Question

For a pyroclastic flow travelling at a constant​ speed, the relationship between distance traveled​ (d), rate​ (speed), and time​ (t) is​ d=rt.
Find the time in minutes at which a pyroclastic flow will reach a town 30 kilometers​ (~19 miles) away if the flow is travelling at a constant rate of 3​ km/min (~100 miles per​ hour). (Give answer to 1 decimal​ place.)
Find a function t that gives the time it takes for a pyroclastic flow to travel 30 kilometers at a constant rate of r kilometers per​ minute, and state the domain of t.
Find a function r that gives the speed of the pyroclastic flow when 30 kilometers are covered in t​ minutes, and state the domain of r.​ (Note however that the maximum speed of a pyroclastic flow is 10​ km/min, that​ is, the maximum value of the function​ r(t) is 10​ km/min).
Find a function d that gives the distance that the flow will travel in 30 minutes at a constant rate of r​ km/min, and state the domain of d.

Answer 1

The time in minutes at which a pyroclastic flow will reach the town  is 0.3 minutes and the functions are t = 30/r, r = 30/t and d = 30r

Find the time in minutes at which a pyroclastic flow will reach a town 30 kilometers​ (~19 miles) away

Here, we have:

Distance = 30 kilometers

Rate = 100 kilometers per minute

The time is calculated as:

Time = Distance/Rate

So, we have

Time = 30/100

Evaluate

Time = 0.30

Hence, the time in minutes at which a pyroclastic flow will reach the town  is 0.3 minutes

Find a function t that gives the time it takes for a pyroclastic flow to travel 30 kilometers

Here, we have

Distance = 30 kilometers

Rate = r

The time is calculated as:

Time = Distance/Rate

So, we have

t = 30/r

Hence, the function for time is t = 30/r and the domain is r > 0

Find a function r that gives the speed of the pyroclastic flow when 30 kilometers are covered in t​ minutes

Here, we have

Distance = 30 kilometers

Time = t

The time is calculated as:

Time = Distance/Rate

So, we have

t = 30/r

The maximum speed is 10 km/min

So, we have:

t = 30/10 = 3

Rewrite t = 30/r as:

r = 30/t

Hence, the function for rate is r = 30/t and the domain is 0 < t < 3

Find a function d that gives the distance that the flow will travel in 30 minutes at a constant rate of r​ km/min

Here, we have

time = 30 minutes

Rate = r

The time is calculated as:

Time = Distance/Rate

So, we have

30 = d/r

So, we have

d = 30r

Hence, the function for time is d = 30r and the domain is r > 0

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