Question

1 pausing on the trail there is a hiking trail near the town where han and jada live that starts at a parking lot and ends at a lake. han and jada both decide to hike from the parking lot to the lake and back, but they start their hikes at different times. at the time that han reaches the lake and starts to turn back, jada is 0.6 miles away from the parking lot and hiking at a constant speed of 3.2 miles per hour toward the lake. han’s distance from the parking lot can be expressed as d = -2.4t + 4.8, where t represents the time in hours since he left the lake. 1 what is an equation for jada’s distance from the parking lot as she heads toward the lake? 2 draw both lines: one representing han’s equation and one representing jada’s equation. it is important to be very precise! be careful, work in pencil, and use a straightedge. 3 find the point where the 2 lines intersect each other. what are the coordinates of this point?

Explanation:

Problem 1: Equation for Jada's Distance

Step 1: Analyze Han's Equation

Han's distance from the parking lot is given by \( d = -2.4t + 4.8 \). When \( t = 0 \) (start of Han's return), this gives the distance from parking lot to lake: \( d = 4.8 \) miles. So the lake is 4.8 miles from the parking lot.

Step 2: Determine Jada's Initial Distance

When Han starts returning ( \( t = 0 \) for Han's return), Jada is 0.6 miles from the lake. Since the lake is 4.8 miles from parking lot, Jada's distance from parking lot at \( t = 0 \) (her start time relative to Han's return) is \( 4.8 - 0.6 = 4.2 \)? Wait, no—wait, Jada is heading toward the lake. Wait, maybe misread. Wait, the problem says "Jada is 0.6 miles away from the lake" when Han starts to turn back. So Jada's distance from parking lot when Han starts returning: lake is 4.8 miles from parking lot (from Han's equation, when \( t = 0 \), \( d = 4.8 \), so that's the lake's distance). So Jada is \( 4.8 - 0.6 = 4.2 \) miles from parking lot? No, wait, no—Jada is heading toward the lake, so her distance from parking lot increases as she moves toward the lake. Wait, maybe the initial time for Jada: when Han starts returning (let's say that's \( t = 0 \) for Jada's journey toward the lake), Jada is 0.6 miles from the lake, so her distance from parking lot is \( 4.8 - 0.6 = 4.2 \)? But the graph shows Jada's line starting at (0, 0.6)? Wait, maybe the problem's context: maybe Han and Jada start at different times. Wait, the graph has Jada's line starting at (0, 0.6), so maybe at \( t = 0 \) (Jada's start time), she is 0.6 miles from parking lot, and moving toward the lake at 3.2 mph. So her distance from parking lot is \( d = 3.2t + 0.6 \). Let's check: when \( t = 0 \), \( d = 0.6 \) (her initial distance from parking lot, heading toward lake, so as \( t \) increases, \( d \) increases). That matches the graph's Jada line (labeled \( d = 0.6 + 3.2t \) or \( d = 3.2t + 0.6 \)).

So Jada's speed is 3.2 mph toward the lake, so her distance from parking lot is linear with slope 3.2, and initial distance (when \( t = 0 \), her start time) is 0.6 miles from parking lot. So equation is \( d = 3.2t + 0.6 \).

Step 1: Set Han's and Jada's Equations Equal

Han's equation: \( d = -2.4t + 4.8 \)
Jada's equation: \( d = 3.2t + 0.6 \)

Set them equal:
\( -2.4t + 4.8 = 3.2t + 0.6 \)

Step 2: Solve for \( t \)

Add \( 2.4t \) to both sides:
\( 4.8 = 5.6t + 0.6 \)

Subtract 0.6 from both sides:
\( 4.2 = 5.6t \)

Divide both sides by 5.6:
\( t = \frac{4.2}{5.6} = 0.75 \) hours

Step 3: Find \( d \) by Substituting \( t = 0.75 \) into Jada's Equation

\( d = 3.2(0.75) + 0.6 = 2.4 + 0.6 = 3 \) miles

So the intersection point is \( (0.75, 3) \).

Answer:

(Problem 1): \( d = 3.2t + 0.6 \)

Problem 3: Intersection Point of the Two Lines