Question

"Mike just bought a new 60" television for his parents for his apartment. The opening in the entertainment center is only 55 inches by 30 inches. If the new television has an aspect ratio of 16:9, will the new TV fit in the old entertainment center? Justify your answer by finding the width and height of the new television."

Will the new TV fit in the old entertainment center, and how do you justify your answer?

a) Yes, it will fit; the width and height match the dimensions of the opening.
b) No, it won't fit; the aspect ratio is incorrect.
c) It depends on the diagonal size of the TV.

Answer 1

To determine if the new TV will fit in the old entertainment center, compare the dimensions of the TV and the opening in the center. Calculate the diagonal of the TV using the aspect ratio and the Pythagorean theorem. Compare the diagonal of the TV to the diagonal of the opening to determine if the TV will fit.

Step-by-step explanation:

To determine if the new TV will fit in the old entertainment center, we need to compare the dimensions of the TV and the opening in the center. The aspect ratio of the TV is 16:9, which means that for every 16 units of width, there are 9 units of height. Let's assume the width of the TV is 16x and the height is 9x. Using the Pythagorean theorem, we can find the diagonal of the TV, which is √((16x)² + (9x)²). We can then compare the diagonal of the TV to the diagonal of the opening in the center to see if the TV will fit.

Let's calculate the diagonal of the TV:

Diagonal = √((16x)² + (9x)²) = √(256x² + 81x²) = √(337x²)

The opening in the center has a width of 55 inches and a height of 30 inches. Let's calculate the diagonal of the opening:

Diagonal = √(55² + 30²) = √(3025 + 900) = √(3925)

So the diagonal of the TV is √(337x²) and the diagonal of the opening is √(3925). We can compare these values to determine if the TV will fit.