There is an arcade in the amusement park. A computer game there uses functions to simulate the paths of an archer’s arrows. The
x-axis represents the level ground on which the archer stands, and the coordinate pair (2,5) represents the top of a castle wall over which he is trying to fire an arrow.
In response to user input, the first arrow followed a path defined by the function f(x)=6-x^2, failing to clear the castle wall.
The next arrow must be launched with the same force and trajectory, so the user must reposition the archer in order for his next arrow to have any chance of clearing the wall.
a. How much closer to the wall must the archer stand in order for the arrow to clear the wall by the greatest possible distance?
b. What function must the user enter in order to accomplish this?
c. If the user can only enter functions of the form f(x+k), what are all the values of k that would result in the arrow clearing the castle wall?
In response to user input, the first arrow followed a path defined by the function f(x)=6-x^2, failing to clear the castle wall.
The next arrow must be launched with the same force and trajectory, so the user must reposition the archer in order for his next arrow to have any chance of clearing the wall.
a. How much closer to the wall must the archer stand in order for the arrow to clear the wall by the greatest possible distance?
b. What function must the user enter in order to accomplish this?
c. If the user can only enter functions of the form f(x+k), what are all the values of k that would result in the arrow clearing the castle wall?
1 answer:
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Answer:
or
(depending on your teacher's format preference)
Step-by-step explanation:
<h3><u>Proportionality background</u></h3>Proportionality is sometimes called "variation". (ex. " 'y' varies inversely as 'x' ")
There are two main types of proportionality/variation:
- Direct
- Inverse.
Every proportionality, regardless of whether it is direct or inverse, will have a constant of proportionality (I'm going to call it "k").
Below are several different examples of both types of proportionality, and how they might be stated in words:
y is directly proportional to x
y is directly proportional to x squared
y is directly proportional to x cubed
y is directly proportional to the square root of x
y is inversely proportional to x
y is inversely proportional to x squared
From these examples, we see that two things:
- things that are <u>directly proportional</u> -- the thing is <u>multipli</u>ed to the constant of proportionality "k"
- things that are <u>inversely proportional</u> -- the thing is <u>divide</u>d from the constant of proportionality "k".
In our question, y is inversely proportional to x, so the equation we're looking at is the following .
It isn't yet clear what the constant of proportionality "k" is for this situation, but we are given enough information to solve for it: "When y=12, x=5."
We can substitute this known relationship pair, and find the "k" that relates this pair of numbers:
<h3><u>Solving for k, and finding the general equation</u></h3>General Inverse variation equation...
Substituting known values...
Multiplying both sides by 5...
Simplifying/arithmetic...
So, for our situation, k=60. So the inverse proportionality relationship equation for this situation is .
The way your question is phrased, they may prefer the form:
Answer:
Her average speed is 48 miles per hour.
Step-by-step explanation:
We solve this question using a system of equations.
The speed equation is:
In which s is the speed, d is the distance, and t is the time.
Gabriella drives her car 320 miles and averages a certain speed.
So
Then
If the average speed has been 6 miles less she could have traveled only 280 miles in the same length of time.
So, which s - 6, d = 280.
From the first equation:
Replacing:
Her average speed is 48 miles per hour.