BuilderBoi

Without loss of generality, let the diameter be 100, and let the height be 100. 

The original volume is: \(250000 \pi\)

With the increased base, the volume of the new cube is: \(462400\pi\)

For the volume to stay the same, we have to multiply the height by \({250000 \over 462400} = {625 \over 1156}\)

This means that the percent decrease in the height is \(\approx \color{brown}\boxed{45.93 \text{%}}\)

The main diagonal is 2. Because it is a cube, all the side lengths are the same, and we have the equation: \(\sqrt{x^2+x^2+x^2}=2\)

Solving, we find \(x^2 = {4 \over3}\). The surface area is \(6x^2\), meaning the surace area of the cube is \(8\).

Now, we have the equation: \(4\pi r=8\)

Diving both sides by 4, we get: \(\pi r = 2\)

Solving for r, we find \(r = \color{brown}\boxed{\sqrt{2\over \pi}}\) 

\(c = \) Number of chicken sandwiches purchased

\(s\) = Number of vegetable soups purchased

\(2c+3s=16.50\)          (1)

\(c+2s=9.50\)              (2)

Subtract \(2s\) from both sides: \(c=9.50-2s\)

Substitute into (1): \(2(9.50-2s)+3s=16.50\)

Simplify: \(19-4s+3s=16.50\)

Solve: \(s = \color{brown}\boxed{2.50}\)

Now we have to wait for Melody to do the honors!!

Happy Birthday!!

I hope your day is going well :)

We know the number of boys who downloaded the game is\(1,916 \times 6 = 11,496\).

But, only half of these bought the full game, so only \(11496 \div 2 = \color{brown}\boxed{5,748}\) boys downloaded the game. 

There are \(94 \times 54 = 5076 \) nails. 

He then distributes the \(5076\) nails among \(9\) houses. 

Thus, each house receives \(5076 \div 9 = \color{brown}\boxed{564}\)

There are 2 cases to consider: 

Even - Odd - Even

Odd - Even - Even

For the first case, there are \(4 \times 5 \times 5 = 100\) numbers. (Remember, the leading digit can't be 0!!)

For the second case, there are \(5 \times 5 \times 5 = 125\) numbers. 

Thus, there are a total of \(100 + 125 = \color{brown}\boxed{225}\) numbers that meet these criteria.

Write in slope intercept form: \(y = -{2 \over3}x+1\)

Add \({2 \over 3}x\) to both sides: \(y+{2 \over3}x=1\)

Multiply the equation by 3: \(\color{brown}\boxed{2x+3y=3}\)

The slope of the line is 2. 

The point \((16,c)\) is 11 units away from \((5,5)\)

Because the slope is 2, the line goes up 22 from 5, meaning \(c = 5+22=\color{brown}\boxed{27}\)