1. What is the base and the height of the rectangle if the area is 216 and the perimeter is 66?

2.The perimeter of an isosceles trapezoid is 40 ft. Thr bases of the trapezoid are 11ft and 19 ft. Find the area of the trapezoid.

3*2^-1+3*6^-1

1. base -- b

height ==h

bh = 216
2b + 2h = 66
b+h = 33
h = 33-b

then b(33-b) = 216
b^2 - 33b + 216=0
(b-24)(b-9) = 0
b = 24 or b=9

if b=24, h = 33-24 = 9
if b = 9 , h = 33-9 = 24

the rectangle is 24 by 9, (you can call either one the base, then the other is the height)

2. Make a sketch, label each of the equal sides x
construct right-angled triangles at each end
So you have a rectangle + 2 equal right-angled triangles
in each of those triangles, the hypotenuse is x
the base is 4
Find the height using Pythagoras,
the rest is easy.

1. To find the base and height of the rectangle, we can use the formulas for area and perimeter.

Let's denote the base of the rectangle as 'b' and the height as 'h'.

Area of a rectangle = length x width = base x height
Perimeter of a rectangle = 2 x (length + width) = 2 x (base + height)

Given:
Area = 216
Perimeter = 66

Using the formulas, we can set up the following equations:

216 = b * h (Equation 1)
66 = 2 * (b + h) (Equation 2)

To solve these equations simultaneously, we can use substitution or elimination.

Let's solve using elimination method:
Multiply Equation 2 by 2 to eliminate the coefficient 2:
132 = 4 * (b + h) (Equation 3)

Now, subtract Equation 1 from Equation 3:
(132 - 216) = (4 * (b + h) - (b * h))
-84 = 4b + 4h - bh

Since we cannot solve this equation directly, we need one more equation. Let's try to find a relationship between b and h.

From Equation 2:
66 = 2(b + h)
33 = b + h (Equation 4)

Now, we can substitute Equation 4 into Equation 1:
216 = b * (33 - b)
216 = 33b - b^2
Rearranging the terms:
b^2 - 33b + 216 = 0

This quadratic equation can be factored as follows:
(b - 24)(b - 9) = 0

So, b = 24 or b = 9.

If b = 24, substituting it into Equation 4:
33 = 24 + h
h = 33 - 24
h = 9

If b = 9, substituting it into Equation 4:
33 = 9 + h
h = 33 - 9
h = 24

Therefore, the possible base and height combinations are:
Base = 24 and Height = 9
OR
Base = 9 and Height = 24.

2. To find the area of the isosceles trapezoid, we can use the formula:

Area = [(base1 + base2) / 2] * height

Given:
Base1 = 11 ft
Base2 = 19 ft
Perimeter = 40 ft

Since the isosceles trapezoid has two equal sides, we can set up the equation:

Perimeter = 2 * base1 + base2
40 = 2 * 11 + 19
40 = 22 + 19
40 = 41
This is not possible, so it seems there may be a mistake in the given information. Please double-check the numbers and provide the correct information, so I can help you calculate the area of the trapezoid accurately.

To find the base and height of a rectangle when given the area and perimeter, you'll need to use a system of equations.

1. Let's assign variables to the base and height of the rectangle:
- Base: Let's use the variable 'b'.
- Height: Let's use the variable 'h'.

2. We know that the area of a rectangle is given by the formula: Area = base * height. Therefore, we have the equation:
- Equation 1: b * h = 216

3. The perimeter of a rectangle is calculated by the formula: Perimeter = 2(base + height). In this case, the perimeter is given as 66. Therefore, we have the equation:
- Equation 2: 2(b + h) = 66

4. We have two equations (Equation 1 and Equation 2) with two unknowns (b and h), so we can solve this system of equations simultaneously.

- From Equation 2, we can simplify it to get: b + h = 33.
- Now, we can subtract h from 33 to get: b = 33 - h.

5. Substitute the value of b from Equation 2 into Equation 1:
(33 - h) * h = 216

6. Solve this quadratic equation to find the value of h. Rearrange Equation 1 to get:
h^2 - 33h + 216 = 0

7. We can solve this quadratic equation by factoring or using the quadratic formula:
- Factoring: (h - 9)(h - 24) = 0
This gives us two possible solutions: h = 9 or h = 24.

8. Plug these values back into Equation 2 to find the corresponding values of b:
- For h = 9: b + 9 = 33 -> b = 24.
- For h = 24: b + 24 = 33 -> b = 9.

Therefore, the possible values for the base and height of the rectangle are:
- Base: 9 units, Height: 24 units
- Base: 24 units, Height: 9 units

Moving to the second question:

To find the area of an isosceles trapezoid, you can use the formula:

Area = ((sum of the bases) * height) / 2

1. Given that the bases of the trapezoid are 11 ft and 19 ft, and the perimeter is given as 40 ft. We can set up an equation to find the height:

11 + 19 + 2h = 40
30 + 2h = 40
2h = 40 - 30
2h = 10
h = 10 / 2
h = 5 ft

2. Now that we know the height is 5 ft, we can calculate the area using the formula:

Area = ((11 + 19) * 5) / 2
Area = (30 * 5) / 2
Area = 150 / 2
Area = 75 square feet

Therefore, the area of the trapezoid is 75 square feet.