Question

The maximum afternoon temperature in a city might be modeled by t = 60 - 30cos(xπ/6) where t represents the maximum afternoon temperature in month, x, with x = 0 representing January, x = 1 representing February, and so on. Describe how to find the maximum temperature in April and state what this would be. How would you have to change the model if the maximum temperature in April, due to global warming, starts rising

Answer 1

1. First, we are going to find the value of $x$ form April. We know form our problem that  the maximum afternoon temperature in month, x, with x = 0 representing January, x = 1 representing February, and so on; therefore:
January         $x=0$
February       $x=1$ 
March            $x=2$
April               $x=3$
Now that we know that the value of $x$ that represent April is 3, we just need to replace $x$ with 3 in our model and evaluate:
$t=60-30cos( \frac{x \pi }{6} )$
$t=60-30cos( \frac{3 \pi }{6} )$
$t=60-30(0)$
$t=60$
We can conclude that the maximum temperature in April will be 60.

2. Remember that in a function of the form $y=C+Acos(Bx)$
$A$ is the amplitude 
$C$ is the vertical shift 

We have tow option to change our model to increase the maximum temperature to global warming:
1. Increase the value of D to increase the vertical shifting of the model. D affects the maximum value of the function; if we increase D, the maximum value of the function will increase as well. We know from our model that $D=60$, so to increase the maximum temperature of our model, we just need to increase the value of 60.
2. Increase the value of A to increase the amplitude of the model. The amplitude, also increases or decreases the maximum value of the function -regardless of the sing of A, so if we increase the value of A, we will increase the value of the function. We know from our model that $A=30$, so to increase the maximum temperature of our model, we just need to increase 30 (without considering the sign).