Question

For the initial quantity Q0​=100 and the growth rate 3% per unit time, give a formula for quantity Q as a function of time t, and find the value of the quantity at time t=10. NOTE: Find exact answers for Q(t). Round your answers for Q(10) to three decimal places. (a) Assume the growth rate is not continuous Q(t)= At t=10,Q= (b) Assume the growth rate is continuous Q(t)= At t=10,Q=

Answer 1

The formula for quantity Q in terms of time t, for a non-continuous growth rate, is Q(t) = 100 * (1.03)^t and for a continuous growth rate is Q(t) = 100 * e^(0.03t). After substituting t=10 into both equations, Q(10) for non-continuous growth equals to 134.391 and for continuous growth equals to approximately 135.339.

Step-by-step explanation:

In the field of mathematics, specifically, exponential growth, we utilize certain formulas considering either continuous or non-continuous growth. The initial quantity (Q0​​) is given as 100 and the growth rate as 3% per time unit.

(a) For non-continuous growth rate, the formula used is Q(t) = Q0 * (1 + r)t where Q0 is the initial quantity, r is the growth rate and t is time. Substituting the given values, we get: Q(t) = 100 * (1 + 0.03)t.

At time t=10, we substitute 10 for t in our equation. The resulting quantity Q(10) = 100 * (1.03)10 = 134.391.

(b) For continuous growth rate, the formula used is derived from the compound interest formula and is expressed as Q(t) = Q0 * ert where 'e' is the base of natural logarithms (approximately 2.71828). Here, r is the growth rate, t is time and Q0​​ is the initial quantity. Substituting in the values, we get: Q(t) = 100 * e0.03t.

At time t=10, we substitute 10 for t in our equation. The resulting quantity Q(10) = 100 * e0.3 approximates to 135.339.

Learn more about Exponential Growth