Integrate the following function from x = pi/2 to x = pi: f(x) = 10sin^4(x)cos(x).
U-substitution is a way to tackle this problem. We see sin(x) with 4th power. We will choose that as our u. then our du becomes cos(x). Then we integrate: 10u^4du. The result becomes 2sin^5(x) from x = pi/2 to pi. Now, plug in for x: 2sin^5(pi) - 2sin^5(pi/2). Recall that sin(pi) is 0 and sin(pi/2) is 1. Thus, the answer of this integration by u-sub is -2.
Find solutions: y = 3x^3+5x^2-28x
Finding solutions means finding zeroes. In other words, what would x be when y equals zero? When y = 0, the equation becomes, 0 = 3x^3+5x^2+28x. We see that we can take out x to make the equation like this: 0 = x(3x^2+5x-28). Then, we see whether we can factor out the equation inside the parenthesis. It becomes 0 = x(3x-7)(x+4). Now, let's see which values of x can make the equation 0. Recall 0 times 0 is 0. Therefore, the solutions or the zeroes are x = 0, 7/3, and -4
On one side of a straight road that is 500m long, a planting community decides to plan trees. Each tree takes up 2m of the side of the road. How many trees can the planting community plant on the road?
The community can plant total of 250 trees on the road. Given that the road is 500m long, let x be the number of trees that the community can plant, and the question says every tree takes up 2m of the the road. That means, every time the community plants a tree, 2m of the distance of the road is covered. Then, you can come up with an equation 2x = 500 (notice the units are the same). Therefore, by dividing 500 by 2, the answer becomes 250 trees that the community can plant.