1. First, we are going to find the value of [tex]x[/tex] 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 [tex]x=0[/tex]
February [tex]x=1[/tex]
March [tex]x=2[/tex]
April [tex]x=3[/tex]
Now that we know that the value of [tex]x[/tex] that represent April is 3, we just need to replace [tex]x[/tex] with 3 in our model and evaluate:
[tex]t=60-30cos( \frac{x \pi }{6} )[/tex]
[tex]t=60-30cos( \frac{3 \pi }{6} )[/tex]
[tex]t=60-30(0)[/tex]
[tex]t=60[/tex]
We can conclude that the maximum temperature in April will be 60.
2. Remember that in a function of the form [tex]y=C+Acos(Bx)[/tex]
[tex]A[/tex] is the amplitude
[tex]C[/tex] 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 [tex]D=60[/tex], 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 [tex]A=30[/tex], so to increase the maximum temperature of our model, we just need to increase 30 (without considering the sign).
