Mathematics

Mathematics

Problem

A chemical reaction produces a reactant at a rate given by g(t)=0.5t2−3t+4g(t)=0.5t^{2}-3t+4 grams per hour, where tt is the time in hours. Determine the net amount of reactant produced, in grams, during the time interval 1≤t≤51\le t\le 5. a) −23-\dfrac{2}{3} b) 43\dfrac{4}{3} c) 23\dfrac{2}{3} d) 00
Intuitive explanation

The key insight is that a rate function tells you how fast something is changing, and to find the total amount produced over a time interval, you need to accumulate all those tiny changes—which is exactly what integration does. You're summing up all the instantaneous production rates from t=1t=1 to t=5t=5 to get the net amount. Think of it as: if g(t)g(t) is grams per hour, then ∫15g(t) dt\int_1^5 g(t)\,dt gives you the total grams.

Intuitive explanation

The key insight is that a rate function tells you how fast something is changing, and to find the total amount produced over a time interval, you need to accumulate all those tiny changes—which is exactly what integration does. You're summing up all the instantaneous production rates from t=1t=1 to t=5t=5 to get the net amount. Think of it as: if g(t)g(t) is grams per hour, then ∫15g(t) dt\int_1^5 g(t)\,dt gives you the total grams.

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