dplostthegame

tip: if you want to actually make your LaTeX show, don't put dollar signs around it, there's a specific tool for it

let the cubic be ax^3 + bx^2 + cx + d, where a, b, c, and d are the coefficients

f(1) = 0 gives a + b + c + d = 0

so the sum of the coefficients is 0

please tell me if there are any errors

I think you made an error... BD^2 = 80, not 16/5. You divided by 5, not multiplied it. But thank you!

Thanks so much for your effort, even to those who failed!

ez

4 hrs at 60 mph = 240 miles

5 hrs at 80 mph = 400 miles

240 + 400 = 640

4 + 5 = 9

his average speed is 640/9 mph(approx. 71.1 mph)

Sorry, but I think you missed the fact that B is on AC. Perhaps you could try again? 

why does this look like a problem from AoPS

i hope you aren't trying to cheat on your homework

if you are i'll still give it but at least read the solution and understand it

kay

so

We don't know any other possible outputs from any possible inputs from f(x), so we can only assume that in order to simplify f(x), we have to make the input equal to 3. This gives 5x + 5 = 3, which simplifies to x = -0.4. Then, since f(3) = 4, we have 1/5 * 4 - 5 = 0.8 - 5 = -4.2.

Since the input is -0.4 and the output is -4.2, a point that must be on the graph of f(x) is (-2/5, -21/5).

The area of the triangle is (12 * 6)/2 = 36.

The area formula for a triangle is 1/2 * bh.

We already know the base; it is XY, which is 12.

So, 1/2 * 12 * h is less than or equal to 24.

This is only true when h is less than or equal to 4, in which the altitude from D to XY is less than 4.

So, the area of the triangle that does not satisfy this condition has height 2 and base 4, which works out to an area of 4.

So, the chance of faliure is 4/36 = 1/9, and the chance of success is 1 - 1/9 = 8/9.

The solution now has 200 + 150 + 250 = 600 mL of water.

As shown in the image, Franklin could be at (0, 1), (0, -1), (1, 0), or (-1, 0) after 1 step, as shown in the blue points in the image.

We add further possibilities to those on the 2nd step, giving the orange points in the image.

Note that we do not count the blue dots in our final answer. So, we count only the orange dots, in which there are 9. Therefore, there are 9 different points that Franklin could possibly be at after 2 steps.