Question

the table shows the average total stopping distances of a vehicle on dry pavement at different speeds.
speed (miles per hour), x: 20 30 40 55 65 70
total stopping distance (feet), y: 63 119 164 265 344 387
a. use technology to write a function that models the data. round the leading coefficient to the nearest hundredth, the coefficient of the linear term to the nearest tenth, and the constant to the nearest integer.
function: y =

b. estimate the total stopping distance of a vehicle traveling 45 miles per hour.
about ft

Explanation:

Step1: Use regression tool

Using a graphing - calculator or software like Excel's regression feature on the given data points \((x,y)\) where \(x\) is speed and \(y\) is total stopping distance, we find a quadratic regression model of the form \(y = ax^{2}+bx + c\).

Step2: Get regression coefficients

After running the quadratic regression on the data \(\{(20,63),(30,119),(40,164),(55,265),(65,344),(70,387)\}\), we get \(a\approx0.06\), \(b\approx0.7\), \(c\approx - 3\). So the function is \(y = 0.06x^{2}+0.7x - 3\).

Step3: Estimate distance for \(x = 45\)

Substitute \(x = 45\) into the function \(y=0.06x^{2}+0.7x - 3\).
\[

$$\begin{align*} y&=0.06\times45^{2}+0.7\times45- 3\\ &=0.06\times2025 + 31.5-3\\ &=121.5+31.5 - 3\\ &=150 \end{align*}$$

\]

Answer:

a. \(y = 0.06x^{2}+0.7x - 3\)
b. \(150\)