NotThatSmart

First, let's note that triangle ABC is a 45-45-90 triangle. 

We also now know that \(AB = BC = 12/\sqrt{ 2} = 6\sqrt{ 2} \)

We have that

\(AD = AD \\ BD = BD \\ AB = BC\)

So by SSS congruence, triangles ABd and CBD are congruent. 

So, this means we have

\( [ ABD ] = (1/2) [ABC] = (1/2) ( 1/2) ( 6\sqrt{ 2})^2 = (1/4) (72) = 18\)

So 18 is our answer!

Thanks! :)

Alright, let's note that the sector has an angle of 83 degrees. 

This means the area of the sector is \(83/360\) of the total area. 

We can write the equation 

\(\frac{83}{360}\pi r^2 = 60\pi\\ 83/360r^2=60\\ r^2=60*360/83\\ r^2 = 21600/83\\ r=\sqrt{216000/83} \approx 16.132\)

Thanks! :)

Alright. Let's first put y in terms of x. 

We get \(y=25-x\). 

Subbing this value back into the second equation, we get 

\(3x+75\). 

Since x can't be negative, the smallest possible value of x is 0. 

When x is 0, we have \(3(0)+75=75\)

So 75 is our final answer. 

Thanks! :)

First, let's calculate how far Will got before Grace even started to row, 

We have \(d=st\), where d is distance, s is speed, and t is time. 

We have \((45 min)(50 m/min) = 2250 m \). Will rowed 2250m before Grace began. 

Now, Will and Grace have to cover \((2800 m) – (2250 m) = 550 m \) together. 

We have 

\( (50)(t) + (30)(t) = 550 \\ 80t = 550 \\ t = 550/80 = 6.875 \)

6.875 minutes is about 6 minutes and 52 seconds. 

The two started at 6:45, so they arrive at about \(6:51:52\) P.M. when they meet. 

Thanks! :)

While it may not seem the most important, let's calculate the semiperimeter. We get \(\frac{3+4+6}{2}=\frac{13}{2}\)

The reaon why we must find the semiperimeter is because we can use Heron's Formula. 

It states that the area is \(\sqrt{s(s-a)(s-b)(s-c)}\) where s is the semiperimeter and a,b, and c are the sidelengths. 

Applying this, we get \(Area = \sqrt{\frac{13}{2}(\frac{13}{2}-3)(\frac{13}{2}-4)(\frac{13}{2}-6)}\)

\(Area = \sqrt{\frac{455}{16}}=\frac{\sqrt{455}}{4}\)

This rounds to about \(5.3327\)

Thanks! :)

Alright. Let's first isolate y. We have \(y=25-x\)

Subbing this into the second equation, we get \(3x + 75\)

Well, the minimum value where x is not negative is when x is 0. 

When x is 0, we get \(3(0)+75=75\)

So 75 is our answer!

Thanks! :)

Well, this isn't as easy as it looks

We have \(2 \text{(number) (in ordinary arithmetic)}\\ 1 \text{(number) (in Boolean algebra with a notation where '+' denotes a logical disjunction)}\\ 0 \text{(number) (in Boolean algebra with a notation where '+' denotes 'exclusive or' operation, or in a quotient ring of numbers modulo 2)}\)

Theoretically, the answer is indeed 2, according to multiple sources. 

So our answer is 2. 

Thanks! :)

Let's first combine all like terms. We get 

\(x^{2}+18x+3\)

Now, completing the square,we get \((x+9)^2-78\)

So the first blank is 1

The second blank is 9

The third blank -78

Thanks! :)