Algebra 2-Finding the diagonal.?

Question

A wide screen TV has a length to width ratio of 16 to 9. Use this information to help you answer the following questions:
1) Write an equation for the length and the width.

2) Write an equation for the diagonal of the wide screen.

3)The size of a tv display is determined by the length and width of the diagonal of the tv. If you have a 42-inch tv, it means the diagonal of the display is 42-inches. Round the following answers to the nearest ten-thousandth.
A) What is the length and width (in inches) of a 42-inch tv?
B) What is the length ans width (in inches) of a 55-inch tv?

4) What is the area of the display of a 42-inch tv? Round to the nearest hundredth.

5) If you have a 42-inch tv and would like to buy another that is doubled the display area of your 42-inch tv, would buying a 55-inch tv be good enough? If not, determine the minimum size tv you would buy. Explain your answer with supporting work.

Answer 1

5) If you'd like to double the display area, you would need 753.76 square inches × 2 = 1507.52 square inches.

A 55 diagonal is 2.996 times our original diagonal. Multiplying our original length and width, we get new dimensions of:

length 16 × 2.996 = 47.936 in

width 9 × 2.996 = 26.964 in

Area = 47.936 in × 26.964 in = 1292.546304 square inches.

So now we must do the extra work.

Remember, we want the viewing area to be 1507.52 square inches. Remember area is length × width.

So length * width must = 1507.52

16x × 9x = 1507.52

144x² = 1507.52

Divide each side by 14.

x² = 10.468888888

Apply square root each side.

x = 3.235566239 times the original ratios

So

The New Diagonal must be 18.357559750 × 3.235566239 = 59.39710055752528025

Basically you would need a 60-in diagonal big screen TV.

Answer 2

1).

Let length: 16x

And width: 9x

Answer 3

2) Must use the Pythagorean Theorem: (9x²) + (16x²) = d²
 

81x² + 256x² = d²

337x² = d²
 

18.357559750x = diagonal (if you square root both sides)

diagonal = 18.357559750x

 

Answer 4

3) A 42-in is 2.287885753880 times our original diagonal

So we must multiply our length and width by this ratio factor:

A) length = 16x = 16 × 2.287885753880 = 36.6062

width = 9x = 9 × 2.287885753880 = 20.5910

 

A 55-in is 2.996 times our original diagonal

So we must multiply our length and width by this ratio factor:

B) length = 16x = 16 × 2.99604090 = 47.9367 in

width = 9x = 9 × 2.99604090 = 26.9644 in

Answer 5

4) Area of display is length × width

= 36.6062 in × 20.5910 in

= 753.76 square inches