Question

tiana and rei are biking in the same direction on the same path.
rei rides at a constant speed of 16 miles per hour.
tiana starts riding a half hour before rei. tiana rides at a constant speed of 12 miles per hour.
use your expressions from the previous screens to determine when rei and tiana meet.

Explanation:

Step1: Define variables

Let \( t \) be the time (in hours) that Rei has been riding when they meet. Then Tiana has been riding for \( t + 0.5 \) hours.

Step2: Set up distance equations

The distance Rei travels is \( d_R = 16t \) (since distance = speed × time, and Rei's speed is 16 mph).
The distance Tiana travels is \( d_T = 12(t + 0.5) \) (Tiana's speed is 12 mph and she rides 0.5 hours more than Rei).

Step3: Set distances equal (they meet when distances are equal)

Set \( 16t = 12(t + 0.5) \)

Step4: Solve the equation

Expand the right side: \( 16t = 12t + 6 \)
Subtract \( 12t \) from both sides: \( 16t - 12t = 6 \)
Simplify: \( 4t = 6 \)
Divide both sides by 4: \( t = \frac{6}{4} = 1.5 \) hours. But this is Rei's time. To find when they meet, we can also consider Tiana's time or the total time since Tiana started. Wait, actually, let's check again. Wait, when they meet, their distances are equal. So solving \( 16t = 12(t + 0.5) \):

\( 16t = 12t + 6 \)

\( 4t = 6 \)

\( t = 1.5 \) hours after Rei starts. But Tiana started 0.5 hours earlier, so total time since Tiana started is \( 1.5 + 0.5 = 2 \) hours. Alternatively, we can think of Tiana's time as \( t \) and Rei's time as \( t - 0.5 \). Let's redefine: Let \( t \) be Tiana's time. Then Rei's time is \( t - 0.5 \). Distance for Tiana: \( 12t \), Distance for Rei: \( 16(t - 0.5) \). Set equal: \( 12t = 16(t - 0.5) \)

\( 12t = 16t - 8 \)

\( -4t = -8 \)

\( t = 2 \) hours. So Tiana rides for 2 hours, Rei rides for \( 2 - 0.5 = 1.5 \) hours. Let's check distances: Tiana: \( 12 \times 2 = 24 \) miles. Rei: \( 16 \times 1.5 = 24 \) miles. So they meet 2 hours after Tiana starts, or 1.5 hours after Rei starts. The question says "determine when Rei and Tiana meet" – probably asking for the time since Rei starts or since Tiana starts. But let's see the problem statement. Wait, the problem says "use your expressions from the previous screens" – but since we don't have those, we can solve it as above. So the time when they meet is 1.5 hours after Rei starts (or 2 hours after Tiana starts). Let's confirm the calculation again.

Alternative approach: Let’s let \( t \) be the number of hours Rei has been riding when they meet. Then Tiana has been riding for \( t + 0.5 \) hours. Distance Rei covers: \( 16t \). Distance Tiana covers: \( 12(t + 0.5) \). Set equal:

\( 16t = 12(t + 0.5) \)

\( 16t = 12t + 6 \)

\( 4t = 6 \)

\( t = 1.5 \) hours. So Rei rides for 1.5 hours, Tiana rides for \( 1.5 + 0.5 = 2 \) hours. Both cover 24 miles. So the answer is 1.5 hours after Rei starts, or 2 hours after Tiana starts. Depending on how the question wants it, but likely 1.5 hours after Rei starts (or 2 hours after Tiana starts). Wait, but let's check the problem again. It says "determine when Rei and Tiana meet" – probably in terms of hours after Rei starts or after Tiana starts. But let's see the equations. So the solution is \( t = 1.5 \) hours after Rei begins, or 2 hours after Tiana begins.

Answer:

1.5 hours after Rei starts (or 2 hours after Tiana starts, with both covering 24 miles)