Question

PLEASE PLEASE HELP WITH THIS, i have 3 problems and I need them solved ASAP

1. Ryan and Kayley launch their rockets at the same time. The height of Ryan’s rocket, in meters, is given by the function
f(x)= -3.4x^2 +64x
where x is the number of seconds after the launch. The height of Kayley’s rocket, in meters, is given by the function
g(x)=-3.4x^2 +25x +45
where x is the number of seconds after the launch.

Algebraically find the moment when the rockets are at the same height, and then use that to calculate the height. Round to the nearest hundredth. Show all work.

2. What is the average rate of change for the function
g(x) for the interval [4, 9]? g(x)=4x^2 + 3x -2

3. The cost per guest of catering an event of no more than 150 people is modeled by the function
f(x)=45+15x
The number of guests is modeled by the function
g(x)=150−x
where x represents the number of guests fewer than 150 that attend
Evaluate
(f∙g)(68)
and interpret what it means in the context of the problem

please please please help me get these done TODAY, I need all work shown and explained!

Answer 1

Sure, I'd be happy to help you with these problems!

1. To find the moment when the rockets are at the same height, we need to set the two functions equal to each other and solve for x:

-3.4x^2 +64x = -3.4x^2 +25x +45

Simplifying this equation, we get:

39x = 45

x = 1.15

So the moment when the rockets are at the same height is 1.15 seconds after the launch. To find the height at this moment, we can plug x = 1.15 into either function (they will give the same answer since they are equal at this point).

f(1.15) = -3.4(1.15)^2 + 64(1.15) = 41.18 meters (rounded to the nearest hundredth)

So the rockets are at the same height of 41.18 meters at 1.15 seconds after the launch.

2. The average rate of change for a function over an interval is the slope of the line connecting the two endpoints of the interval. To find this, we can use the formula:

average rate of change = (g(9) - g(4)) / (9 - 4)

g(9) = 4(9)^2 + 3(9) - 2 = 331

g(4) = 4(4)^2 + 3(4) - 2 = 62

average rate of change = (331 - 62) / 5 = 53.8

So the average rate of change of g(x) over the interval [4, 9] is 53.8.

3. (f ∙ g)(68) means we need to plug 68 into g(x) to find the number of guests and then plug that into f(x) to find the cost per guest, and then multiply them together.

g(68) = 150 - 68 = 82

f(82) = 45 + 15(82) = 1275

(f ∙ g)(68) = 68 * 1275 = 86700

So the cost of catering an event with 68 guests is $86,700.

I hope that helps! Let me know if you have any questions or need further explanation.