Water is flowing from a major broken water main at the intersection of two streets. the resulting puddle of water is circular and the radius r of the puddle is given by the equation r=5t feet, where t represents time in seconds elapsed since the the main broke. (a) when the main broke, a runner was located 6 miles from the intersection. the runner continues toward the intersection at the constant speed of 17 feet per second. when will the runner's feet get wet? (b) suppose, instead, that when the main broke, the runner was 6 miles east, and 5000 feet north of the intersection. the runner runs due west at 17 feet per second. when will the runner's feet get wet?

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

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;

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;

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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

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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

y=mx+b

step-by-step explanation:

m is slope, b is y-intercept

the answer is c

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