Jayden wants to lay sod on his front yard and on half of his back yard. His front yard has a length of 50 feet and a width of 70 feet. His back yard has a length of 20 feet and a width of 40 feet. How many square feet of sod does Jayden need to purchase?
First, calculate the area of the front yard:
Area of front yard = length x width = 50 ft x 70 ft = 3,500 square feet
Then, calculate the area of half of the back yard (since Jayden only wants to lay sod on half of it):
Area of back yard = (length x width) / 2 = (20 ft x 40 ft) / 2 = 400 square feet
Finally, add the area of the front yard and half of the back yard together to get the total area of sod needed:
Total area of sod = 3,500 square feet + 400 square feet = 3,900 square feet.
Therefore, Jayden needs to purchase 3,900 square feet of sod.
Julie printed out a picture of each of the five members of her favorite band to decorate her bedroom door. Each picture measures 8 inches by 10 inches. Once decorated, how many square inches of Julie's door will be covered by these pictures?
Since there are five pictures, we need to find 5 times the area of one picture.
Area of one picture = length x width = 8 inches x 10 inches = 80 square inches
Therefore, the total area covered by the five pictures would be:
Total area = number of pictures × area of one picture
Total area = 5 × 80 square inches = 400 square inches
So, Julie's door will be covered by 400 square inches of pictures.
The rectangle has an area of 128 square inches and a width of 8 inches. Find the perimeter of the rectangle.
To find the length of the rectangle, we need to divide the area by the width:
Length = Area / Width = 128 sq in / 8 in = 16 in
Now that we know the length is 16 inches and the width is 8 inches, we can use the formula for perimeter:
Perimeter = 2 × (Length + Width)
Plugging in the values we found:
Perimeter = 2 × (16 in + 8 in) = 2 × 24 in = 48 in
Therefore, the perimeter of the rectangle is 48 inches.
Joey is creating a rectangular garden. The garden is 9 feet long and 7 feet wide.
Joey calculated he needs
feet of fencing to close in the garden and
square feet of soil to cover the garden.
To find the amount of fencing needed, we need to calculate the perimeter of the rectangular garden.
Perimeter = 2 × (Length + Width) = 2 × (9 ft + 7 ft) = 2 × 16 ft = 32 ft
Therefore, Joey needs 32 feet of fencing to close in the garden.
To find the amount of soil needed, we need to calculate the area of the rectangular garden.
Area = Length × Width = 9 ft × 7 ft = 63 square feet
Therefore, Joey needs 63 square feet of soil to cover the garden.
The total area of the figure is 77 square centimeters. The overall height of the figure is 9 centimeters.
Drag the correct measurements to label the unknown lengths in the figure.
There are different ways to approach this problem, but one possible method is to use the fact that the figure can be split into two smaller rectangles and a triangle, as shown:
```
x 7
┌────┴────┐ ┌─────┴─────┐
| | | |
| x | | 10 |
| | | |
├─────┬───┤ x├────┬────┤
| 5 | h | | x | h |
└─────┴───┘ └────┴────┘
```
The area of the figure can then be expressed as:
Area = 5x + 7h + (1/2) * 10 * h
Since we know that the total area is 77 square centimeters, we can substitute this value and simplify the equation:
77 = 5x + 7h + 5h
77 = 5x + 12h
x = (77 - 12h) / 5
We are also given that the overall height of the figure is 9 centimeters, so we can substitute this value and solve for h:
9 = h + h
h = 4.5
Now we can plug in the value of h and solve for x:
x = (77 - 12h) / 5
x = (77 - 12 * 4.5) / 5
x = 2.7
Therefore, the lengths of the unknown sides are approximately x = 2.7 centimeters and h = 4.5 centimeters.
Try a different method and a different answer
Sure! Another way to approach this problem is to use the fact that the figure can be split into a rectangle and a triangle, as shown:
```
x 7
┌─────┴─────┐ ┌─────┴─────┐
| | | |
| | | |
| | | |
| | | |
| 5 | | 10 |
| | | |
| | | |
| | x | |
└─────┬─────┘ └─────┴─────┘
(9-h)
The area of the rectangle can be expressed as:
Area_rect = 5 * (9 - h) = 45 - 5h
The area of the triangle can be expressed as:
Area_tri = (1/2) * 10 * h = 5h
The total area is given as 77 square centimeters, so we can write:
45 - 5h + 5h = 77
Simplifying, we get:
45 = 77
This is not a valid equation, since the left-hand side is smaller than the right-hand side. Therefore, there must be an error in the problem statement or in the calculations.
