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 2 stalks, each plant yields 70 ounces of beans. When she plants 8 stalks, each plant yields 160 ounces of beans.

Which of these equations correctly represents this situation?

Select the correct answer below:


Y(n)=19n+32

Y(n)=11n+72

Y(n)=11n+48

Y(n)=19n+8

Y(n)=15n+30

Y(n)=15n+40

Answer 1

The linear equation representing the relationship between the number of bean stalks planted and the yield is Y(n) = 190n - 240, derived from the slope calculated using the given points. However, this equation does not match any of the provided options, which suggests there may be an error in the question or the choices given.

Step-by-step explanation:

To find the equation that represents the linear relationship between the number of bean stalks n and the yield Y, you can use the two points provided: (2, 140) and (8, 1280). Here, the yield Y is calculated as the product of the number of ounces per plant and the number of plants.

Lets calculate the slope (m) of the linear equation:

Y2 - Y1 / n2 - n1 = (1280 - 140) / (8 - 2) = 1140 / 6 = 190

So, the slope m is 190 ounces per stalk. We can now use one of the points to solve for the y-intercept (b).

Y = mn + b
140 = 190(2) + b
140 = 380 + b
b = 140 - 380
b = -240

Therefore, the correct equation is Y(n) = 190n - 240. However, this equation is not among the provided options. It seems there may have been a mistake in either the question or the options given. The student should recheck the data or contact the instructor for clarification, as none of the options match the results of our calculation.

Answer 2

Y(n)=15n+40

Step-by-step explanation:

Lets represent the number of bean stalks she plant by n and

the yield by Y.

It says in the question that when she plants 2 stalks, each plant yields 70 ounces of beans. When she plants 8 stalks, each plant yields 160 ounces of beans.

Which means 2 stalks correspond to 70 ounces and 8 stalks corresponds to 160 ounces we could write the data Mathematically like:

(2 , 70) and (8 , 160) are our coordinates where the x-axis would be n the number of beans stalks planted and y-axis would be Y the yield in ounces.

So we have two coordinates and to find the linear relationship we must find out the slope/gradient(m) of the linear function by the formula of slope:

`m=(y_2-y_1)/(x_2-x_1)\\\\m=(160-70)/(8-2) \\\\m=(90)/(6) \\\\m=15`

so we plug in one of the coordinates and our slope in the point-slope formula :

`y-y_1=m(x-x_1)`

where,

`(x_1,y_1)\ represent\ the\ coordinate\ (2,70)`

so plugging in the coordinate and the value of m we get:

`y-y_1=m(x-x_1)\\y-70=15(x-2)\\y-70=15x-30\\y=15x-30+70\\y=15x+40\\`

so our the answer is the last option

Y(n)=15n+40

here x corresponds to n and y corresponds to Y(n)