An illustration shows a polygon where the top, right, and bottom sides are perpendicular to each other. A smaller rectangular piece is missing from the center of the left side of the polygon inward. The dimensions are as follows: the top side is 6, the right side is 8, and the bottom side is 6. The left side is divided into three parts. It shows a vertical width of 2 from the top
Solve the following problem using what you know about composite figures. What is the area of the figure?
To find the area of the figure, we need to break it up into smaller shapes that we can calculate the area of and then add them together.
First, we can find the area of the larger rectangle which is (6+2+6) x 8 = 56.
Next, we need to find the area of the missing rectangle on the left side. We know that the vertical width is 2 and we don't know the horizontal length. However, we do know that the left side of the polygon is perpendicular to the bottom side, which means that the missing rectangle is a right-angled triangle with height 2 and hypotenuse 6. Using the Pythagorean theorem, we can find the length of the missing rectangle:
a^2 + b^2 = c^2
a^2 + 2^2 = 6^2
a^2 + 4 = 36
a^2 = 32
a = sqrt(32) = 4sqrt(2)
So the missing rectangle has an area of 2 x 4sqrt(2) = 8sqrt(2).
Finally, we add the areas of the two shapes together:
56 + 8sqrt(2)
This is the area of the composite figure.
An illustration shows a rectangle at the center with a right triangle each on the left and right sides. The bottom side of the rectangle extends 3 inches to the left to form the base of the triangle. Similarly, the top side of the rectangle extends 3 inches to the right to the form the
The parallelogram in the image above has an area of 40 square inches. If the area of a parallelogram is A=bh, what is the length of the base of the rectangle?
We can start by using the formula for the area of a parallelogram: A = bh, where b is the length of the base and h is the height.
We know that the area of the parallelogram is 40 square inches, so we can write:
40 = b * h
However, we don't know the height of the parallelogram, so we need to find it in terms of the base.
Looking at the figure, we can see that the height of the parallelogram is equal to the height of the rectangle, which we can call h1, plus the height of the two right triangles, which we can call h2.
So: h = h1 + h2
The height of the rectangle is not given, but we do know that the bottom and top sides of the rectangle each extend 3 inches out to form the bases of the right triangles.
So:
h1 = base of right triangle + base of rectangle + base of right triangle
h1 = 3 + b + 3 = b + 6
Now we need to find the height of the right triangles, h2. We know that the hypotenuse of each right triangle is equal to the height of the rectangle, so we can use the Pythagorean theorem to find h2:
h2^2 + 3^2 = h1^2
h2^2 + 9 = (b + 6)^2
h2^2 + 9 = b^2 + 12b + 36
h2^2 = b^2 + 12b + 27
Now we can substitute our expressions for h1 and h2 into the formula for the area of the parallelogram:
40 = b * (h1 + h2)
40 = b * (b + 6 + sqrt(b^2 + 12b + 27))
This is a quadratic equation in b. We can simplify it to:
sqrt(b^2 + 12b + 27) = (40 - 6b) / b
Squaring both sides:
b^2 + 12b + 27 = (40 - 6b)^2 / b^2
Multiplying both sides by b^2:
b^4 + 12b^3 + 27b^2 = 1600 - 480b + 36b^2
Bringing all the terms to one side:
b^4 + 12b^3 - 63b^2 + 480b - 1600 = 0
We can use a calculator or software to solve for b, or we can factor the equation using synthetic division or trial and error. It turns out that b = 5 is one solution, so we can divide by (b-5):
b^3 + 17b^2 - 8b + 320 = 0
This is also a quadratic equation, and we can factor it as:
(b-5)(b^2 + 22b + 64) = 0
The quadratic factor has two solutions, but only one of them is positive:
b = -11 + sqrt(57)
We can use a calculator to approximate this value:
b ≈ 2.58
Therefore, the length of the base of the rectangle is approximately 2.58 inches.
Solve the following problem: A town has a swimming pool shaped as in the diagram.
What is the area of the swimming pool in square yards?
We first need to break up the swimming pool into two separate shapes: a rectangle and a trapezoid.
To find the area of the rectangle, we need to multiply the length by the width. The length of the rectangle is 40 feet, which is equivalent to 13.33 yards (since 1 yard equals 3 feet). The width of the rectangle is 15 feet, which is equivalent to 5 yards. Therefore, the area of the rectangle is 13.33 yards x 5 yards = 66.65 square yards.
To find the area of the trapezoid, we need to use the formula:
Area = [(a + b) / 2] x h
where a and b are the lengths of the parallel sides, and h is the height.
In this case, a is equal to the width of the rectangle (5 yards). b is equal to the length of the trapezoid's top side, which we need to calculate. Looking at the diagram, we can see that the length of the top side is equal to the length of the rectangle's top side minus two half-circles. The diameter of each half-circle is 10 feet, or 3.33 yards, so the radius is half of that, or 1.67 yards. Therefore, each half-circle has an area of 1.67^2 x pi = 8.75 square yards (to approximate pi as 3.14). So the combined area of both half-circles is 17.5 square yards. Subtracting this from the length of the rectangle's top side (30 feet or 10 yards) gives us:
b = 10 yards - 17.5 square yards / 5 yards
b = 6.5 yards
Finally, we need to find the height of the trapezoid. Looking at the diagram, we can see that the height is equal to the difference in the lengths of the two horizontal sides. These two sides are each 10 feet long, or 3.33 yards. So the height of the trapezoid is 3.33 yards - 5 yards = -1.67 yards. However, we can't have a negative height, so we take the absolute value of this quantity:
h = |-1.67 yards| = 1.67 yards
Now we can find the area of the trapezoid:
Area = [(a + b) / 2] x h
Area = [(5 yards + 6.5 yards) / 2] x 1.67 yards
Area = 5.75 yards x 1.67 yards
Area = 9.6 square yards
Adding the area of the rectangle and the area of the trapezoid, we get:
66.65 square yards + 9.6 square yards = 76.25 square yards
Therefore, the area of the swimming pool in square yards is approximately 76.25 square yards.
To find the area of the figure, we need to break it down into smaller components and calculate their individual areas.
First, let's calculate the area of the larger polygon. Since the top, right, and bottom sides are perpendicular, we can consider it as a rectangle with dimensions 6, 8, and 6.
Area of the larger rectangle = Length × Width
= 6 × 8
= 48 square units
Now, let's find the area of the missing rectangular piece. We are given that it has a vertical width of 2 and it is missing from the left side of the polygon. Therefore, the height of the missing piece is 6 units (same as the height of the larger rectangle).
Area of the missing rectangular piece = Length × Width
= 6 × 2
= 12 square units
To find the area of the figure, we subtract the area of the missing piece from the area of the larger polygon.
Area of the figure = Area of the larger polygon - Area of the missing piece
= 48 - 12
= 36 square units
Therefore, the area of the figure is 36 square units.