amazingxin777

In Vonnie's 350 gallons of lemonade, she has \(272\frac{2}{9}\) gallons of water \(77 \frac{7}{9}\) gallons of lemon juice (in the ratio of 7:2). If we want to add more water to make it the same ratio as Lonnie's lemonade we need to write an equation. \(\frac{272 \frac{2}{9}+x}{77 \frac{7}{9}}=\frac{6}{1}\). I am going to leave it up to you to solve the equation.          

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If you are familiar with the standard form it is \(y=ax+b\) where \(a\) is slope and \(b\) is a constant. Let's put our slope in to change the equation to: \(y=\frac{9}{8}x+b\). Now all we have to do is solve for \(b\) since we are given the point (9,-5). Let's plug it in to form the equation: \(-5=\frac{9}{8}\times9+b\). Now we solve it: \(-5=\frac{81}{8}+b\). So \(b\) is equal to \(-5-\frac{81}{8}\Rightarrow\frac{-121}{8}\). Now we rewrite the equation: \(y=\frac{9}{8}x-\frac{121}{8}\)

Average speed is equal to total distance divided by total time. Total distance is 5 miles (going and returning). The formula is distance equals speed times time. Try figuring out the rest. 

I am not sure wha the x3 at the front means 

To put it into standard form (and to make it look better), multiply by -1. This is the resulting equation: \(-1(4x^2+3x-2)\). For the whole next part we will totally disregard the -1.Then we complete the square. But to do that we need to divide by four. \(4(x^2+\frac{3}{4}x-\frac{1}{2})\). Now we can easily complete the square. \(4[(x^2+\frac{3}{4}x+(\frac{3}{8})^2)-\frac{1}{2}]-\frac{9}{16}\). Then write it out into a simpler form: \(4[(x+\frac{3}{8})^2-\frac{1}{2}]-\frac{9}{16}\). Now expand what is within the parentheses. \(4(x+\frac{3}{8})^2-2-\frac{9}{16}\). Combine the constants \(\Rightarrow 4(x+\frac{3}{8})^2-\frac{41}{16}\).

But wait!!! If you remember what I said: we have to multiply by the negative one since we disregarded it back in the first paragraph. So we multiply the whole thing by negative one. This is what it will look like: \(-4(x+\frac{3}{8})^2+\frac{41}{16}\). By the Trivial Inequality, we know that that square right in the middle must be positive or 0. But if you look back at the equation: you have to multiply by a negative number (-4). So the smaller the square is the better. If the square can only be 0 or positive. We definitely want it to be 0. If you plug that into the equation, you will realize that it is only just \(\frac{41}{16}\). 

Edit: I realized I made 2 stupid mistakes! (Thanks EP!!!!)
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Ur Welcome

The last digit is 4. There’s a pattern: 4,6.... so just divide 35849 by 2. There’s a remainder of 1. So the answer is \(4\)

Great answer TLB!!!

first compare A and B. At first glance we con obviously tell that A is larger because B is only \(999^{1000}\)while A is 998 \(\times\) B. So we have one inequality already A > B.

Now let's compare B and C. B is \(999^{1000}\) and C is definetely greater than that so another inequality:

C > B.

Now let's compare C and D: If you estimate, then C is greater.

C > D. 

but we aren't done!!! A may be greater than B but it may be smaller than C or D. When we compare A and C then C is greater by a lot.

so C > A. 

Now we compare all our inequalities: B < A < C and D < C. 

Let's compare A and D. and we found that A is larger. 

D < A.

Last one: is D smaller than B? D is in fact smaller than B. 

This is the final solution: \(\boxed{D < B < A < C}\)

Yikes!! (I may have done some estimation wrong.)