Adam started to run at a rate of 6 mph from city A towards city B, which is 18 miles away. Ten minutes later, Benjamin started to run from city B towards city Aand simultaneously, Caroline started to bicycle at a rate of 11 mph from city A towards city B. If Caroline caught up to Benjamin 50 minutes after passing Adam, what was Benjamin's rate in mph?

Benjamin's rate in mph was 6.44

Step-by-step explanation:

1. Let's review the information given to us to answer the question correctly:

2. If Caroline caught up to Benjamin 50 minutes after passing Adam, what was Benjamin's rate in mph?

A. Let's calculate when Caroline caught up Adam:

Adam's rate = 6 mph

Caroline's rate = 11 mph

10 minutes = 1/6 of an hour

Distance covered by Adam when Caroline started to bike = 6 * 1/6 = 1 mile

t = Time it took Caroline to catch up to Adam

Therefore, we can solve for t, this way:

11t = 6t + 1

11t - 6t = 1

5t = 1

t = 1/5 of an hour

t = 12 minutes

Caroline passed Adam after biking 12 minutes, therefore she caught up to Benjamin 50 minutes later, or after 62 minutes.

B. Let's calculate the distance she covered after 62 minutes, this way:

62 minutes = 1 hour and 2 minutes or 1 2/60 or 1.03

Distance covered by Caroline = Time * rate

Replacing with the values we have:

Distance covered by Caroline = 1 2/60 * 11

Distance covered by Caroline = 11 11/30 or 11.37 miles

Let's recall the distance from city A to city B is 18 miles, therefore Benjamin covered 6.63 miles (18 - 11.37) when Caroline caught up to him

C. Finally let's calculate Benjamin's rate, this way:

Rate = Distance/Time

Replacing with the values we know:

Rate = 6.63/1.03

Rate = 6.44 mph (rounding to the next hundredth)

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

b