Question

but a change from 40 to 20 is a 50% decrease.

1. mp modeling real life the table shows population

data for a community.
a. what is the percent of change from 2011 to 2017?
b. predict the population in 2023. explain
your reasoning.

1. geometry suppose the length and the width

of the sandbox are doubled.
a. find the percent of change in the perimeter.
b. find the percent of change in the area.

year	population
2017	138,000

(image of a sandbox with 6 ft and 10 ft dimensions, and two toys inside)

Explanation:

Part 29a: Percent of Change from 2011 to 2017

Step 1: Recall Percent Change Formula

The formula for percent change is $\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$. Here, the old value (2011 population) is 118,000, and the new value (2017 population) is 138,000.

Step 2: Calculate the Difference

First, find the difference between the new and old values: $138000 - 118000 = 20000$.

Step 3: Apply the Percent Change Formula

Now, divide this difference by the old value and multiply by 100%: $\frac{20000}{118000} \times 100\% \approx 16.95\%$.

Step 1: Determine the Time Interval

From 2011 to 2017 is 6 years (2017 - 2011 = 6). From 2017 to 2023 is also 6 years (2023 - 2017 = 6), so the time interval is consistent.

Step 2: Analyze the Growth Pattern

We assume the population grows at a constant rate (linear growth) since we're predicting. First, find the annual growth rate or the total growth over 6 years. From 2011 to 2017, the population grew by 20,000 (as calculated in 29a).

Step 3: Predict the Next 6 Years

If the growth is linear (constant amount per 6 years), we add the same growth amount (20,000) to the 2017 population. So, $138000 + 20000 = 158000$. Alternatively, using the percent growth rate from 29a ($\approx 16.95\%$), the growth factor is $1 + 0.1695 = 1.1695$. Over 6 years, the population in 2023 would be $138000 \times 1.1695 \approx 158391$ (close to 158,000 with linear approximation).

Step 1: Recall Perimeter Formula for a Rectangle

The perimeter of a rectangle is $P = 2(l + w)$, where $l$ is length and $w$ is width. Original length ($l$) = 10 ft, original width ($w$) = 6 ft.

Step 2: Calculate Original Perimeter

Original perimeter: $P_{\text{original}} = 2(10 + 6) = 2(16) = 32$ ft.

Step 3: Calculate New Dimensions

Length and width are doubled, so new length ($l'$) = $2 \times 10 = 20$ ft, new width ($w'$) = $2 \times 6 = 12$ ft.

Step 4: Calculate New Perimeter

New perimeter: $P_{\text{new}} = 2(20 + 12) = 2(32) = 64$ ft.

Step 5: Find Percent of Change in Perimeter

Using the percent change formula: $\frac{64 - 32}{32} \times 100\% = \frac{32}{32} \times 100\% = 100\%$.

Answer:

Approximately $16.95\%$ (or $17\%$ when rounded)

Part 29b: Predict Population in 2023