Solve for x: e^(2x) +2e^(x)-15=0
Here we can substitute another variable in for e^x. Call this variable y, so y=e^x. Therefore we have y^2+2y-15=0. This factors to (y+5)(y-3)=0. So, y=-5 and y=3. Substituting back in we have e^x=-5 and e^x=3. e^x has a range of 0 to infinity so it will never equal -5 and we can reject that answer. As for e^x=3, we can solve this by taking the natural log of both sides, giving us x=ln(3) since the natural log of e^x equals x.
A farmer has 60 feet of fencing and wants to fence in rectangular pen up against a barn. (the barn side does not need fencing, so we are only fencing three sides) What is the maximum amount of area that the farmer can fence in?
This is an optimization equation. First we need to figure out what equation we can use as a constraint and what equation we are optimizing. The constraint is the perimeter equation which here is 60= 2x+y (only 1 y because y here is going to represent the side opposite the barn that we don't need to fence). Our optimizing equation is going to be the area equation, A=x*y. Using the constraint equation we can get this in terms of one variable. Take 60=2x+y and solve for y by subtracting 2x from both sides, leaving us with y=60-2x. Substituting in for y in the area equation now we have A=x*(60-2x) which we can simplify to A=60x-2x^2. Now we can derive this, giving us A'=60-4x. Set this equal to zero to find our critical value, giving us 0=60-4x. Algebra then solves x=15. Normally in optimization problems we also check the end points, but here the end points would realistically be ridiculous to consider, being x=0 and x=30 which both clearly would give us an area of 0. So we know x=15 gives us our maximum. Plugging this into the area equation of A=60x-2x^2 gives us A=450 so the maximum area that can be enclosed here is 450 feet.
Find x: 24x-20=-4(1-5x)
x=4 First we look at the right side of the equation and distribute the -4 to the (1-5x), giving us 24x-20=-4+20x, then we can add 20 to both sides, giving us 24x=16+20x. Next, we can subtract 20x from both sides, giving us 4x=16. Finally we reach our goal of x being alone by dividing both sides by 4, giving us x=4.