Question

from 2000 to 2020, a world health organization collected data on the percentage of a countrys population that had access to safe drinking water. the function $w(t) = 0.5t + 47$ models the percentage of the population in guatemala that has safe drinking water where $t$ is the number of years since 2000 and $w$ is the percentage of the population that has access.

1. explain why the values of $t$ should have a constraint.
2. write the constraint for the values of $t$ using set builder notation.
3. write the constraint for the values of $w$ using set builder notation.
4. graph the function using the constraints from questions 6 and 7.

graph with x-axis labeled years since 2000 and y-axis labeled percentage

1. use the graph to find the solution $w(t) = 55$. label the point that represents the solution.
2. solve $0.5t + 47 = 55$. show your work.

Explanation:

Step 5: Explain t constraint

The data is only collected from 2000 to 2020, so \(t\) (years since 2000) cannot be negative or exceed 20, as those values fall outside the study period.

Step 6: Set builder for t

Define valid \(t\) values: integers/real numbers between 0 and 20 inclusive.
\(\{ t \mid t \in \mathbb{R}, 0 \leq t \leq 20 \}\)

Step 7: Set builder for W

Find min/max W values:

Step7.1: Calculate min W

Substitute \(t=0\): \(W(0)=0.5(0)+47=47\)

Step7.2: Calculate max W

Substitute \(t=20\): \(W(20)=0.5(20)+47=57\)

Step7.3: Write set notation

\(\{ W \mid W \in \mathbb{R}, 47 \leq W \leq 57 \}\)

Step 8: Graph the function

Step8.1: Identify endpoints

When \(t=0\), \(W=47\) (point \((0,47)\)); when \(t=20\), \(W=57\) (point \((20,57)\)).

Step8.2: Plot and connect

Draw a straight line segment between \((0,47)\) and \((20,57)\) on the given grid, with the x-axis labeled "years since 2000" and y-axis labeled "percentage".

Step 9: Graph solution for W(t)=55

Step9.1: Locate W=55 on y-axis

Find the horizontal line at \(W=55\), then find its intersection with the graphed segment.

Step9.2: Identify the point

The intersection point is \((16,55)\); label this point on the graph.

Step10: Solve algebraically

Step10.1: Subtract 47 from both sides

\(0.5t + 47 - 47 = 55 - 47\)
\(0.5t = 8\)

Step10.2: Divide by 0.5

\(t = \frac{8}{0.5}\)
\(t = 16\)

Answer:

1. The function only models data collected from 2000 to 2020, so \(t\) must be within the range of years corresponding to this period (cannot be negative or greater than 20).
2. \(\{ t \mid t \in \mathbb{R}, 0 \leq t \leq 20 \}\)
3. \(\{ W \mid W \in \mathbb{R}, 47 \leq W \leq 57 \}\)
4. A line segment connecting the points \((0, 47)\) and \((20, 57)\) on the provided grid (x-axis: years since 2000, y-axis: percentage).
5. The solution is the point \((16, 55)\) on the graphed line segment.
6. \(t = 16\)