One possible error could be that the overall height of the figure is not 9 centimeters, but a different value that makes the equation valid. Another possible error could be a mistake in the calculation of the areas or in the substitution of values. Without further information, it is difficult to determine the correct answer.
If x = 5 units, y = 15 units, and h = 9 units, find the area of the parallelogram shown above using decomposition.
To find the area of the parallelogram, we can use the formula:
Area = base x height
The base is given as x = 5 units, but the height is not directly given. However, we can decompose the parallelogram into a rectangle and two right triangles, as shown:
```
15 units
┌──────┐
│\ │
│ \ │
│ \ │
h │ \ │
│ \ │
│_____\\│ x = 5 units
5 units
```
The height of the parallelogram is the same as the height of the rectangle, which is h = 9 units. The length of the rectangle is the same as the base of the parallelogram, which is x = 5 units. Therefore, the area of the rectangle is:
Area_rect = base x height = 5 units x 9 units = 45 square units
The two triangles are congruent, so we can find the area of one and then double the result. The base of each triangle is half of x, which is 2.5 units, and the height is h = 9 units. Therefore, the area of one triangle is:
(1/2) x base x height = (1/2) x 2.5 units x 9 units = 11.25 square units
The total area of the parallelogram is the sum of the area of the rectangle and twice the area of one triangle:
Area = Area_rect + 2 x Area_tri = 45 square units + 2 x 11.25 square units = 67.5 square units
Therefore, the area of the parallelogram is 67.5 square units.
Try again
Actually, I apologize for my mistake. I realized that the base of the parallelogram is not given as x = 5 units, but as y = 15 units. Here's the correct solution using decomposition:
To find the area of the parallelogram, we can use the formula:
Area = base x height
The base is given as y = 15 units, but the height is not directly given. However, we can decompose the parallelogram into a rectangle and two right triangles, as shown:
```
15 units
┌────┐
│\ │ h
│ \ │ ┌─┐
│ \ │ │ │
h │ \│ └─┘ x
│___\\│
x = 5
```
We can use the Pythagorean Theorem to find the height h of the parallelogram:
h^2 = x^2 - (1/2 * y)^2
h^2 = 5^2 - (1/2 * 15)^2
h^2 = 25 - 56.25
h^2 = -31.25
This is not possible, since the square of a real number cannot be negative. Therefore, there is no solution to this problem with the given values.
I apologize for any confusion caused by my earlier incorrect response.
Malik bought chargers that are shaped like regular octagons. If the chargers have the dimensions shown below, what is the area of one charger?
The regular octagon can be split into 8 congruent isosceles triangles with a vertex angle of 45 degrees, as shown below:
```
4 cm
┌─────────────┐
│ 45° │
5 cm│ │5 cm
│ │
│ │
└─────────────┘
4 cm
```
The area of one triangle can be found using the formula:
Area_tri = (1/2) x base x height
The base of the triangle is one side of the octagon, which is given as 4 cm. The height of the triangle is the distance from the side to the center of the octagon, which can be found by drawing a line from the vertex of the angle to the center of the circle circumscribed around the octagon:
```
4 cm
┌─────────────┐
│ 45° │
5 cm│ │5 cm
│ │
│ O │
└─────────────┘
4 cm
```
The line from the vertex to the center divides the angle into two congruent angles of 22.5 degrees each. Therefore, we can draw a line from the vertex perpendicular to the base, forming a right triangle with a hypotenuse of 5 cm:
```
4 cm
┌─────────────┐
│ 45° │
5 cm│ │5 cm
│ │
│ O │
└───┬─────────┘
h │ 2 cm
│
```
Using trigonometry, we can find that:
h = (5/2) x tan(22.5°)
h ≈ 1.961 cm
Therefore, the area of one triangle is:
Area_tri = (1/2) x base x height = (1/2) x 4 cm x 1.961 cm ≈ 3.922 cm²
Since there are 8 triangles in the octagon, the total area of one charger is:
Area_charger = 8 x Area_tri ≈ 31.376 cm²
Therefore, the area of one charger is approximately 31.376 square centimeters.
