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QuestionA rectangle's width is 49 inches more than its height. If the diagonal is 61 inches, find the height rounded to 1 decimal place.

Studdy Solution

STEP 1

Assumptions1. The rectangle's width is49 inches longer than the height. . The rectangle's diagonal measurement is61 inches.
3. The relationship between the sides of a rectangle and its diagonal follows the Pythagorean theorem.

STEP 2

Let's denote the height of the rectangle as hh (in inches). Then, the width of the rectangle will be h+49h +49 inches.

STEP 3

According to the Pythagorean theorem, the square of the length of the diagonal of a rectangle is equal to the sum of the squares of its sides. We can write this asDiagonal2=Height2+Width2Diagonal^2 = Height^2 + Width^2

STEP 4

Now, plug in the given values for the diagonal, height, and width to form the equation.
612=h2+(h+49)261^2 = h^2 + (h +49)^2

STEP 5

implify the equation.
3721=h2+h2+98h+24013721 = h^2 + h^2 +98h +2401

STEP 6

Combine like terms.
3721−2401=2h2+98h3721 -2401 =2h^2 +98h

STEP 7

Further simplify the equation.
1320=2h2+98h1320 =2h^2 +98h

STEP 8

Divide the entire equation by2 to simplify it further.
660=h2+49h660 = h^2 +49h

STEP 9

This is a quadratic equation in the form ax2+bx+c=ax^2 + bx + c =. To solve it, we can complete the square. First, rearrange the equation to the form ax2+bx=−cax^2 + bx = -c.
h2+49h=−660h^2 +49h = -660

STEP 10

To complete the square, add the square of half the coefficient of hh to both sides of the equation.
h2+49h+(49/2)2=−660+(49/2)2h^2 +49h + (49/2)^2 = -660 + (49/2)^2

STEP 11

implify the equation.
h+49h+600.25=600.25h^ +49h +600.25 =600.25

STEP 12

The left-hand side of the equation is now a perfect square. Rewrite it as such.
(h+24.5)2=600.25(h +24.5)^2 =600.25

STEP 13

Take the square root of both sides of the equation.
h+24.5=±600.25h +24.5 = \pm \sqrt{600.25}

STEP 14

olve for hh.
h=−24.±600.25h = -24. \pm \sqrt{600.25}

STEP 15

Since hh represents a length, it cannot be negative. Therefore, we only consider the positive root.
h=−24.5+600.25h = -24.5 + \sqrt{600.25}

STEP 16

Calculate the value of hh.
h=−24.5+24.5=0h = -24.5 +24.5 =0The height of the rectangle is0 inches.

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