The dimensions of a door are 3 ft wide by 7 ft 6 in high. If the same amount is added to each dimension of the door, the area is increased by 5.5 ft2. Find the dimensions of the new door. (Round your answer to one decimal place.)
(3+x)(7.5+x) = 3*7.5 + 5.5
solve for x, and then the new dimensions (in feet) are the factors on the left side above.
can you show to do it
yeah, but you clearly need the practice. expand the left side, collect terms, and then just solve the quadratic equation, using the quadratic formula.
are the answers:
x=−0.218847
x=−10.2812
not likely to have negative dimensions for the door, are you?
Once you get an answer, check to see whether it makes sense!
Specifically, they said x was added to each dimension.
I get x = 0.5
How did you get your answers?
To solve this problem, we can set up an equation using the given information.
Let x be the amount added to each dimension of the door.
The original width of the door is 3 ft, so the new width would be 3 + x ft.
The original height of the door is 7 ft 6 in, which is equivalent to 7.5 ft. So the new height would be 7.5 + x ft.
The area of the original door can be calculated by multiplying the width and height:
Area = width * height
Area = 3 ft * 7.5 ft
Area = 22.5 ft^2
The area of the new door can be calculated in the same way:
New Area = (width + x) * (height + x)
According to the problem, the new area is increased by 5.5 ft^2. So we can set up the equation:
New Area - Original Area = 5.5 ft^2
(width + x) * (height + x) - Area = 5.5 ft^2
Substituting the values we know, we get:
(3 + x) * (7.5 + x) - 22.5 = 5.5
Now we can solve this equation to find the value of x, and then use it to determine the dimensions of the new door.
Expanding the equation:
(22.5 + 3x + 7.5x + x^2) - 22.5 = 5.5
Combine like terms:
x^2 + 10.5x + 22.5 - 22.5 = 5.5
Simplifying:
x^2 + 10.5x = 5.5
Move 5.5 to the left side:
x^2 + 10.5x - 5.5 = 0
Now we can solve this quadratic equation. We can factor it or use the quadratic formula:
After solving the quadratic equation, we find that x ≈ -11.9 or x ≈ 0.4.
Since the dimensions cannot have a negative value, the new dimensions can be found by adding 0.4 ft to each dimension of the original door.
The new width would be 3 + 0.4 = 3.4 ft, and the new height would be 7.5 + 0.4 = 7.9 ft.
Therefore, the dimensions of the new door are approximately 3.4 ft wide by 7.9 ft high.