Question

Televisions are like pizzas; the price tends to go up with the area but not linearly. Prices in May 2021 at a national store were $190, $270, $348, and $498 for diagonal measurement 32-in., 43-in., 50-in., and 65-in. models of a particular brand and style of TV. The TVs have screens all the same shape, for which the area for diagonal measurement is approximately 0.45d²

a. What is the linear scaling factor for a 65-in. TV compared to a 32-in. one?

b. How many times as large in area is the 65-in. TV compared with the 32-in. one?

c. Which of the four models offers the lowest cost per square inch of screen?

Answer 1

A 65-in TV has a linear scaling factor of approximately 2.03 compared to a 32-in TV and is about 4.15 times larger in area. To find the TV with the lowest cost per square inch, divide the price by the area for each model. The relationship between size and price is not linear.

Step-by-step explanation:

Understanding TV Prices and Area Relationships

When examining the relationship between a television's price and its screen size, one must consider both the linear scaling and the area difference. To respond to the student's question regarding a 65-in. TV compared to a 32-in. one:

1. The linear scaling factor is the ratio of the diagonal measurements, which is 65/32 or approximately 2.03.
2. To find how many times as large in area a 65-in. TV is compared to a 32-in. one, we use the formula for area provided (0.45d²). For the 32-in. TV, the area is 0.45 × 32² = 460.8 square inches. For the 65-in., it is 0.45 × 65² = 1912.5 square inches. The ratio of their areas is 1912.5 / 460.8 ≈ 4.15.
3. Calculating the lowest cost per square inch involves dividing the price by the area for each model. For example, the 32-in. TV's cost per square inch is $190/460.8 ≈ $0.41. Following this calculation for each model will identify which offers the best value.

Important to note is that the relationship between screen size and price does not appear to be linear, as larger TVs can have additional features and technologies that disproportionately increase their price. To examine this further, regression analysis that incorporates the diagonal measurements and prices could be utilized, which may result in a formula with a base price and a multiplier per additional inch, such as the given formula d = -745.252 + 54.75569x.