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Answer :
The rate of change defines the change of a value of a variable over a unit of time
(a) The time at which the runners feet will get wet is 44 minutes
(b) The time the runner shoes get wet is after approximately 24.415 minutes
Reason:
Known parameters;
The radius of the puddle, r = 5·t
The time after which the pump broke = t
Initial distance of the runner from the intersection, d = 6 miles = 31,680 ft.
The speed of the runner = 17 ft./s
(a) Required:
The time at which the runners feet will get wet
Solution;
At the time the runner's feet get wet, we have;
5·t + 17·t = 31,680
[tex]t = \dfrac{31680}{12} = 2,640[/tex]
The time at which the runners feet will get wet is 2,640 seconds = 44 minutes
(b) Location of the runner = 6 miles East and 5,000 feet North
The direction the runner runs = Due West
Speed of the runner = 17 ft. per second
The time at which the runners feet will get wet
Solution:
At the point the runner's feet get wet, we have;
The vertical component of the radius, y = 5,000 = The direction north from the runner's path
The horizontal component of the radius, x = 31,680 - 17·t
Therefore;
The magnitude of the radius of the puddle at the point the runner's feet get wet is given as follows;
[tex]5 \cdot t = \sqrt{5000^2 + (31680 - 17 \cdot t)^2}[/tex]
Which gives;
25·t² = 5000² + (31,680 - 17·t)²
264·t² - 1077120·t + 1028622400 = 0
Solving for t
t = 1524.924 seconds = 25.415 minutes or t = 2,555.076 seconds = 42.8846 minutes
Therefore;
- The time the runner shoes get wet is after, t ≈ 24.415 minutes
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Rewritten by : Barada
Answer: a = 24 minutes, b = 25 minutes
Step-by-step explanation:
There are two equations (puddle = circle & runner = line). Need to find where they intersect.
Puddle Equation: x² + y² = r² → x² + y² = (5t)² → x² + y² = 25t²
Runner Equation: x = 6(5280) - 17t since y = 0 (note that 6 is miles so multiplied by 5280 to convert it into feet) ⇒ x = 31,680 - 17t
Use substitution to get:
(31,680 - 17t)² + (0)² = 25t²
Use a quadratic calculator to discover that t = 2,641 or 1440
use the smaller number (since we want to know the FIRST time his feet get wet. Next, convert 1440 seconds into minutes by dividing by 60
t = 24 minutes
***************************************************************************************
For part (b), follow the same steps as part (a) except the y-coordinates of the runner will change: x = 31,680 - 17t, y = 5000
x² + y² = 25t²
(31,680 - 17t)² + (5000)² = 25t²
use the calculator to discover that t = 2555 or 1525
use the smaller number, then convert 1525 into minutes by dividing by 60
t = 25 minutes
