A rectangle is 4 inches wide by 7 inches long. When the length and width are increased by the same amount, the area is incresed by 26 square inches. What are the dimensions of the new rectangle?
Let x = the increase.
(4+x)(7+x) - 4*7 = 26
Solve for x.
Let's assume the increase in both length and width is x inches.
The original rectangle has a width of 4 inches and a length of 7 inches. Therefore, its area is:
Area = Length × Width
Area = 7 × 4
Area = 28 square inches
After increasing the length and width by the same amount, the new rectangle's dimensions will be:
Width = 4 + x
Length = 7 + x
The area of the new rectangle is increased by 26 square inches, so:
New Area = Original Area + 26
(Width + x) × (Length + x) = 28 + 26
(4 + x) × (7 + x) = 54
Now we can solve this quadratic equation for x. Multiplying the brackets, we get:
4x + 7x + x^2 + 28 = 54
x^2 + 11x + 28 = 54
x^2 + 11x - 26 = 0
By factoring or using the quadratic formula, we find that:
(x + 13)(x - 2) = 0
Therefore, either x + 13 = 0 or x - 2 = 0. Solving for x gives us:
x = -13 or x = 2
Since a length or width cannot be negative, we discard the solution x = -13. The new dimensions of the rectangle are:
Width = 4 + x = 4 + 2 = 6 inches
Length = 7 + x = 7 + 2 = 9 inches
Therefore, the new rectangle is 6 inches wide and 9 inches long.
To find the dimensions of the new rectangle, we need to determine the amount by which both the length and width are increased. Let's denote this amount as "x".
Given that the original rectangle has a width of 4 inches and a length of 7 inches, its original area can be calculated by multiplying these two dimensions together: Area_1 = 4 inches * 7 inches.
Next, we need to calculate the new area of the rectangle by adding 26 square inches to the original area: Area_2 = Area_1 + 26 square inches.
Since increasing both the length and width by the same amount will result in a rectangular shape, the dimensions of the new rectangle will be the original dimensions (4 inches and 7 inches) plus the increase "x".
Using the new dimensions, we can calculate the new area of the rectangle: Area_2 = (4 inches + x) * (7 inches + x).
Setting the expressions for Area_2 equal, we then have the equation:
Area_1 + 26 square inches = (4 inches + x) * (7 inches + x).
Now, we can substitute the values we know: Area_1 = 28 square inches (since 4 inches * 7 inches = 28 square inches).
The equation now becomes:
28 + 26 square inches = (4 inches + x) * (7 inches + x).
Expanding the equation:
54 square inches = 28 square inches + 11 inches * x + 4 inches * x + x * x.
Combining like terms and simplifying:
26 square inches = 15 inches * x + x^2.
Rearranging the equation into a quadratic form:
x^2 + 15 inches * x - 26 square inches = 0.
Solving this quadratic equation will yield the value(s) of "x", which represents the increase in dimensions.
Using the quadratic formula: x = [ -15 inches ± √(15 inches)^2 - 4(1)(-26 square inches) ] / (2 * 1).
Calculating further: x = [ -15 inches ± √(225 inches^2 + 104 square inches) ] / 2 inches.
Simplifying: x = [ -15 inches ± √(50641 inches^2) ] / 2 inches.
Taking the positive square root: x = ( √50641 inches ) / 2 inches ≈ 35.64 inches / 2 inches ≈ 17.82 inches.
Since we are dealing with lengths of a rectangle, the increase should be positive. Therefore, x ≈ 17.82 inches.
Finally, we can find the dimensions of the new rectangle by adding this increase to the original dimensions:
New width = 4 inches + x ≈ 4 inches + 17.82 inches ≈ 21.82 inches.
New length = 7 inches + x ≈ 7 inches + 17.82 inches ≈ 24.82 inches.
Therefore, the dimensions of the new rectangle are approximately 21.82 inches wide by 24.82 inches long.