What size should be the frame be?
Suppose an artist has a circular picture with a diameter 6 inches which she wishes to frame by a uniform width frame with covers 1/2 inch width uniformity of the painting and has total area equal to that of the exposed painted surface when the frame is in place.
To determine the size of the frame, we need to find the total area of both the circular picture and the frame. Let's break it down step by step:
1. Start by finding the area of the circular picture:
- The diameter of the circular picture is given as 6 inches, which means the radius (r) is half of the diameter, i.e., r = 6/2 = 3 inches.
- The area of a circle is calculated using the formula A = πr^2, where π is a mathematical constant approximately equal to 3.14159.
- Plugging in the radius, we get A = 3.14159 * (3 inches)^2.
- Evaluating the expression, we find the area of the circular picture to be A = 3.14159 * 9 square inches ≈ 28.274 square inches.
2. Now, let's calculate the dimensions of the frame:
- The cover of the frame has a uniform width of 1/2 inch on each side. So, we need to add 1 inch to both the length and width of the circular picture to account for the frame.
- The new diameter of the frame will be 6 inches + 1 inch (both sides) = 8 inches.
- This means the radius of the frame will be r = 8/2 = 4 inches.
- The area of the frame can be calculated using the same formula A = πr^2 as before.
- Substituting the radius, we get A = 3.14159 * (4 inches)^2.
- Solving the expression, we find the area of the frame to be A = 3.14159 * 16 square inches ≈ 50.265 square inches.
3. Finally, we need to determine the size of the frame:
- The total area of the frame and the circular picture combined is equal to the area of the exposed painted surface when the frame is in place.
- Therefore, the total area is 28.274 square inches + 50.265 square inches = 78.539 square inches.
Thus, the size of the frame should be 78.539 square inches.