A rectangle is drawn so the width is 7 inches longer than the height. If the rectangle's diagonal measurement is 46 inches, find the height.
Let's assume the height of the rectangle is h inches.
According to the problem, the width of the rectangle is 7 inches longer than the height. So, the width can be represented as h + 7 inches.
Using the Pythagorean theorem, we can relate the height, width, and diagonal of the rectangle. The theorem states that for any right-angled triangle, the square of the length of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides.
In this case, the diagonal is 46 inches, the height is h inches, and the width is h + 7 inches.
Applying the Pythagorean theorem, we get:
(h^2) + ((h + 7)^2) = (46^2)
Expanding the equation:
h^2 + (h^2 + 14h + 49) = 2116
Combining like terms:
2h^2 + 14h + 49 = 2116
Rearranging the equation:
2h^2 + 14h - 2067 = 0
Simplifying the equation is a bit complex because it cannot be factored easily. To solve this equation, we can use the quadratic formula. For a quadratic equation of the form ax^2 + bx + c = 0, the quadratic formula states that the solutions are given by:
h = (-b ± √(b^2 - 4ac)) / (2a)
In our case,
a = 2, b = 14, and c = -2067.
Using the quadratic formula, we can calculate the two possible values for h.
To find the height of the rectangle, we can set up a Pythagorean theorem equation using the given information about the rectangle's width, height, and diagonal.
Let's denote the height of the rectangle as "h" inches. According to the problem, the width of the rectangle is 7 inches longer than the height, which means its width is h + 7 inches.
According to the Pythagorean theorem, the square of the length of the hypotenuse (diagonal) of a right triangle is equal to the sum of the squares of the other two sides. In this case, we can consider the height, width, and diagonal of the rectangle as the three sides of a right triangle.
Using this information, we can set up the equation as follows:
h^2 + (h + 7)^2 = 46^2
Next, we simplify and solve the equation:
h^2 + (h^2 + 14h + 49) = 2116
Combining like terms:
2h^2 + 14h + 49 = 2116
Rearranging the equation:
2h^2 + 14h - 2067 = 0
Since this is a quadratic equation, we can solve it by factoring, completing the square, or using the quadratic formula. Factoring may be challenging in this case, so we'll use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
Using the quadratic formula where:
a = 2
b = 14
c = -2067
Substituting the values into the formula:
h = (-14 ± √(14^2 - 4 * 2 * -2067)) / (2 * 2)
Simplifying further:
h = (-14 ± √(196 + 16536)) / 4
Calculating inside the square root:
h = (-14 ± √16732) / 4
Now, we can calculate the square root:
h = (-14 ± 129.378) / 4
Simplifying further:
h ≈ (-14 + 129.378) / 4 or h ≈ (-14 - 129.378) / 4
h ≈ 115.378 / 4 or h ≈ -143.378 / 4
h ≈ 28.845 or h ≈ -35.8445
Since a height cannot be negative in this context, we discard the negative solution.
Therefore, the height of the rectangle is approximately 28.845 inches.
Use the Pythagorean theorem:
a^2 + b^2 = c^2
x = height
x + 7 = width
Therefore:
x^2 + (x + 7)^2 = 46^2
Expand and try to factor. If this does not factor easily, use the quadratic formula to solve.