i need help with this problem please!

Question

person taking a 30-hour review course to prepare for a standardized exam average a score of 620 on that exam. Persons taking a 70-hour review course average a score of 765. find a linear function, S(t), which fits this date, and which expresses score as a function of time. Use this function to predict an average score for persons taking a 57-hour review course. Round your answer to the tenths place.

Answer 1

Because it is a linear function S(t) will take the form S(t) = at + b, where a is the slope of the equation, and b is the y-intercept. To find the slope, we can use the "rise over run" method. a = (765-620)/(70-30) = 3.625

To find b, we simply plug a into the equation, and solve for one of the points we know. Here, I'll use the 30 hour, 620 points example.

S(t) = at + b

620 = 3.625 * 30 + b

620 = 108.75 + b

511.25 = b

To verify, we need to check that 765 = 3.625 * 70 + 511.25, which it does

To predict the score for someone taking a 57 hour review course, we simply plug 57 in for t:

S(t) = 3.625 * 57 + 511.25 = 717.875 -> 717.9

Answer 2

We can express the hours, scores as x,y values (30,620) and (70,765) and use those to come up with a linear equation of the form y = mx + b which has score (y) as a function of time (x).

We then plug 57 into the equation as the time (x) value to predict the corresponding score (y)

Answer 3

The input variable (the x) is the time, the output variable (the y) is the score. So the data provided gives two points: (30, 620) and (70, 765). The slope of the linear function is (y₂ - y₁)/(x₂ - x₁); plugging in the numbers we get (765 - 620)/(70 - 30) = 145/40 = 29/8 (by the way, plugging in the values the other way (620- 765)/(30 - 70) gives the same result).

Now that we have the slope, we use either point to get the equation of the line using the point-slope format:

y - b = m(x - a)

And we get y - 620 = 29/8 (x - 30). Solving for y, we get the following:

y - 620 = 29/8x - 435/4

y = 29/8 x + 2045/4 or, if you want decimals, y = 3.625 x + 511.25

To find the result of a 57 hour review course, put 57 in for x:

y = 29/8 (57) + 2045/4 = 5743/8 = 717.875 which rounds to a score of 718