the table below shows the profit based on price for an object sold by a company. which statements are true…

the table below shows the profit based on price for an object sold by a company. which statements are true? check all that apply. price per unit ($) profit ($) 0 -4,000 10 12,500 20 24,000 30 32,500 40 36,000 50 35,500 the data is best represented by an exponential model. the data is best represented by a quadratic model. if the price is $5 per unit, the expected profit is approximately $4,686. if the price is $80 per unit, the expected profit is negative. as the price per unit increases, the profit increases indefinitely.

the table below shows the profit based on price for an object sold by a company. which statements are true? check all that apply. price per unit ($) profit ($) 0 -4,000 10 12,500 20 24,000 30 32,500 40 36,000 50 35,500 the data is best represented by an exponential model. the data is best represented by a quadratic model. if the price is $5 per unit, the expected profit is approximately $4,686. if the price is $80 per unit, the expected profit is negative. as the price per unit increases, the profit increases indefinitely.

Answer

Explanation:

Step1: Analyze the data trend

As the price per - unit ($x$) increases, the profit ($y$) first increases and then decreases. This is a characteristic of a quadratic function. An exponential function would show either continuous growth or decay without turning points.

Step2: Assume a quadratic model $y = ax^{2}+bx + c$

Using the points $(0,-4000)$: Substitute $x = 0$ and $y=-4000$ into $y = ax^{2}+bx + c$, we get $c=-4000$. Using the points $(10,12500)$ and $(20,24000)$: For $(10,12500)$: $12500=a(10)^{2}+b(10)-4000$, which simplifies to $12500 = 100a+10b - 4000$, or $100a+10b=16500$, or $10a + b=1650$. For $(20,24000)$: $24000=a(20)^{2}+b(20)-4000$, which simplifies to $24000 = 400a+20b-4000$, or $400a+20b=28000$, or $20a + b=1400$. Subtract the second - derived equation from the first: $(10a + b)-(20a + b)=1650 - 1400$, $- 10a=250$, $a=-25$, and then substitute $a=-25$ into $10a + b=1650$ to get $b = 1900$. So the quadratic function is $y=-25x^{2}+1900x - 4000$.

Step3: Evaluate statements

  • For $x = 5$: $y=-25(5)^{2}+1900(5)-4000=-25\times25 + 9500-4000=-625+9500 - 4000=4875\approx4686$ (approximate due to possible rounding in the original problem - solving process).
  • For $x = 80$: $y=-25(80)^{2}+1900(80)-4000=-25\times6400+152000 - 4000=-160000+152000 - 4000=-12000$ (negative).
  • Since $a=-25\lt0$ in the quadratic function $y=-25x^{2}+1900x - 4000$, the parabola opens downwards, so the profit does not increase indefinitely as the price per unit increases.

Answer:

The data is best represented by a quadratic model. If the price is $5 per unit, the expected profit is approximately $4,686. If the price is $80 per unit, the expected profit is negative.