algebra questiion

Question

Cally plans to paint a retangular mural of a mountain waterfall image in the west side of her garage. The height of the mural will be 7 feet shorter than the width. The diagnonnal is 17 feet. Find the dimensions of the mural.

Answer 1

For the width of the mural we will use "w".

Therefore the height h = w -7.

Drawing a diagonal into a rectangle creates two right triangle that both have the that diagonal as their hypotenuse. A right triangle follows the Pythagorean Theorem of a² + b² = c²

Now we can put it together...

w² + (w-7)² = 17²

w² + (w-7)(w-7) = 17²

w² + w²-14w +49 = 17²

2w² -14w +49 = 17²

2w² -14w +49 = 289

2w² -14w -240 = 0

From the form ax² + bx +c = 0 (as used above) we get a = 2; b = -14 ; c = -240

Now we can use the quadratic formula to solve...

-b ± √(b² - 4ac)

x = -------------------------

2a

14 + √( 196 + 1920)

x = ------------------------------

4

14 + √2116

x = ------------------

4

14 + 46

x = ------------

4

60

x = ----- = 15

4

This "x" represents the "w" from our initial Pythagorean setup, which means our rectangle is 15 wide and 15 - 7 = 8 tall.

Verify...

a² + b² = c²

15² + 8² = 17²

225 + 64 = 289

Checks out!