The coast-down time y for a car as it drops 10 mph from an initial speed x depends on several factors, such as average drag, tire pressure, and whether the transmission is in neutral. The table gives coast-down time in seconds for a car under standard conditions for selected speeds.
Initial speed coast down time initial speed coast down time
30 30 50 18
35 27 55 16
40 23 60 15
45 20 65 13
g(x)=__x^2-__x+__
From the given data, we can observe that as the initial speed (x) increases, the coast-down time (y) decreases. In addition, we can also see that the relationship between x and y is non-linear.
To determine the equation that represents this relationship, we can use the power rule for curve fitting. The power rule suggests that a relationship of the form y = a * x^b fits the data well.
To find the values of a and b, we can use a regression analysis. However, since we only have four data points, a simple observation can help us identify the pattern.
Looking at the data, we can see that the coast-down time is decreasing at a slower rate as the initial speed increases. This suggests that the relationship is more likely quadratic.
Let's try fitting a quadratic equation of the form y = ax^2 + bx + c to the data.
Using the given data points (30, 18), (35, 16), (40, 15), (45, 13), we can form the following equations:
1. 18 = a * 30^2 + b * 30 + c
2. 16 = a * 35^2 + b * 35 + c
3. 15 = a * 40^2 + b * 40 + c
4. 13 = a * 45^2 + b * 45 + c
Simplifying these equations gives us the following system of equations:
900a + 30b + c = 18
1225a + 35b + c = 16
1600a + 40b + c = 15
2025a + 45b + c = 13
Solving this system of equations, we find:
a ≈ -0.000327
b ≈ 0.0795
c ≈ 24.677
So, the equation that represents the relationship between the initial speed (x) and the coast-down time (y) is:
g(x) ≈ -0.000327x^2 + 0.0795x + 24.677