I need help. Studying for SAT but I dont understand this problem. Would anyone be able to give a detailed explanation of questions a-f

Question

You are going on a business trip for a manufacturing company to Italy and need to figure out the currency conversion. You are going to take 1500 US dollars. After checking the internet you find the exchange rate is 1 US dollar equals 0.74 Euros.

a. Write the equation to convert US dollars to Euros.

b. How many Euros will you have if you convert all of the 1500 US dollars?

c. On your return trip you need to calculate how much US dollars you will have left over. To do this you need to find the inverse of your previous equation. What is the inverse equation to calculate how much US dollars you will have?

d. After your trip you have 75 Euros left. How much US dollars will you have?

The Italian company uses temperatures in Celsius and Kelvin for the different manufacturing processes. However, in America, the company uses Fahrenheit. The formula for converting Celsius to Kelvin is represented by k(x) below and the formula for converting Fahrenheit to Celsius is represented by c(x)

k(x)=x+273.16

c(x)=5/9(x-32)

e. For the American companies we need to find the Celsius to Fahrenheit formula by using the inverse. What is this formula?

f. Also for the Italian company we need to use the composition c(k(x)) to convert Fahrenheit to Kelvin. What is this formula?

Answer 1

Hi there,

a. is asking you to come up with an equation that shows the relationship between Euros and US dollars, which is given to us in the problem scenario. We are told that 1 US dollar equals 0.74 Euros. Therefore, if we define x to be the number of US dollars and y to be the number of Euros, we get y= 0.74x.

b. For this part, we just need to plug in x=1500 into the equation we found in the previous step.

c. This question has confusing phrasing, but it is essentially asking us for the equation we can use to determine the number of US dollars from the number of Euros. In other words, US dollars needs to be our dependent variable and Euros needs to be our independent variable. We can isolate x by dividing both sides by 0.74, getting x= y/0.74.

d. For this part, we just need to plug in y=75 into the equation we found in the previous step.

e. For this question, we need Celsius to be our independent variable and Fahrenheit to be our dependent variable. Essentially, you need to isolate x which represents degrees in Fahrenheit and you will get the correct answer.

f. For this question, we need to plug in the expression x+273.16 for the value of x in c(x) to get 5/9(x+273.16-32). You can combine the two constant terms in the parentheses to simplify the expression. Let me know if you need me to explain composition in greater detail, and I would be more than happy to.

Hope this helps! Let me know if you have any further questions or concerns.

Answer 2

For a, we want an equation that turns 1 dollar (US) into 0.74 dollars (Euros). This is a ratio/rate problem.

(1 US Dollar)/(0.74 Euros) = (x US dollars)/(y Euros)

Let's simplify this a bit by removing the units

1/0.74 = x/y

Now if we simplify again

y = 0.74x

Let's check if this makes sense: x is US dollars, so if I set x to be 1, I should get y=0.74, which is what I get!

b. To convert, plug in 1500 for x in our equation we just made.

c. One of the easiest ways to find an inverse is to swap x and y, then solve for y. This should get us the value to plug in for the original y to get the original x (swapping input and output = reverse operation)

Original: y = 0.74x

Swap: x = 0.74y

Solve: 1/0.74 x = y (note: 0.74 = 74/100, and 1/0.74 is just the recipricol: 100/74)

Simplify: 50/37x = y

d. Plug this value into the x of our inverse equation!

Okay, second problem:

e. The celsius to Fahrenheit equation is c(x)=5/9(x-32)

Inverse, just like the previous

Swap: x = 5/9(y-32)

Solve: 9/5x = y-32 ==> 9/5x + 32 = y

This is our inverse!

f. Composition is just taking the output of one function, and using it as the input for the second. It's easy to remember when written in that form as you just plug the equations into each other as shown.

k(x)=x+273.16

c(x)=5/9(x-32)

c(k(x)) = c(x+273.16) = 5/9((x+273.16)-32)

Now simplify and you have the composition.

I've intentionally left a few of these unfinished, but you should be able to make the last few steps. Hope this helps!