Question

A farmer notices that there is a linear relationship between the number of bean stalks, n, she plants and the yield, Y. When she plants 3 stalks, each plant yields 115 ounces of beans. When she plants 8 stalks, each plant yields 190 ounces of beans.

Answer 1

a) Write an equation that represents the linear relationship between the number of bean stalks and the yield.

b) Use the equation to find the yield when the farmer plants 15 stalks.

a) Let's use the point-slope form of the equation of a line to represent the linear relationship between the number of bean stalks and the yield:

Y - Y1 = m(n - n1)

where Y1 and n1 are the coordinates of a point on the line, m is the slope of the line.

We can use the two data points to find the slope of the line:

m = (Y2 - Y1) / (n2 - n1)
= (190 - 115) / (8 - 3)
= 15

Using the first data point (3, 115), we can find the equation of the line:

Y - 115 = 15(n - 3)

Expanding the right side:

Y - 115 = 15n - 45

Adding 115 to both sides:

Y = 15n + 70

Therefore, the equation that represents the linear relationship between the number of bean stalks and the yield is Y = 15n + 70.

b) To find the yield when the farmer plants 15 stalks, we substitute n = 15 into the equation:

Y = 15n + 70
Y = 15(15) + 70
Y = 295

Therefore, the yield when the farmer plants 15 stalks is 295 ounces of beans.

Answer 2

Explanation:

We can use the information given to find the equation of the line that represents the relationship between the number of bean stalks and the yield. The equation of a line is typically given by the slope-intercept form, y = mx + b, where y is the dependent variable (in this case, the yield), x is the independent variable (the number of bean stalks), m is the slope, and b is the y-intercept.

To find the slope of the line, we can use the formula:

m = (Y2 - Y1) / (n2 - n1)

where (n1, Y1) and (n2, Y2) are two points on the line. We can use the two data points given in the problem to find the slope:

m = (190 - 115) / (8 - 3) = 15

To find the y-intercept, we can use the point-slope form of a line, which is:

y - Y1 = m(x - n1)

where (n1, Y1) is one of the points on the line. We can use the point (3, 115):

y - 115 = 15(x - 3)

Simplifying:

y = 15x - 20

Therefore, the equation that represents the relationship between the number of bean stalks and the yield is:

Y = 15n - 20

This equation tells us that for each additional bean stalk planted, the yield increases by 15 ounces, and the y-intercept of -20 indicates that even if no bean stalks were planted, there would still be a yield of -20 ounces (which doesn't make physical sense in this case, but is a mathematical artifact of the linear regression).