The area of a rectangular plot 24 feet long and 16 feet wide will be doubled by adding an equal distance to each side of the plot. What is the distance added to each side?
solve
(24+x)(16+x) = 2(24)(16)
You of course will have to know how to solve quadratic equations.
To find the distance added to each side of the plot, we need to first calculate the current area of the plot and then determine the amount by which it needs to be increased.
The current area of the rectangular plot can be calculated by multiplying its length by its width. Therefore, the current area is:
Area = Length × Width = 24 feet × 16 feet = 384 square feet
To double this area, we need to determine the amount by which it needs to be increased. We know that the new area is twice the current area, so:
New Area = 2 × Current Area
New Area = 2 × 384 square feet
New Area = 768 square feet
Now, we have the new area and need to find the distance added to each side to achieve this. Since the length of the plot is increased by an equal distance on both sides, the new length is:
New Length = Current Length + 2 × Added Distance
Similarly, the new width is:
New Width = Current Width + 2 × Added Distance
We can substitute the current dimensions and solve for the added distance:
New Area = New Length × New Width = (Current Length + 2 × Added Distance) × (Current Width + 2 × Added Distance)
768 square feet = (24 feet + 2 × Added Distance) × (16 feet + 2 × Added Distance)
Now we can solve this quadratic equation to find the value of the added distance:
768 square feet = (24 feet + 2 × Added Distance) × (16 feet + 2 × Added Distance)
768 square feet = 384 square feet + 48 feet × Added Distance + 4 × (Added Distance)^2
Rearranging the equation and simplifying, we get a quadratic equation:
4 × (Added Distance)^2 + 48 feet × Added Distance + (384 square feet - 768 square feet) = 0
4 × (Added Distance)^2 + 48 feet × Added Distance - 384 square feet = 0
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Plugging the values into the quadratic formula:
Added Distance = (-48 feet ± √((48 feet)^2 - 4 × 4 × -384 square feet)) / (2 × 4)
Simplifying further, we get:
Added Distance = (-48 feet ± √(2304 square feet + 6144 square feet)) / 8
Added Distance = (-48 feet ± √8448 square feet) / 8
Added Distance = (-48 feet ± √(16 × 528) feet) / 8
Added Distance = (-48 feet ± (4 × √528) feet) / 8
Added Distance = -6 feet ± √528 feet
Therefore, the distance added to each side of the plot can be either -6 feet + √528 feet or -6 feet - √528 feet. However, since distance cannot be negative in this context, the answer is -6 feet + √528 feet.