr²; 169 = r²; r = 13

From this, it is easy to solve for the volume of the sphere.  Recall the equation:

V = (4/3)πr³

For our data, this is: V = (4/3)π * 13³ = (4π * 2197)/3 = (8788π)/3

Surface Area = 4πr² = 4 * π * 6² = 144π

Volume = 4πr³/3 = 4 * π * 6³ / 3 = 288π

Volume – Surface Area = 288π – 144π = 144π

To begin, we must determine the dimensions of the cube. This is done by solving the simple equation:

We know the volume is 216, allowing us to solve for the length of a side of the cube.

Taking the cube root of both sides, we get s = 6.

The diameter of the sphere will be equal to side of the cube, since the question states that the sphere is perfectly contained. The diameter of the sphere will be 6, and the radius will be 3.

We can plug this into the equation for volume of a sphere:

We can cancel out the 3 in the denominator.

Simplify.

First, let's find the surface area of the hemisphere. Because the hemisphere is basically a full sphere cut in half, we need to find half of the surface area of a full sphere. However, because the hemisphere also has a circular base, we must then add the area of the base.

 (surface area of sphere) + (surface area of base)

The surface area of a sphere with radius r is equal to . The surface area of the base is just equal to the surface area of a circle, which is .

The volume of the hemisphere is going to be half of the volume of an entire sphere. The volume for a full sphere is .

 (volume of sphere)

Ultimately, the question asks us to find the ratio of S to V. To do this, we can write S to V as a fraction.

In order to simplify this, let's multiply the numerator and denominator both by 3.

=

The answer is .

In order to figure out how many cubic centimeters can fit into the box, we need to figure out the volume of the box in terms of cubic centimeters. However, the measurements of the box are given in meters. Therefore, we need to convert these measurements to centimeters and then determine the volume of the box.

There are 100 centimeters in one meter. This means that in order to convert from meters to centimeters, we must multiply by 100.

The length of the box is 2 meters, which is equal to 2 x 100, or 200, centimeters.

The width of the box is 0.5(100) = 50 centimeters.

The height of the box is 3.2(100) = 320 centimeters.

Now that all of our measurements are in centimeters, we can calculate the volume of the box in cubic centimeters. Remember that the volume of a rectangular box (or prism) is equal to the product of the length, width, and height.

V = length x width x height

V = (200 cm)(50 cm)(320 cm) = 3,200,000 cm³

To rewrite this in scientific notation, we must move the decimal six places to the left.

V = 3.2 x 10⁶ cm³

The answer is 3.2 x 10⁶.

Let l be the length, w be the width, and h be the height of the prism. We are told that the length is twice the width, and that the width is twice the height. We can set up the following two equations:

l = 2w

w = 2h

Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:

surface area = 2lw + 2lh + 2wh = 252

We now have three equations and three unknowns. In order to solve for one of the variables, let's try to write w and l in terms of h. We know that w = 2h. Because l = 2w, we can write l as follows:

l = 2w = 2(2h) = 4h

Now, let's substitute w = 2h and l = 4h into the equation we wrote for surface area.

2(4h)(2h) + 2(4h)(h) + 2(2h)(h) = 252

Simplify each term.

16h² + 8h² + 4h² = 252

Combine h² terms.

28h² = 252

Divide both sides by 28.

h² = 9

Take the square root of both sides.

h = 3.

This means that h = 3. Because w = 2h, the width must be 6. And because l = 2w, the length must be 12.

Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.

volume of a prism = l • w • h

volume = 12(6)(3)

= 216 cubic units

The answer is 216.

The volume of B is 7920 in³. The volume of A is 7722 in³. Treasure Chest B can hold more treasure.