Question

The sales of a small company were $26,000 in its second year of operation and $74,000 in its sixth year. Let y represent sales in the xth year of operation. Assume that the data can be approximated by a straight line. (a) Find the slope of the sales line, and give an equation for the line in the form y=mx+b (b) Use your answer from part (b) to find out how many years must pass before the sales surpass $110.000. (a) The slope is and the equation is y (Type integers or decimals.) (b) The sales will surpass $110,000 in years. (Round up to the nearest year.)

Answer 1

Explanation:

(a)Let x = number of years after the start of the businessLet y = sales in the xth year of operationUsing the given data, (x₁, y₁) = (2, 26,000) and (x₂, y₂) = (6, 74,000)We know that the equation of a line is of the form y = mx + b, where m is the slope and b is the y-intercept.To find the slope, we use the formula: m = (y₂ - y₁) / (x₂ - x₁)Substituting the values, we get: m = (74,000 - 26,000) / (6 - 2) = 12,000Therefore, the slope of the sales line is 12,000.Using the point-slope form of a line, the equation of the line is given by: y - y₁ = m(x - x₁)Substituting the values, we get: y - 26,000 = 12,000(x - 2)Simplifying the equation, we get: y = 12,000x + 2,000(b)We want to find the number of years it takes for the sales to surpass $110,000. Substituting y = 110,000 in the equation we found in part (a), we get:110,000 = 12,000x + 2,000Solving for x, we get: x = 9.23Therefore, it will take approximately 10 years for the sales to surpass $110,000 (rounded up to the nearest year).Answer: (a) The slope is 12,000 and the equation is y = 12,000x + 2,000. (b) The sales will surpass $110,000 in 10 years.

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