Find the area of the largest rectangle that fits inside a semicircle of radius 10 (one side of the rectangle is along the diameter of the semicircle).
Hmmm. If you imagine the semicircle being made into a circle, this largest rectangle turns into a square of diagonal 2r. Hmmm. But since it is a square, then each side must be equal to diagonal divided by sqrt 2.
each side=2r/sqrt2
area of square= s^2= 4r^2/2=2r^2
area of the rectangle in the semicircle is half this, or r^2, in this case, 100.
area of semicircle = PI/2 * 100=157
percent of area in the semicircle occupied by square: 100/157=.637
Answer
To find the area of the largest rectangle that fits inside a semicircle of radius 10, we need to determine the dimensions of the rectangle.
Let's consider the semicircle with its diameter as the base of the rectangle. The maximum area of the rectangle will be obtained when its height is equal to the radius of the semicircle.
So, the height of the rectangle will be 10.
Now, let's calculate the length of the base of the rectangle. Since the base of the rectangle is along the diameter of the semicircle, it is twice the radius.
Therefore, the length of the base of the rectangle is 2 * 10 = 20.
Finally, we can calculate the area of the rectangle using the formula: Area = Length * Height.
So, the area of the largest rectangle that fits inside the semicircle of radius 10 is 20 * 10 = 200 square units.
To find the area of the largest rectangle that fits inside a semicircle of radius 10, one side of the rectangle needs to be along the diameter of the semicircle. Let's solve this step by step:
Step 1: Draw a diagram
Start by drawing a semicircle with radius 10. Then draw a rectangle inside the semicircle with one side along the diameter. Label the dimensions of the rectangle, considering the height, width, and length of the rectangle along the diameter.
Step 2: Identify the dimensions
Let the width of the rectangle be 'w' and the length along the diameter be 'l'. The height of the rectangle, which is the other side along the diameter, will also be 'l' since it is on the diameter of the semicircle.
Step 3: Write equations using known values
Since the rectangle lies inside the semicircle, the diagonal of the rectangle will be the diameter of the semicircle, which is 2 multiplied by the radius (2 * 10 = 20). We can use the Pythagorean theorem to find the relationship between the width, length, and diagonal of the rectangle.
According to the Pythagorean theorem, the sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse (diagonal in our case).
So, we can write this equation based on the width (w), length (l), and diagonal (d) of the rectangle:
w^2 + l^2 = d^2
Substituting the known values, we have:
w^2 + l^2 = 20^2
Step 4: Express one variable in terms of the other
Since we are interested in finding the maximum area, we need to express the area of the rectangle in terms of a single variable. We have the equation w^2 + l^2 = 400, which can be rearranged to l^2 = 400 - w^2.
Step 5: Maximize the area
The formula for the area of a rectangle is given by A = w * l. To maximize the area, we need to find the maximum value of A.
Substitute l^2 = 400 - w^2 into the area formula:
A = w * (400 - w^2)
Take the derivative of A with respect to w, set it equal to zero, and solve for w to find the critical points. Then, check the endpoints of the interval (0 to 10) to determine the maximum value of A.
Step 6: Calculate the maximum area
Solve A' = 0 for w:
(400 - w^2) - 3w^2 = 0
400 - 4w^2 = 0
4w^2 = 400
w^2 = 100
w = 10 or w = -10
Since we are interested in the width, which cannot be negative, the width is 10.
Calculate the length (l) using the equation l^2 = 400 - w^2:
l^2 = 400 - 10^2
l^2 = 400 - 100
l^2 = 300
l = √300 ≈ 17.32
Finally, calculate the area of the largest rectangle:
A = 10 * 17.32 ≈ 173.2 square units
Therefore, the area of the largest rectangle that fits inside the semicircle of radius 10 is approximately 173.2 square units.