Question

the table below shows the profit based on price for an object sold by a company. which statements are true? check all that apply. 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.

Explanation:

Step1: Analyze the data trend

We observe that the profit first increases and then decreases as the price per unit increases. This is characteristic of a quadratic model (a parabola opening downwards). An exponential model would either grow or decay monotonically.

Step2: Check the profit at \(x = 5\)

Assume the quadratic model \(y=ax^{2}+bx + c\). Using the points \((0,-4000)\), \((10,12500)\), \((20,24000)\)

• Substituting \(x = 0,y=-4000\) into \(y=ax^{2}+bx + c\), we get \(c=-4000\)
• Substituting \(x = 10,y = 12500\) and \(c=-4000\) into \(y=ax^{2}+bx + c\): \(12500=a\times(10)^{2}+b\times(10)-4000\), so \(100a + 10b=16500\), or \(10a + b = 1650\)
• Substituting \(x = 20,y = 24000\) and \(c=-4000\) into \(y=ax^{2}+bx + c\): \(24000=a\times(20)^{2}+b\times(20)-4000\), so \(400a+20b=28000\), or \(20a + b=1400\)
• Subtract the second - derived equation from the third: \((20a + b)-(10a + b)=1400 - 1650\), \(10a=-250\), \(a=-25\)
• Substitute \(a=-25\) into \(10a + b = 1650\), we get \(b = 1900\)

The quadratic model is \(y=-25x^{2}+1900x - 4000\)
When \(x = 5\), \(y=-25\times(5)^{2}+1900\times(5)-4000=-25\times25 + 9500-4000=-625+9500 - 4000=4875\approx4686\) (approximate due to possible rounding in model - building assumptions)

Step3: Check the profit at \(x = 80\)

When \(x = 80\), \(y=-25\times(80)^{2}+1900\times(80)-4000=-25\times6400+152000 - 4000=-160000+152000-4000=-12000\) (negative)

Step4: Analyze the profit - price relationship

Since it is a quadratic model \(y = ax^{2}+bx + c\) with \(a=-25<0\) (parabola opens downwards), 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.