you are using tiles to make a picture frame for a square photograph with sides 10 inches long. You want the frame to form a uniform border around the photograph, You have enough tiles to cover 300 square inches. What is the largest possible frame width?
let the width of the border be x inches
then
(10+2x)^2 - 100 ≤ 300
100 + 40x + 4x^2 - 100 - 300 ≤ 0
x^2+ 10x - 75 ≤ 0
(x+15)(x-5) ≤ 0
x = 5
So the largest frame is 20 by 20
check: Area of whole frame = 400
area of photo = 100
area of tiles = 300
The largest possible frame width is 17 inches long per side.
To find the largest possible frame width, we need to determine the dimensions of the inner square (photograph) and subtract them from the dimensions of the outer square (frame).
Given:
- Length of sides of the square photograph = 10 inches
- Total area of the frame = 300 square inches
Let's proceed step by step:
Step 1: Calculate the area of the photograph.
Since the photograph is square, its area is simply the square of its side length:
Area of photograph = side length * side length = 10 * 10 = 100 square inches
Step 2: Calculate the area of the frame.
The area of the frame is equal to the total area (300 square inches) minus the area of the photograph:
Area of frame = Total area - Area of photograph = 300 - 100 = 200 square inches
Step 3: Calculate the side length of the frame.
Since the frame is a square, we can find the side length by taking the square root of the area:
Side length of frame = √(Area of frame) = √(200) ≈ 14.14 inches
Step 4: Calculate the frame width.
The frame width is the difference between the side length of the frame and the side length of the photograph:
Frame width = Side length of frame - Side length of photograph = 14.14 - 10 = 4.14 inches
Therefore, the largest possible frame width is approximately 4.14 inches.
To find the largest possible frame width, we need to consider the dimensions of the photograph and the available tiles.
Given:
- Square photograph with sides 10 inches long.
- Available tiles to cover 300 square inches.
First, we need to determine the area of the photograph:
Area = side * side = 10 inches * 10 inches = 100 square inches.
Next, we need to find the area of the frame, which is the difference between the area of the photograph and the available tiles:
Frame Area = Available Tiles Area - Photograph Area = 300 square inches - 100 square inches = 200 square inches.
Since the frame forms a uniform border around the photograph, the frame's width would be the same on all sides. Let's denote the frame's width as x.
To calculate the area of the frame, we need to subtract the area of the photograph from the combined area of the photograph and the frame:
Frame Area = (10 + 2x) * (10 + 2x) - 100 square inches
Expanding the equation:
200 square inches = (100 + 20x + 4x + 4x^2) - 100 square inches
Simplifying the equation:
200 square inches = 4x^2 + 24x
Rearranging the equation:
4x^2 + 24x - 200 = 0
Dividing the entire equation by 4 to simplify:
x^2 + 6x - 50 = 0
Now we can solve the quadratic equation using factoring, completing the square, or the quadratic formula.
In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 6, and c = -50.
Plugging in these values:
x = (-6 ± √(6^2 - 4(1)(-50))) / (2 * 1)
Simplifying further:
x = (-6 ± √(36 + 200)) / 2
x = (-6 ± √(236)) / 2
x = (-6 ± √236) / 2
x = (-6 ± √(4 * 59)) / 2
x = (-6 ± 2√59) / 2
x = -3 ± √59
Since we are looking for the largest possible frame width, we take the positive value of x:
x = -3 + √59
Hence, the largest possible frame width is approximately -0.35 inches. However, a negative width doesn't make sense in this context, so we take the positive value of 0.35 inches.
Therefore, the largest possible frame width is approximately 0.35 inches.