Please EXPLAIN the steps on how to solve these problems. Please provide detailed steps.
A rectangle picture is 12 by 16 inches. If a frame of uniform width contains an area of 165 square inches, what is the width of the frame?
The length of a rectangular garden is 6 feet more than its width. A walkway is 3 feet wide surrounds the outside of the garen. The total area of the walkway is 288 square feet. Find the dimensions of the garden.
The first problem:
Sketch this problem to make it easier to visualize. There is a rectangular frame with the inside dimensions of 12 * 16 inches. The frame has a width of W. You need to divide the frame into pieces and then write an equation for those pieces to equal 165 in^2. One approach would be to extend the lines of the picture to form 4 rectangles of the frame along the edges and 4 squares at the corners.
W
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| | | | W
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| | | |
| | 12 | |
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| | 16 | |
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| | | | W
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W
There are 2 rectangles that are W * 16.
There are 2 rectangles that are W * 12.
There are 4 corner squares that are W * W.
Add these 8 items and set the total to 165. Then solve for W. You will need to use the quadratic formula.
Sorry, the diagram didn't survive the font change.
For the rectangle problem, let w be the frame width. 165 in^2 is considered to be the frame area, not including the picture hole in the middle.
165 + (12*16) = (12+2w)(16+2w)
= 192 + 56w + 4w^2
4w^2 + 56w -165 = 0
(2w -5)(2w+33) = 0
w = 5/2 inch (ignore the negative root)
Use a similar approach for the second problem. Someone will be glad to critique your work.
To solve the first problem, finding the width of the frame, you can follow these steps:
1. Start by determining the area of the rectangle picture. Use the formula for the area of a rectangle, which is length multiplied by width. In this case, the length is 12 inches and the width is 16 inches, so the area of the picture is 12 * 16 = 192 square inches.
2. Next, you need to subtract the area of the picture from the total area that the frame contains. This will give you the area of the frame itself. In this case, the area of the frame is given as 165 square inches, so subtracting the area of the picture (192 square inches) from it would give you 165 - 192 = -27 square inches.
3. Since a negative area doesn't make sense in this context, it indicates that there is an error in the problem or the given areas are not consistent. Double-check the information and the areas provided to ensure accuracy.
For the second problem, finding the dimensions of the garden, you can follow these steps:
1. Let's assume the width of the garden is "W." According to the problem, the length of the garden is 6 feet more than its width. So, the length can be represented as "W + 6."
2. First, calculate the area of the garden by multiplying its length and width. The area of a rectangle is length multiplied by width. In this case, the area of the garden is W * (W + 6).
3. Now, let's add the walkway surrounding the garden to this area. The walkway has a width of 3 feet, so we need to add 3 feet to both the length and width of the garden. This gives us the new length as (W + 6 + 2 * 3) = (W + 12) and the new width as (W + 2 * 3) = (W + 6).
4. Calculate the area of the garden with the walkway included. This area is equal to the area of the garden plus the total area of the walkway, which is given as 288 square feet. So, the equation becomes (W + 12) * (W + 6) = 288.
5. Simplify and solve the quadratic equation. Expand the equation to get W^2 + 18W + 72 = 288.
6. Bring all terms to one side of the equation, resulting in W^2 + 18W + 72 - 288 = 0.
7. Combine like terms to get W^2 + 18W - 216 = 0.
8. Solve for W by factoring, completing the square, or using the quadratic formula. In this case, the quadratic can be factored as (W - 6)(W + 36) = 0.
9. Set each factor equal to zero and solve for W. So, we have two possible solutions: W - 6 = 0 (W = 6) or W + 36 = 0 (W = -36). Since a negative width doesn't make sense, we take W = 6 as the valid width.
10. With the width known, we can find the length by adding 6 feet to it. So, the length of the garden is 6 + 6 = 12 feet.
Therefore, the dimensions of the garden are 6 feet (width) by 12 feet (length).