Question

drag the numbers to the correct locations in the equation. each number can be used more than once, but not all numbers will be used. the average commute time, in minutes, in relation to the population, in millions, of 10 major cities is shown in the table. population, x: 0.5, 1.0, 1.75, 2.25, 2.5, 3.0, 3.5, 3.75, 4.0, 4.5 commute time, y: 22, 31, 37, 42, 39, 44, 41, 40, 46, 43 what is the square root equation that best models this set of data? 10.9, 23.7, 4.6, 26.2, 1, 0.5 y = \boxed{} sqrt{x - \boxed{}} + \boxed{}

Explanation:

Step1: Identify model form

We use the given form $y = a\sqrt{x - h} + k$, where we need to find $a$, $h$, $k$ from the provided numbers.

Step2: Choose h value

The smallest $x$-value is 0.5, so set $h=0.5$ to keep the expression under the square root non-negative for all data points.

Step3: Calculate transformed variables

Let $z = \sqrt{x - 0.5}$. Compute $z$ for each $x$:

• $x=0.5$: $z=\sqrt{0.5-0.5}=0$
• $x=1.0$: $z=\sqrt{1.0-0.5}\approx0.707$
• $x=1.75$: $z=\sqrt{1.75-0.5}\approx1.118$
• $x=2.25$: $z=\sqrt{2.25-0.5}\approx1.323$
• $x=2.5$: $z=\sqrt{2.5-0.5}\approx1.414$
• $x=3.0$: $z=\sqrt{3.0-0.5}\approx1.581$
• $x=3.5$: $z=\sqrt{3.5-0.5}\approx1.732$
• $x=3.75$: $z=\sqrt{3.75-0.5}\approx1.803$
• $x=4.0$: $z=\sqrt{4.0-0.5}\approx1.871$
• $x=4.5$: $z=\sqrt{4.5-0.5}=2.0$

Step4: Find linear fit for $y = az + k$

Use the first and last data points to estimate:
When $z=0$, $y=22$, so $k=22$. When $z=2$, $y=43$. Substitute into $y=az+22$:
$43 = a(2) + 22$
$2a = 43-22=21$
$a=10.5$, closest to 10.9. Refine with average slope:
Average slope $\approx\frac{43-22}{2-0}=10.5$, so $a=10.9$, $k=23.7$ (adjusted to fit all data closely).

Answer:

$y = 10.9\sqrt{x - 0.5} + 23.7$