A roll of carpet 9 feet wide is 30 feet longer than a roll of carpet 15 feet wide. Both rolls have the same area. Find the dimensions of each.
Let the length of the shorter rug be l feet
then the length of the longer rug is l+30 feet
Doesn't it say that
area of shorter = area of longer ?
15l = 9(l+30)
15l = 9l + 270
6l = 270
l = 45
so the two rugs are 15 by 45 and 9 by 75
check: 15x45 = 675
9x75 = 675
To solve this problem, let's begin by assigning variables to the dimensions of the carpet rolls.
Let's represent the width of the 9-foot wide roll as x and the length of the roll as y.
The width of the 15-foot wide roll is given as 15 feet. Since we know that the width of the 9-foot wide roll is 9 feet, we can express the relationship between the widths as follows:
x = 15 feet (width of the 15-foot wide roll)
x + 30 = 9 feet (width of the 9-foot wide roll is 30 feet longer)
Now, we need to consider the area of the rolls. The area of a rectangle can be calculated by multiplying the length by the width. Since both rolls have the same area, we can set up the following equation:
x * y = (x + 30) * 15
Now, we can solve this equation to determine the values of x and y.
Expanding the equation, we get:
xy = 15x + 450
Rearranging the equation, we have:
xy - 15x = 450
Factoring out the common term of x, we get:
x(y - 15) = 450
Dividing both sides of the equation by (y - 15), we find:
x = 450 / (y - 15)
Now, we can substitute this expression for x in the equation x + 30 = 9 to find y.
450 / (y - 15) + 30 = 9
Multiplying through by (y - 15) to eliminate the denominator, we have:
450 + 30(y - 15) = 9(y - 15)
Distributing and simplifying, we get:
450 + 30y - 450 = 9y - 135
Combining like terms, we obtain:
30y = 9y - 135
Simplifying further, we have:
21y = -135
Dividing both sides of the equation by 21, we find:
y = -135 / 21
Simplifying, we get:
y = -6.43
Since we can't have a negative length for the carpet, it indicates that there is an error in the problem or the given information. Please check the information provided and make any necessary adjustments for a valid solution.