Question

The normal monthly temperatures (°F) for Omaha, Nebraska, are recorded below.

Answer 1

To answer this question the first thing we will do is to make a graph of the dataset, to understand the periodic behavior of the temperature

We will do the first part of the qeustion.

Write a sinusoidal function that models the Ohmana's monthly temperature

Basically, we have to choose between two options, the sin(x) function and the cos(x) function, let us see the graph of them to have a better idea of which one resemble more accurately the Ohama's temperature

From the graph for the sin(x) and the cos(x) functions, we see that the cos(x) from zero is in a top position and then decrease, and the sin(x) function from 0 is in the lowest position and then starts to decrease

Looking at the points of the dataset, we see that the points start from a low position and then starts to increase, therefore the function sin(x) describe better this behavior

We are looking for a function of the form

`F(t)=Asin(Bt+C)+D`

We have to find now, which values for the parameters,A,B,C and D, make sense for the data presented.

The first parameters that we will analyze is the parameter A, which is the factor before the sin(x) function, this parameter is called amplitude and has the following graphical interpretation

In other words, the amplitute is the distance from the higher temperature to the mean temperature, in the present case the higher temperature is 77, that is the temperature of July, and the mean temperature is 49 , that is the middle point between the higher temperature and the lower temperature. We conclude that:

`A\approx77-49=28`

Looking at the options, we discover that there is only one option, that is sinusoidal following the sin(x) function, and such that the amplitude is 28. We conclude that the correct option for this question is the option C

.