Ian's house is located 20km north of Aida's house. At 9am, one saturday, Ian leaves his house and jogs south at 8km/hour. At the same time Aida leaves her house and jogs east at 6km/h. When are Ian and Aida closest together given that they both run for 2.5 hours? The answer is t=1.6 hours @ 10:36 am
PLEASE SHOW sTEP bY STEP!! i don't understand how to do this
draw a sketch of the situation,
at a time of t hours after 9:00 am
let A1 be the position of Aida
let I1 be the position of Ian
Join A1 and I1, to get a right-angled triangle
let D be the distance between them
D^2 = (6t)^2 + (20-8t)^2
= 36t^2 + 400 - 320t + 64t^2
= 100t^2 - 320t + 400
2D dD/dt = 200t - 320
dD/dt = (200t-320)/(2D)
= 0 for a minimum of D
200t = 320
t = 1.6
So they are closest after 1.6 hours or 1 hour and 36 minutes past 9:00 am
making it 10:36 am
This is a standard question for this topic of Calculus.
In my opinion, you should really know how to do this question
To find when Ian and Aida are closest together, we need to calculate their distances from each other over the span of 2.5 hours. Let's break down the problem into smaller steps:
Step 1: Set up a coordinate system
To simplify the problem, let's set up a coordinate system. We can choose Ian's house as the origin (0,0) and label north as the positive y-direction and east as the positive x-direction.
Step 2: Determine Aida's position at each given time
Aida starts at her house (0,0) and jogs east at a speed of 6 km/h for 2.5 hours. Using the formula distance = speed × time, we can calculate Aida's distance using the equation: distance = 6 km/h × 2.5 h = 15 km.
So after 2.5 hours, Aida is located at (15,0) on the coordinate system.
Step 3: Determine Ian's position at each given time
Ian starts at his house (0,0) and jogs south at a speed of 8 km/h for 2.5 hours. Using the same formula as before, we can calculate Ian's distance using the equation: distance = 8 km/h × 2.5 h = 20 km.
So after 2.5 hours, Ian is located at (0,-20) on the coordinate system.
Step 4: Find the distance between Ian and Aida
To find the distance between two points, we can use the distance formula: d = √[(x2 - x1)^2 + (y2 - y1)^2]
Using the coordinates from Step 2 and Step 3, we can plug them into the distance formula:
d = √[(15 - 0)^2 + (0 - (-20))^2]
= √[225 + 400]
= √625
= 25 km
So after 2.5 hours, the distance between Ian and Aida is 25 km.
Step 5: Find when Ian and Aida are closest together
To find when Ian and Aida are closest together, we need to find the time at which their distance is minimized.
We can calculate their distance at each given time during the 2.5-hour period by calculating the distance between (0,0) and (15t,-20t), where t represents the time elapsed.
Using the distance formula, we have:
d = √[(15t - 0)^2 + (-20t - 0)^2]
= √[225t^2 + 400t^2]
= √(625t^2)
= 25t
This gives us the distance between Ian and Aida at any given time.
To minimize this distance, we need to find when the derivative of the distance function equals zero.
d/dt (25t) = 25 = 0
Solving for t, we find t = 1. This means that Ian and Aida are closest together at t = 1 hour.
To convert this time into a specific time of the day, we need to add the starting time (9 am) and calculate the time it takes to reach that point. Since Ian and Aida jogged for 2.5 hours, the closest distance between them occurs at 9 am + 1 hour + 1.6 hours (rounded), which is approximately 10:36 am.
So Ian and Aida are closest to each other at 10:36 am.