Wanda will catch up to Hector when their distances completed are equal.
B
Compare the length of the race to the distance that Wanda and Hector travel before their meeting.
C
Calculate the time from the moment Wanda started the race until Hector completed the race. Use the formula for Hector's distance found in Part A.
A
Equation: 16t = 18 + 12t When Will Wanda Catch Up to Hector? After 4.5 hours
B
No, see solution.
C
24 miles per hour, see solution.
Practice makes perfect
We want to write and solve an equation to find when Wanda will catch up to Hector. We want to find when, so our unknown value is time. In any equation, the unknown is represented by a variable. In our equation, time will be represented by t.
t - variable representing time
An equation shows the equality of two quantities. Therefore, we need to find two expressions representing equal quantities. We want to find the time when Wanda will catch up to Hector or, in other words, the time they meet. Notice that they meet when their distances are equal.
Wanda's distance = Hector's distanceTo write this equation we need to find expressions describing Wanda's and Hector's distances. Recall the distance-time equation.
d=r * t
In the equation, d represents distance, r is the rate, or speed, of a moving object, and t represents time. We know that Wanda is traveling at a constant speed of 16 miles per hour.
Wanda's distance = 16 t
We also know that Hector's speed is 12 miles per hour and that he already completed 18 miles before Wanda even started.
Hector's distance = 18 + 12 t
Now we can complete our equation!
16 t = 18 + 12 t
To find the time from the moment Wanda began the race until their meeting, we need to solve the equation for t.
Wanda will catch up to Hector after 4.5 hours, which is 4 hours and 30 minutes.
We want to know whether Wanda will catch up to Hector before the end of the race. To find the answer, we will compare the length of the race to the distance that Wanda and Hector travel before they meet. The length of the race is 42 miles.
distance before the meeting ? < 42 miles
To find the distance they traveled before meeting, we can use the information found in Part A. We found that Wanda needs to travel for 4.5 hours to catch up to Hector and that the following formula describes her distance.
Wanda's distance = 16 t
At the moment of their meeting, the variable t is equal to 4.5. We can substitute it into the expression to calculate the distance that Wanda needs to travel to catch up to Hector.
To catch up to Hector, Wanda needs to ride 72 miles. This is a much longer distance than the length of the race. Therefore, Wanda will not catch up to Hector before the end of the race.
72 miles ≮ 42 miles
Now we want to find at what constant speed Wanda could travel to catch up with Hector at the finish line. In Part A we found formulas describing Wanda's and Hector's distance.
Wanda's distance &= 16 t
Hector's distance &= 18 + 12 t
We wrote these formulas knowing that Wanda traveled with constant speed of 16 miles per hour. But now we should forget about this value and find the new speed r that allows Wanda to catch up with Hector at the finish line.
Wanda's distance &= 16t = rt
Hector's distance &= 18 + 12tFor Wanda to catch up to Hector at the finish line, they both need to finish 42 miles race at the same time.
Wanda's distance &= r t = 42 miles
Hector's distance &= 18 + 12 t = 42 miles
We can use the formula for Hector's distance to calculate the time from Wanda's start until the end of the race. To do this, we will solve the equation for t.
Hector completes the race 2 hours after Wanda started the race. Therefore, Wanda has 2 hours to catch up to Hector at the finish line. We can partially write the formula for Wanda's distance.
Wanda's distance = r * 2 = 42 miles
Finally, we can find the speed that will allow Wanda to catch up to Hector at the finish line. Let's solve this equation for r!
