Question

A fruit company recently released a new applesauce. By the end of its first​ year, profits on this product amounted to $25,000. The anticipated profit for the end of the fourth year is $72,700. After the first​ year, the ratio of change in time to change in profit is constant. Let ( x ) be years and ( y ) be profit in dollars. *HINT: The line must pass through (1, 25000) and (4, 72700)!* Write a linear function (in slope-intercept form) that expresses profit as a function of time. Then, use this function to predict at the end of what year will the company’s profit reach $104,500.

Answer 1

The linear function expressing profit as a function of time is y = 15900x + 9100. Using this function, it is predicted that the company's profit will reach $104,500 by the end of the 6th year.

Step-by-step explanation:

To find a linear function that expresses profit (y) as a function of time (x), we need to use two points through which the line passes: (1, 25000) and (4, 72700). The slope (m) of the line is the change in profit divided by the change in time: m = (72700 - 25000) / (4 - 1) = 47700 / 3 = 15900. Therefore, the slope of the function is 15900.

The slope-intercept form of a linear equation is y = mx + b, where b is the y-intercept. Plugging in the slope and one point into the equation, we can find the y-intercept: 25000 = 15900(1) + b, resulting in b = 9100. Hence, the linear function is y = 15900x + 9100.

To predict when the company's profit will reach $104,500, set y to 104500 and solve for x: 104500 = 15900x + 9100. Subtracting 9100 from both sides gives 95400 = 15900x, and dividing by 15900 results in x ≈ 6. So, the company's profit is expected to reach $104,500 by the end of the 6th year.