NotThatSmart

I'm not great at direct addition, so let's convert everything to base 10. We have

\((221)_3 = (2 × 3^2) + (2 × 3^1) + (1 × 3^0) = (25)_{10} \)

\((122)_3 = (1 × 3^2) + (2 × 3^1) + (2 × 3^0) = (17)_{10} \)

Adding these two numbers up, we get \(25+17=42\)

So now we want to convert 42 back to base 3. 

First, we do 42/3, which has a quotient of 14 and remainder of 0.

Next, we take 14 and have 14/3, which has a quotient of 4 and remainder of 2. 

Next, we do 4/3, which has a quotient of 1 and remainder 1. 

Lastly, we do 1/3, which has quotient 0 and remainder 1. 

So our number is \(42_{10}=1120_3\)

Our final answer is \(1120_3\)

Thanks! :)

We know that \(f(3)=4\), this will be very important. 

Now, in the second equation, notice that we have \(f(2x/3)\). The only value of f(x) we know is when the x is 3. 

So, we want \(2x/3=3\). 

Solving for this, we get \(x=9/2\)

So, now we have 

\(y=(5 + f(3))/11\\ y=(5 + 4)/11\\ y=9/11\)

This means that \((9/2, 9/11)\) must be on the graph.

So our answer is \((9/2, 9/11)\)

Thanks! :)

I think the problem is meant to be \(0.1101_2\)

To solve this problem, let's note something really important. We have

\(\\0.1_2=\frac{1}{2}\\\\ 0.01_2=\frac{1}{2^2}=\frac{1}{4}\\\\ 0.0001_2=\frac{1}{2^4}=\frac{1}{16}\\\\\)

This is important because notice that \(0.1101 = 0.1+0.01+0.0001\)

So, now we just have \(\frac{1}{2}+\frac{1}{4}+\frac{1}{16}=\frac{8}{16}+\frac{4}{16}+\frac{1}{16} = 13/16\)

So 13/16 is our answer. 

Thanks! :)

First, let's note something really important for the problem. 

We have \(P_1 P_2 P_3 \dotsb P_{10}\), which actually just forms a regular decagon!

\(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\)is just the perimeter of that decagon. 

The length of a decagon can be found with formula \(\frac{\text{radius}}{2( -1 + \sqrt {5})}\)

Thus, the perimeter is \(10 (1/2) ( -1 + \sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)

So our answer is approximately 6.18

Thanks! :)

There are 2 ways to do the problem. 

First, we could take a look at all the possible conbinations we could have to give us x^2. 

First off, we have 8x*x = 8x^2. 

Next, we have 7x*x=x^2

Lastly, we have x^2*1 = x^2

Adding all the coefficients up, we get \(8+7+1=16\)

We could also just factor it, which isn't reccomended, but we get

\(x^{7}+4x^{6}+4x^{5}+19x^{4}+19x^{3}+16x^{2}+16x+1\)

Which shows the coefficient of x^2 is 16. 

So our final answer is 16

Thanks! :)

We could probably do this really hard way. 

By distributing every number, we get

\({x^{3}\left(x^{4}+3x^{3}+4x^{2}+8x+7\right)+x^{2}\left(x^{4}+3x^{3}+4x^{2}+8x+7\right)+x\left(x^{4}+3x^{3}+4x^{2}+8x+7\right)+1\left(x^{4}+3x^{3}+4x^{2}+8x+7\right)}\)

Simplfying, we have

\(x^{7}+4x^{6}+8x^{5}+16x^{4}+22x^{3}+19x^{2}+15x+7\)

As it clear shows\(15\) is the coefficient of x. 

So our final answer is 15. 

Thanks! :)

Notice that \(x^2 + y^2 = 4x + 5y\) is very similar to the equation of a circle. 

We have \(x^2-4x+y^2-5y=0\). 

Now, completing the square for both x and y, we get

\((x-2)^2+(y-5/2)^2=25/4+16\)

This is the official equation of a circle. 

It has the center at \((2, 5/2)\) and a radius of \(\sqrt{89/4}\)

The largest value of x is basically the x value of the center added to the radius. 

Thus, we have \(2+\frac{\sqrt{89}}{2} = \frac{4+\sqrt{89}}{2}\)

So our final answer is \(\frac{4+\sqrt{89}}{2}\)

Thanks! :)

So, first let's analyze the question. 

Note that when Dirk finishes, Edith has ran 14km and Foley has run 9km. 

Well, this means Foley runs \(9/14\) as fast as Edith. 

So when Edith runs the entire 24km, Foley has ran \(24 (9/14) ≈ 15.43 km\)

Which means that Foley is \(24 - 15.43 ≈ 8.57 km \) behind Edith. 

So 8.57 km is our answer!

Thanks! :)

So, right now there are \(5\) workers. They need to get the job completed 20 days earlier.

Since each additional worker adds on \(4\) days, we simply have

\(\frac{\text{Number Of Days}}{\text{Days Saved Per Worker}}\). Thus, we can plug in \(20/4 = 5\) workers.

So our answer is 5 workers. 

Thanks! :)

We can write an equation to solve this problem. We have

\((3p+12)(t-7)=2tp\)

Now, factoring the left side, we get

\(3pt-21p+12t-84=2tp\\ 12t+pt-21p=84\\ pt=21p-12t+84\\ pt=7(3p+12)-12t\\ pt+12t=7(3p+12)\\ t(p+12)=7(3p+12)\\ t/7=\frac{3p+12}{p+12}\)

We could test some values here, since no time was given. t has to be less than 24 and a multiple of 7. 

When t is 7, p is 0. 

When t is 14, p is 12.

When t is 21, p is undefined. 

Thanks! :)