The function C (t) = 0.16t/t^2 + t + 2 models the concentration, C, in milligrams per cubic centimetre, of a drug entering the bloodstream over time, t, in minutes.
a) Sketch the graph of the relation.
b) Explain the shape of the graph in the context of the concentration of the drug in the bloodstream.
a) To sketch the graph of the relation C(t) = 0.16t / (t^2 + t + 2), we can follow these steps:
1. Determine the domain: In this case, t can take any real value except when the denominator (t^2 + t + 2) equals zero. To find the values of t that make the denominator zero, we solve the quadratic equation t^2 + t + 2 = 0. However, the discriminant (b^2 - 4ac) is negative, which means there are no real solutions. Therefore, the domain is all real numbers.
2. Find the y-intercept: To find the y-intercept, substitute t = 0 into the equation: C(0) = 0.16(0) / (0^2 + 0 + 2) = 0 / 2 = 0. The point (0, 0) is the y-intercept.
3. Determine the behavior as t approaches positive and negative infinity: As t approaches positive infinity, the numerator (0.16t) grows, but the denominator (t^2 + t + 2) increases faster. Therefore, the ratio approaches zero. As t approaches negative infinity, the numerator (0.16t) decreases, but the denominator (t^2 + t + 2) also increases faster. Again, the ratio approaches zero.
4. Find the x-intercepts: To find the x-intercepts, we need to solve the equation C(t) = 0, which becomes 0.16t / (t^2 + t + 2) = 0. This equation is satisfied when the numerator is zero. So set 0.16t = 0, which gives t = 0. Hence, the graph intersects the x-axis at (0, 0).
5. Sketch the graph: Based on the information obtained in the previous steps, we can sketch the graph of the function C(t) = 0.16t / (t^2 + t + 2). The graph starts at the origin (0,0) and approaches the x-axis asymptotically as t goes to positive and negative infinity.
b) The shape of the graph indicates the concentration of the drug in the bloodstream over time. In the given function, C(t) = 0.16t / (t^2 + t + 2), the numerator 0.16t represents the increase in concentration as time passes. However, the denominator (t^2 + t + 2) increases faster than the numerator, causing the concentration to decrease. This behavior is seen in the graph as it starts at the origin and approaches the x-axis asymptotically, indicating that the concentration declines over time. The graph also suggests that the drug's effect diminishes as time goes on, eventually becoming negligible.