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QuestionA rectangle has a width 1 inch longer than its height. Its diagonal is 29 inches. Find the height rounded to 1 decimal place.

Studdy Solution

STEP 1

Assumptions1. The rectangle's width is1 inch longer than the height. . The rectangle's diagonal measurement is29 inches.
3. We are looking for the height of the rectangle.

STEP 2

We can use the Pythagorean theorem to find the height of the rectangle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In our case, the diagonal of the rectangle is the hypotenuse and the height and width are the other two sides.
Diagonal2=Height2+Width2Diagonal^2 = Height^2 + Width^2

STEP 3

Now, plug in the given values for the diagonal and the relationship between the width and height to set up the equation.
292=Height2+(Height+1)229^2 = Height^2 + (Height +1)^2

STEP 4

implify the equation by squaring the diagonal and expanding the right side of the equation.
841=Height2+Height2+2∗Height+1841 = Height^2 + Height^2 +2*Height +1

STEP 5

Combine like terms on the right side of the equation.
841=2∗Height2+2∗Height+1841 =2*Height^2 +2*Height +1

STEP 6

Subtract841 from both sides to set the equation to0, which will allow us to solve for Height.
0=2∗Height2+2∗Height+1−8410 =2*Height^2 +2*Height +1 -841

STEP 7

implify the equation.
0=2∗Height2+2∗Height−8400 =2*Height^2 +2*Height -840

STEP 8

Divide the entire equation by2 to simplify further.
0=Height2+Height−4200 = Height^2 + Height -420

STEP 9

This is a quadratic equation in the form of ax2+bx+c=ax^2 + bx + c =. We can solve it using the quadratic formula, which is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}.

STEP 10

Plug in the values for a, b, and c into the quadratic formula.
Height=−±2−4∗∗(−420)2∗Height = \frac{- \pm \sqrt{^2 -4**(-420)}}{2*}

STEP 11

implify the equation inside the square root.
Height=−±+1680Height = \frac{- \pm \sqrt{ +1680}}{}

STEP 12

implify the equation further.
Height=−±1682Height = \frac{- \pm \sqrt{168}}{2}

STEP 13

Calculate the square root.
Height=−±412Height = \frac{- \pm41}{2}

STEP 14

We have two possible solutions for the height. However, since the height cannot be negative, we discard the negative solution.
Height=−+412Height = \frac{- +41}{2}

STEP 15

implify the equation to find the height.
Height=402=20Height = \frac{40}{2} =20The height of the rectangle is20 inches.

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