Answer:
The new width of the map after enlargement would be (8/5) * 3 = 4.8 inches (assuming the aspect ratio is maintained).
So, the perimeter would be (4.8 + 8) * 2 = 25.6 inches.
The area would be 4.8 * 8 = 38.4 square inches.
Explanation:
Here's an explanation of how to approach problems like these, along with some examples of easy and hard math problems:
When enlarging a map, you need to consider how the scale of the map changes. In this case, the map is being enlarged from 5 inches to 8 inches in length, which is a scale factor of 8/5. This means that every distance on the original map will be scaled up by a factor of 8/5 on the enlarged map.
To find the perimeter and area of the enlarged map, you'll need to use this scale factor to determine the new dimensions of the map. The new width of the map will be the same as the original width, which is 3 inches. The new length of the map will be 8 inches, since that is the length to which the map is being enlarged. So the new dimensions of the map are 3 inches by 8 inches.
To find the new perimeter, you just need to add up the lengths of the four sides of the map: P = 2w + 2l. Using the new dimensions of the map, we get:
P = 2(3) + 2(8) = 6 + 16 = 22 inches
To find the new area, you just need to multiply the new length and width of the map: A = wl. Using the new dimensions of the map, we get:
A = 3(8) = 24 square inches
Examples:
Easy:
The index card is 3 inches wide and 5 inches long, so the perimeter is:
P = 2(3 in) + 2(5 in) = 6 in + 10 in = 16 in
To find the area, we use the formula:
A = L x W
A = 3 in x 5 in = 15 square inches
Now we want to enlarge the map to 8 inches long, but we're not told how wide it will be after enlargement. Let's assume that the width also increases by the same proportion as the length. That means the new dimensions will be:
Length = 8 in
Width = (8/5) x 3 in = 4.8 in
The perimeter will now be:
P = 2(8 in) + 2(4.8 in) = 16 in + 9.6 in = 25.6 in
The area will be:
A = 8 in x 4.8 in = 38.4 square inches
Hard:
A rectangular garden has an area of 320 square feet. If the width of the garden is 10 feet less than the length, find the dimensions of the garden and the length of the diagonal.
Let's use the formula for the area of a rectangle:
A = L x W
We know that the area is 320 square feet, so we can write:
320 = L x (L - 10)
Expanding the right side gives:
320 = L^2 - 10L
Moving all the terms to one side gives:
L^2 - 10L - 320 = 0
We can solve for L using the quadratic formula:
L = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 1, b = -10, and c = -320, so:
L = (10 ± sqrt(10^2 - 4(1)(-320))) / 2(1)
L = (10 ± sqrt(1440)) / 2
L = (10 ± 12√10) / 2
We can simplify this to:
L = 5 + 6√10 or L = 5 - 6√10
The dimensions of the garden are either:
Length = 5 + 6√10 feet
Width = 5 - 6√10 feet
or
Length = 5 - 6√10 feet
Width = 5 + 6√10 feet
We can check that the area is 320 square feet for either set of dimensions.
To find the length of the diagonal, we can use the Pythagorean theorem:
d^2 = L^2 + W^2
We already know the values of L and W for both cases, so we can calculate:
d = sqrt((5 + 6√10)^2 + (5 - 6√10)^2) = 10 sqrt(2)
or
d = sqrt((5 - 6√10)^2 + (5 + 6√10)^2) = 10 sqrt(2)
So the length of the diagonal is 10 times the square root of 2.
Hope this helped! Sorry if it didn't, or it's wrong. If you need more help on questions like these, ask me! :]