The flight paths of two Thunderbird jets are plotted on a Cartesian coordinate plane, and the
equations of the jets’ flight paths are represented by y=3^x+2 and y = (1/4)^x. Where do the flight paths intersect?
3^x+2 = (1/4)^x
very nasty, what method of solving do you have?
The way you typed the equation, they do not intersect for positive values of x and y
http://www.wolframalpha.com/input/?i=plot+y%3D3%5Ex%2B2+,+y+%3D+(1%2F4)%5Ex
did you mean:
3^(x+2) = (1/4)^x ?? If so, then
log (3^(x+2)) = log ( (1/4)^x )
(x+2)log3 = xlog .25
x log3 + 2log3 = xlog.25
xlog3 - xlog.25 = -2log3
x(log3 - log.25) = -2log3
x = -.884228..
y = 3.4068
see: http://www.wolframalpha.com/input/?i=solve+y%3D3%5E(x%2B2)+,+y+%3D+(1%2F4)%5Ex
To find the intersection point of the two flight paths, we need to solve the system of equations formed by the two equations:
1) y = 3^x + 2
2) y = (1/4)^x
To start, let's equate the two equations and solve for x:
3^x + 2 = (1/4)^x
Now, we can rewrite (1/4)^x as 4^(-x):
3^x + 2 = 4^(-x)
Next, let's subtract 2 from both sides:
3^x = 4^(-x) - 2
We can rewrite 4^(-x) as 1/4^x:
3^x = 1/(4^x) - 2
Now, multiplying both sides by 4^x will eliminate the fractions:
3^x * 4^x = 1 - 2 * 4^x
Using the properties of exponents, we can simplify this equation:
(3 * 4)^x = 1 - 2 * 4^x
12^x = 1 - 2 * 4^x
Since 12 is not a power of 4, we cannot easily combine the terms on the right-hand side. Therefore, to find an approximate solution, we can use a numerical method such as the Newton-Raphson method or guess-and-check.
Let's use a guess-and-check approach to find an approximate intersection point:
1) When x = 0, we have:
12^0 = 1 - 2 * 4^0
1 = 1 - 2
1 = -1 (Not true)
2) When x = 1, we have:
12^1 = 1 - 2 * 4^1
12 = 1 - 8
12 = -7 (Not true)
3) When x = -1, we have:
12^(-1) = 1 - 2 * 4^(-1)
1/12 = 1 - 2/4
1/12 = 1/2 (Not true)
From this, we can see that none of these values satisfy the equation. Therefore, there is no exact intersection point based on this equation.
However, if you would like to find an approximate intersection point, you can use a graphing calculator or software to plot the two equations and identify their intersection point.