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.
5. use the graph to find the solution ( w(t) = 55 ). label the point that represents the solution.

Explanation:

Step1: Explain t's constraint need

The data is collected 2000-2020, so \(t\) (years since 2000) can't be outside this range; negative \(t\) is before 2000, \(t>20\) is after 2020, which is outside the data period.

Step2: Find t's valid range

2000 corresponds to \(t=0\), 2020 corresponds to \(t=2020-2000=20\). So \(t\) must be between 0 and 20, inclusive.

Step3: Write t's set-builder notation

\(\{ t \mid t \in \mathbb{R}, 0 \leq t \leq 20 \}\)

Step4: Find W's minimum value

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

Step5: Find W's maximum value

Substitute \(t=20\) into \(W(t)=0.5t+47\): \(W(20)=0.5(20)+47=10+47=57\)

Step6: Write W's set-builder notation

Since \(W(t)\) is linear and increasing, \(W\) ranges from 47 to 57, inclusive: \(\{ W \mid W \in \mathbb{R}, 47 \leq W \leq 57 \}\)

Step7: Solve \(W(t)=55\) algebraically

Set \(0.5t+47=55\), rearrange to solve for \(t\):
\(0.5t=55-47\)
\(0.5t=8\)
\(t=\frac{8}{0.5}=16\)
This corresponds to the point \((16, 55)\) on the graph.

Step8: Graph the function

Plot the endpoints: when \(t=0\), \(W=47\) (point \((0,47)\)); when \(t=20\), \(W=57\) (point \((20,57)\)). Draw a straight line connecting these two points, as the function is linear. Mark the point \((16,55)\) as the solution to \(W(t)=55\).

Answer:

1. The data is only collected from 2000 to 2020, so \(t\) (years since 2000) cannot be negative or greater than 20, as those values fall outside the period of data collection.
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. (Graph: A straight line connecting \((0, 47)\) and \((20, 57)\), restricted to \(0 \leq t \leq 20\) and \(47 \leq W \leq 57\))
5. The solution is \(t=16\), represented by the point \((16, 55)\) on the graph.