Use the following diagram of a flowerbed of petunias to answer the following questions. This diagram is a rectangle with ½ a circle on each end. The circle diameter is 22 feet, while the length of the rectangle is 60 feet. If you draw an X in the middle of the rectangle, you have 4 different flowerbeds. In the top triangle you will put Petunias, in the bottom triangle you will put rock and shrubs, and in the two side triangles, you will have roses.
(8 points)
A. Find the perimeter of the entire display.
B. The two end sections are filled with assorted wildflowers. Find the combined area of these two sections.
C. Find the perimeter of the section that has rocks and shrubs in it if the top point of bed is exactly in the middle of the flowerbed.
D. Assume you get frustrated with growing flowers, so you decide to CEMENT the ENTIRE flowerbed!! How many cubic yards of cement should you order if you are going to cement this flowerbed 3 inches thick.
A:
22π + 2*60
B:
if by "end section" you mean a semi-circle, π*11^2
If the triangle is included, then π*11^2 + 60*22/2
C:
the diagonal of the rectangle has length √(60^2+22^2) = 2√1021
That should enable you to figure the triangle perimeters.
D:
(121π+1320)(1/4)/27
A. To find the perimeter of the entire display, we need to calculate the sum of the lengths of all sides.
The rectangle has two sides with a length of 60 feet each, so the total length of the rectangle is 2 * 60 = 120 feet.
The circles at the ends have a diameter of 22 feet, which means the circumference of each circle is π * 22 = 22π feet. Since there are two circles at the ends, the total length of the circles is 2 * 22π = 44π feet.
Now, to get the perimeter of the entire display, we add the lengths of the rectangle and the circles: 120 + 44π.
So, the perimeter of the entire display is 120 + 44π feet.
B. The two end sections that are filled with assorted wildflowers are semicircles. To find their combined area, we need to calculate the area of one semicircle and then multiply it by 2.
The radius of each semicircle is half the diameter, which is 22/2 = 11 feet.
The area of a semicircle is given by (π * r^2) / 2, where r is the radius. Substituting the values, we get (π * 11^2) / 2 = 121π / 2 square feet.
Now, to find the combined area of both end sections, we multiply this area by 2: 2 * (121π / 2) = 121π square feet.
So, the combined area of the two end sections is 121π square feet.
C. The section that has rocks and shrubs in it is a trapezoid. To find its perimeter, we need to calculate the sum of all four sides.
The top side of the trapezoid has a length of 22 feet, which is the diameter of the circles.
The bottom side of the trapezoid has a length of 60 feet, which is the length of the rectangle.
The two side triangles have a base length of 22 feet each, which is the diameter of the circles.
We don't have the height of the trapezoid, so we can't calculate the exact perimeter. However, we can approximate it by assuming the height is equal to the radius of the circles (11 feet).
Using this approximation, the perimeter of the trapezoid is 22 + 60 + 2 * 11 = 104 feet.
So, the perimeter of the section with rocks and shrubs is approximately 104 feet.
D. To calculate the volume of cement needed to cover the entire flowerbed, we first need to find the area of the flowerbed. As mentioned, the flowerbed is a rectangle with two semicircles at the ends.
The area of the rectangle is length * width, which is 60 * (22 - 11) = 60 * 11 = 660 square feet.
The area of each semicircle is (π * r^2) / 2, where r is the radius (11 feet). The total area of both semicircles is 2 * (π * 11^2) / 2 = 121π square feet (as calculated in part B).
Adding the area of the rectangle and the two semicircles, we get the total area of the flowerbed: 660 + 121π square feet.
Now, to calculate the volume of cement needed, we multiply the total area by the thickness of the cement layer (in feet). The thickness is given as 3 inches, which is 3/12 = 1/4 foot.
So, the volume of cement needed is (660 + 121π) * (1/4) = (165 + 121π/4) cubic feet.
To convert to cubic yards, we divide by 27 (since there are 27 cubic feet in a cubic yard): (165 + 121π/4) / 27 cubic yards.
Therefore, the number of cubic yards of cement that should be ordered is approximately (165 + 121π/4) / 27.