Which has a larger area, a 4:3 aspect ratio 32 inch TV or a 16:9 aspect ratio 32 inch TV? Find the side lengths of each of the TV’s an

Question

Which has a larger area, a 4:3 aspect ratio 32 inch TV or a 16:9 aspect ratio 32 inch
TV? Find the side lengths of each of the TV’s and the area of each TV to compare.
Explain your reasoning and show all mathematical calculations.

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niczorrrr 5 years 2021-08-14T22:51:09+00:00 1 Answers 280 views 0

Answers ( )

  1. Sigrid
    0
    2021-08-14T22:52:56+00:00
    Reply

    Answers:

    • The 4:3 tv has the larger area
    • The 4:3 tv has width = 25.6 inches and height = 19.2 inches. The area is exactly 491.52 square inches
    • The 16:9 tv has width = 27.8904 inches (approximate) and height = 15.68835 inches (approximate). The area is approximately 437.55 square inches.

    ================================================

    Explanation:

    Let x be some positive real number

    The 4:3 aspect ratio means the width (horizontal) portion of the tv is 4x inches while the height (vertical) portion is 3x inches

    The ratio 4x:3x reduces to 4:3 after dividing both parts by x.

    The 4x by 3x rectangle has the diagonal 32 inches as the instructions state. The tv size is always measured along the diagonal.

    So effectively, we have two identical right triangles with legs 4x and 3x, and hypotenuse 32.

    Apply the pythagorean theorem to find x

    a^2+b^2 = c^2

    (4x)^2+(3x)^2 = 32^2

    16x^2+9x^2 = 1024

    25x^2 = 1024

    x^2 = 1024/25

    x = sqrt(1024/25)

    x = 32/5

    x = 6.4

    Recall that x is positive, so we ignore the negative square root here.

    This 4x by 3x tv then has dimensions of

    • horizontal width = 4x = 4*6.4 = 25.6 inches
    • vertical height = 3x = 3*6.4 = 19.2 inches

    These values are exact.

    The area is therefore base*height = 25.6*19.2 = 491.52 square inches

    This of course only applies to the 4:3 tv that’s 32 inches in diagonal.

    ——————————-

    Now onto the 16:9 tv.

    We’ll follow the same steps as the last section. We’ll use y this time

    The 16:9 ratio becomes 16y:9y

    a^2+b^2 = c^2

    (16y)^2 + (9y)^2 = 32^2

    256y^2 + 81y^2 = 1024

    337y^2 = 1024

    y^2 = 1024/337

    y = sqrt(1024/337)

    y = 1.7431510742491 which is approximate

    y = 1.74315

    So,

    • horizontal width = 16y = 16*1.74315 = 27.8904
    • vertical height = 9y = 9*1.74315 = 15.68835
    • area = base*height = 27.8904*15.68835 = 437.55435684

    the area is approximate since the width and height are approximate. It rounds to about 437.55 square inches

    ——————————-

    To recap, we found the following:

    • The 4:3 tv has width and height of 25.6 inches and 19.2 inches respectively. Those values are exact. The area is exactly 491.52 square inches.
    • The 16:9 tv has width and height of approximately 27.8904 inches and 15.68835 inches respectively. The area is approximately 437.55 square inches.
    • The 4:3 tv has larger area.

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