Question

1. expected admission to the zoo (z) can be represented by ( z(t) = 140 + 45t ), where t is the number of hours since opening. if the zoo opened at 10am, how many attendees are expected by 2pm?
2. the measure of an angle exterior to ( angle f ) of triangle def measures ( 16x + 12 ). if ( mangle f = 8x ), what is the measure of angle f, in degrees?

Explanation:

Problem 5

Step1: Calculate hours since opening

From 10 am to 2 pm, \( t = 2 - 10 + 12 = 4 \) (using 12 - hour to 24 - hour conversion logic, or simply counting: 10 am to 11 am is 1h, ..., to 2 pm is 4h).

Step2: Substitute t into the function

Given \( z(t)=140 + 45t \), substitute \( t = 4 \):
\( z(4)=140+45\times4 \)
\( z(4)=140 + 180 \)
\( z(4)=320 \)

Step1: Recall exterior angle theorem

The measure of an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. But in a triangle, the exterior angle is also equal to \( 180^{\circ}-\) the adjacent interior angle. Wait, actually, the exterior angle is equal to the sum of the two remote interior angles, but here, the exterior angle to \( \angle F \) and \( \angle F \) are supplementary? Wait, no, the exterior angle of a triangle at a vertex is equal to the sum of the two opposite interior angles. But if we consider the exterior angle adjacent to \( \angle F \), then the exterior angle plus \( \angle F=180^{\circ} \)? Wait, no, the exterior angle of a triangle is formed by one side and the extension of another side. The exterior angle is equal to the sum of the two non - adjacent interior angles. But in this case, the problem says "the measure of an angle exterior to \( \angle F \) of triangle \( DEF \) measures \( 16x + 12 \). If \( m\angle F = 8x \)". So the exterior angle and \( \angle F \) are supplementary (since they form a linear pair), so \( (16x + 12)+8x=180 \)? Wait, no, the exterior angle is equal to the sum of the other two interior angles. Wait, maybe the problem is that the exterior angle is equal to \( 180 - \angle F \)? No, no. Wait, the exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. But if we have an exterior angle at \( F \), then the two non - adjacent interior angles are the other two angles of the triangle. But maybe the problem is considering the exterior angle adjacent to \( \angle F \), so the exterior angle and \( \angle F \) are supplementary, so \( \text{exterior angle}+\angle F = 180^{\circ} \). Wait, let's check the problem again: "The measure of an angle exterior to \( \angle F \) of triangle \( DEF \) measures \( 16x + 12 \). If \( m\angle F = 8x \), what is the measure of angle \( F \), in degrees?"

So, the exterior angle and \( \angle F \) are supplementary (they form a linear pair), so:

\( 16x + 12+8x=180 \)

Step2: Solve for x

Combine like terms: \( 24x+12 = 180 \)
Subtract 12 from both sides: \( 24x=180 - 12=168 \)
Divide both sides by 24: \( x=\frac{168}{24}=7 \)

Step3: Find \( m\angle F \)

Since \( m\angle F = 8x \), substitute \( x = 7 \):
\( m\angle F=8\times7 = 56 \)

Answer:

320

Problem 6