Malik bought chargers that are shaped like regular octagons. If the chargers have the dimensions shown below, what is the area of one charger?
4inch, 4inch, 4inch, 4inch, 6inch, 6inch,
To find the area of one charger, we can use the formula for the area of a regular octagon:
Area_octagon = 2 (1 + √2) s^2, where s is the side length
In this case, the side length of the octagon can be found by taking an average of the given values:
s = (4 in + 4 in + 4 in + 4 in + 6 in + 6 in) / 6 = 4.33 in (rounded to two decimal places)
Plugging this value into the formula, we get:
Area_charger = 2 (1 + √2) s^2 = 2 (1 + √2) (4.33 in)^2 ≈ 72.06 in²
Therefore, the area of one charger is approximately 72.06 square inches.
A pentagon is formed by connecting the tips of a starfish's legs, as shown below.
10cm,6cm,11cm,5cm,4cm,8cm
*Picture not drawn to scale
What is the area of the pentagon?
To find the area of the pentagon, we can divide it into five triangles, as shown:
```
10cm 8cm
┌───────┐ ┌─────┐
│\ /│ / /│
│ \ / │ / / │
│ \ / │ / / │
4cm───6cm─11cm 5cm─┐───/─────┘───┤ 10cm
│ / \ │ \ / 2 │
│ / \ │ \ / │
│/ \│ \/ │
└───────┘ 4cm └───────┘
```
We can use Heron's formula to find the area of each triangle, which depends only on the length of its sides. The formula is:
Area = √(s(s-a)(s-b)(s-c)), where s is the semiperimeter (half of the perimeter), and a, b, c are the lengths of the sides.
For example, the area of the triangle with sides 10 cm, 6 cm, and 4 cm is:
s = (10 cm + 6 cm + 4 cm) / 2 = 10 cm
Area_1 = √(s(s-a)(s-b)(s-c)) = √(10 cm × 4 cm × 4 cm × 6 cm) ≈ 24 cm²
Similarly, we can find the areas of the other triangles:
Area_2 = √(10 cm × 5 cm × 5 cm × 2 cm) ≈ 31.62 cm²
Area_3 = √(11 cm × 5 cm × 5 cm × 1 cm) ≈ 27.46 cm²
Area_4 = √(5 cm × 4 cm × 4 cm × 1 cm) ≈ 8.94 cm²
Area_5 = √(8 cm × 6 cm × 6 cm × 2 cm) ≈ 51.96 cm²
The total area of the pentagon is the sum of the areas of the triangles:
Area_pentagon = Area_1 + Area_2 + Area_3 + Area_4 + Area_5 ≈ 144.98 cm²
Therefore, the area of the pentagon is approximately 144.98 square centimeters.
If x = 8 units, y = 3 units, and h = 10 units, find the area of the trapezoid shown above using decomposition.
To find the area of the trapezoid, we can use the formula:
Area = (1/2) x (base_1 + base_2) x height
The two bases of the trapezoid can be decomposed into two congruent triangles with a shared vertex at the center of the trapezoid, as shown below:
```
x
┌────────┐
/ \ / \
h / \ / \
/ \ / y \
────────┐/──────\┌────────
y _/_\ /_\_ y
x \ / x
3 8
```
Each triangle can be split in half by a line from the shared vertex to the midpoint of the opposite side, as shown:
```
x
┌────────┐
/ \ / \
h / \ h / \
/_____\/__y___\
┌──────┐┌──────┐
x w x w
```
The base of each triangle is y + w, where w is half of x, or 4 units. Using the Pythagorean theorem, we can find w:
w^2 + h^2 = x^2/4 + y^2
w^2 = x^2/4 + y^2 - h^2
w^2 = 8^2/4 + 3^2 - 10^2
w^2 = -41
Since the square of a real number cannot be negative, we have no real value for w, thus no real solution exists.
Therefore, there must be an error in the problem statement or in the calculations.