Question

A sphere with a diameter of 2 inches is resting on a table. If the distance from the center of the sphere to the edge of the table is 10 inches, what is the distance, in inches, between the point where the sphere contacts the table and the table edge? A. \(4\sqrt{6}\) B. \(3 \sqrt{11}\) C. 10 D. \(\sqrt{101}\)

Short answer

Option B: \(3 \sqrt{11)\

Step-by-step solution

- Understand the Problem

Visualize the problem. A sphere rests on a table and forms a right triangle with the table's edge and the center of the sphere. The radius of the sphere is half its diameter, which is 1 inch.

- Define Variables

Let the center of the sphere be C. The edge of the table is point A. The point of contact between the sphere and the table is point B. We need to find AB.

- Apply the Pythagorean Theorem

Since CA is the hypotenuse of the right triangle CAB, use the Pythagorean Theorem. The distance CA = 10 inches, and CB is the radius = 1 inch. We need to find AB: \( CA^2 = AB^2 + CB^2 \) Alternatively: \(10^2 = AB^2 + 1^2 \) So: \(100 = AB^2 + 1 \)

- Solve for AB

Rearrange the equation to isolate AB: \(AB^2 = 100 - 1 \) \(AB^2 = 99 \) Take the square root of both sides to find AB. \(AB = \sqrt{99} \) \( AB = 3\sqrt{11} \).

- Choose the Correct Answer

Our calculated distance is AB = \(3\sqrt{11}\), corresponding to option B.

Key concepts

right triangle

In SAT geometry problems, understanding the right triangle is crucial. A right triangle has one angle of 90 degrees. In our problem, the triangle is formed by the table edge, the center of the sphere, and the point where the sphere contacts the table. Points C (center of the sphere), A (edge of the table), and B (contact point) form the vertices. The hypotenuse is the longest side, opposite the right angle. It connects A (table edge) and C (center). Recognizing the right triangle helps use the Pythagorean Theorem to solve the problem.

Pythagorean Theorem

The Pythagorean Theorem is fundamental for solving right triangle problems. The theorem states: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse. In our problem:

• \( CA \) (10 inches) is the hypotenuse.
• \( CB \) (1 inch) is one leg.
• We need to find \( AB \) (another leg).

Using the theorem, we set up the equation for \( AB \): \[ 10^2 = AB^2 + 1^2 \]We solve to find \( AB \).

radius and diameter of sphere

The radius and diameter are key geometric properties of a sphere. The diameter spans from one side of the sphere, through the center, to the other side. The radius is half of this distance. For our sphere:

• Diameter = 2 inches
• Radius = 1 inch

This radius becomes one leg of our right triangle. It measures from the center to the contact point with the table. These measurements help set up our geometry problem accurately.

problem-solving steps

Solving any geometry problem requires clear steps:

• Understand the Problem: Visualize and identify the shapes and measurements involved.
• Define Variables: Label points and distances in the problem. Here, we used points C, A, and B.
• Apply Theorems: Use known formulas like the Pythagorean Theorem to set up the equation.
• Solve for Unknowns: Isolate the variable you need to find. Simplify and solve the equation.
• Check and Choose: Match your solution with the given options to select the correct answer.

Following these steps ensures a logical approach and helps avoid mistakes.