Fuel consumption A small plane in flight consumes fuel at a rate (in gal/min) given by
R'(t) ={ 4t^{1/3} if 0 ≤ t ≤ 8 (take-off)
2 if t> 0 (cruising)
a. Find a function R that gives the total fuel consumed, for 0≤t≤8.

Verified step by step guidance

1. Identify the given rate of fuel consumption function for the time interval 0 \leq t \leq 8, which is R'(t) = 4t^{1/3}. This represents the rate of fuel consumption in gallons per minute during take-off.
2. Recall that to find the total fuel consumed function R(t), you need to integrate the rate function R'(t) with respect to time t over the interval from 0 to t.
3. Set up the integral: R(t) = \int 4t^{1/3} \, dt. This integral will give the total fuel consumed from time 0 up to time t during take-off.
4. Perform the integration by applying the power rule for integrals: \int t^{n} \, dt = \frac{t^{n+1}}{n+1} + C. Here, n = \frac{1}{3}, so integrate accordingly and include the constant of integration C.
5. Use the initial condition R(0) = 0 (since no fuel is consumed at time zero) to solve for the constant C, ensuring the total fuel consumed function R(t) correctly models the situation.

Key Concepts

Rate of Change and Derivatives

The rate of change represents how a quantity changes over time, often expressed as a derivative. In this problem, R'(t) is the rate of fuel consumption, showing how many gallons are used per minute at time t. Understanding derivatives helps interpret and work with rates in real-world contexts.

Integration to Find Accumulated Quantity

Integration is the reverse process of differentiation and is used to find the total accumulated amount from a rate function. Here, integrating R'(t) over time gives the total fuel consumed, R(t), between 0 and 8 minutes. This concept connects rates to total quantities.

Piecewise Functions

Piecewise functions define different expressions over different intervals. The fuel consumption rate R'(t) changes form at t=8, requiring careful handling of each interval separately. Understanding piecewise functions ensures correct application of integration and interpretation of the problem.