Bob has 60 feet of fencing to enclose a rectangular garden. if one side of the garden is x feet long, he wants the other side to be (30 – x) feet wide. what value of x will give the largest area, in square feet, for the garden?

1. To understand this, consider first that the 60 feet of fencing goes all the way around a rectangle and is the perimeter. So, if one side is x, so is the opposite side; and if another side is (30-x), so is its opposite side, so the total perimeter, 60 = 2x + 2(30-x). So, they've arranged it so that sides x and (30-x) always provide the 60 foot perimeter, even though you can change x.
Secondly, to maximize the area, A, consider that A = x (30-x), or -x^2 +30x, a quadratic equation, a parabolic curve facing down (the x^2 term is negative, so it faces down), and has a maximum, which will be where the derivative is zero. So, take the derivative: -2x + 30, set it to zero and solve and you find: x = 15. Plunk the 15 back into the original equation for area A, -x^2 + 30x, and you find it's -15*15 + 30*15 = 15(30-15) = 15*15 = 225 square feet. (I hope... feel free to check!) But it's x they wanter, which will be the 15 feet.
2. 3x^2=15x-2 They ask for the discriminant: that's the part of the Quadratic Equation under the square root sign, isn't it? so first put this into standard form and it should become obvious, just read it from the equation: 3x^2 - 15x + 2 = 0, the discriminant (call it N) is b^2 -4ac if my memory's working, a = 3, b = -15, c = 2, so N = 225 - 4*3*2 = 225 - 24 = 201 (do please check, I'm hurring 'cause I have to put supper on soon!). Since that's positive, there should be two solutions. Use sthe Quadratic Formula: [ -b ± √(D) ] / 2a. should be the 2 solutions, [15 ± √(201)] / 6, take it from there.
3. Those four choices should make it obvious to you, as they're all in slope-intercept form, all have slope 100. Trick is to substitute x = 2,000 for that point, and find out what y would be, the 100*x would be 200,000, and you're subtracting something from a choice, to get 3,000 for that point, so the something had better be 197,000 ... :-) Or you could say they're giving you y = 100x - S (S for "something" to tell which answer to choose), and it has got to work for x = 2,000 and y = 3,000, so solve it for S, that is
S = 100x - y, then put in your point (2,000, 3,000) and you'll have S = 100*(2,000) - (3,000) and from that you'd find your answer.
But you're right: aim to understand how they work, then the method's easy to remember, and easy to use in a zillion similar questions. The methods do make sense, and work quite well;)

Answer: 10pi

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.