On Day 1 of the state fair, Austin sold 15 handmade stools. On each of
the next 9 days, he sold 5 stools. The linear function f(d) can be used to
represent the total number of stools Austin sold, where d is the day of
the fair. Solve the problems below. The first one is done for you.
10. What is the slope of the line representing f(d)?
5
11. Find f(2) and f(5).
12. Write the function f(d) as a linear equation.
11. The slope of the line representing f(d) is 5.
12. The function f(d) can be written as a linear equation:
f(d) = 15 + 5d
11. To find f(2), we know that on Day 1, Austin sold 15 stools, and on each of the next 9 days, he sold 5 stools. So, on Day 2, he sold a total of 15 + 5 = 20 stools.
To find f(5), we know that on Day 1, Austin sold 15 stools, and on each of the next 9 days, he sold 5 stools. So, on Day 5, he sold a total of 15 + 5*4 = 35 stools.
12. To write the function f(d) as a linear equation, we need to determine the equation of the line. The slope of the line is given as 5, which means for every day, the number of stools sold increases by 5.
We also know that on Day 1, Austin sold 15 stools. So, to determine the y-intercept (the number of stools sold on Day 1), we substitute the values into the point-slope formula: y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of a point on the line.
Using (x1, y1) = (1, 15) and m = 5, the equation becomes: y - 15 = 5(x - 1).
Simplifying the equation: y - 15 = 5x - 5.
Finally, we can rewrite the equation in the form f(d) = mx + b, where f(d) represents the total number of stools sold, m is the slope, d is the day of the fair, and b is the y-intercept or the number of stools sold on Day 1:
f(d) = 5d + 10.
To find the values of f(2) and f(5), we can use the given information and the linear function f(d).
From the given information, we know that on Day 1, Austin sold 15 handmade stools, and on each of the next 9 days, he sold 5 stools.
Using this, we can determine the number of stools sold on Day 2 and Day 5.
On Day 2:
Since Austin sold 5 stools each day from Day 2 to Day 10, the total number of stools sold on Day 2 would be:
15 (from Day 1) + 5 (from Day 2) = 20
Therefore, f(2) = 20.
On Day 5:
Since Austin sold 5 stools each day from Day 2 to Day 10, the total number of stools sold on Day 5 would be:
15 (from Day 1) + 5 (from Day 2) + 5 (from Day 3) + 5 (from Day 4) + 5 (from Day 5) = 35
Therefore, f(5) = 35.
Now, let's write the function f(d) as a linear equation.
Based on the given information, we know that on Day 1, Austin sold 15 stools, and on each of the next 9 days, he sold 5 stools. So, the slope of the line representing f(d) is 5.
The linear equation representing f(d) can be written as: f(d) = 5d + 10.
This equation represents the relationship between the day of the fair (d) and the total number of stools sold (f(d)), where the slope (5) represents the rate at which the total number of stools sold increases each day, and the constant term (10) represents the initial number of stools sold on Day 1.
clearly, f(1) = 15
since he sold 5 stools/day, the slope of the line is 5, so
f(d) = 15+5d for d > 1