BuilderBoi

Draw a vertical line from B. Label the point where it intersects AC point D.

 \(\triangle{ABD}\) is a 30-60-90 triangle. This means that BD is 10.

Using the Pythagorean Theorem, you find that AD is equal to \(\sqrt{300} = 10\sqrt3\).

To find DC, we need to use trig.

We use sin \(({\text{opposite}\over\text{hypotenuse}})\), and have: \({\sin(40)}={10\over{BC}}\)

This means that \(\color{brown}\boxed{BC \approx 15.55723}\)

Note that \(1 \over {1 \over3}\) is 3. When \(x\) gets bigger, the value of the expression gets smaller. 

Likewise, \(1 \over -{1 \over 4}\) is -4. As \(x\) gets smaller than \( -{1 \over 4}\), the value of the equation gets bigger, but never is greater than 0. 

So, the 2 ranges that work are \(\color{brown}\boxed{{x<-{1\over4} \space \text {and} \space{x>{1\over 3}}}}\)

I don't think there is a point...

You have the right idea guest, here is how you solve it: \(x+y+\sqrt {x^2+y^2}=30\)

Subtarct \(x+y\) from both sides: \(\sqrt{x^2+y^2}=30-x-y\)

Square both sides: \(x^2+y^2=x^2+y^2+2xy-60x-60y+900\)

Subtract \(x^2+y^2\) from both sides:\(0=2xy-60x-60y+900\)

Subsitute \(xy\) for 60:\(1020-60x-60y=0\)

Rewrite as: \(60x+60y=1020\)

Factor the left side: \(60(x+y)=1020\)

Divide both sides by 60: \(x+y = 17\)

From the other equation, divide by \(y\): \(x = {60\over y}\)

Plug into the other equation: \({60\over y}+y=17\)

Multiply by \(y\): \(60+y^2=17y\)

Subtract \(17y\) from both sides: \(y^2-17y+60=0\)

Use quadratic formula, and you find \(y = 12\)

Subsitute into the other equation, and \(x = 5\)

This means that the third side is \(13\), since the triangle must have a perimeter of 30. 

Thus, the side legnths are: \(\color{brown}\boxed{5,12,13}\)

The distance between \((-1,5)\) and \((-6,-2)\)i is 5 units across and 7 units down.

This means that the midpoint is 2.5 units to the right and 3.5 units above \((-6,-2)\). 

This means that the coordinates are: \(\color{brown}\boxed{(-3.5,1.5)}\)

Just as proyaop said :)

Yay!!! You're back!!!

I'll give this problem a shot...

There are 1000 x 999 x 998 x 997 options. 

There are 200 x 199 x 198 x 197 successes. 

So the answer is (200 x 199 x 198 x 197)/ (1000 x 999 x 998 x 997)

Write as an equation: \(36+x = 4x\)

Subtract \(x\) from both sides:

\(36 = 3x\)

Divide each side by \(3\)

This means that after \(\color{brown}\boxed{12}\) seconds, both the adults and the children ran \(\color{brown}\boxed{48}\) meters.

Multiply:

First, simplify the mixed numbers to a fraction. This can be accomplished by multiplying the whole number by the number beneath the fraction sign and adding it to the number on top of the fraction sign. This number would now be your numerator over the original denominator. For example, \(1 {1 \over 2}\) would be \((1 \times 2) + 1\). This means that the improper fraction, in this case, the product would be \({3 \over 2}\). 

Repeat this to the other number, so you would have \(9 \over 4\). To multiply fractions, you multiply the numerators and the denominators. So, in this case, \({ 3 \over 2 }\times {9\over4} =\color{brown}\boxed{{ 27\over 8}}\).

To divide fractions, you need to multiply by the reciprocal. To get the reciprocal, you swap the numerator and the denominator of the SECOND fraction ONLY. In this case, \({3 \over 2 } \div {9\over4}={3\over2}\times {4\over9}\). To multiply both these fractions, you do as stated above, and you recieve \(12 \over 18\) or \(\color{brown}\boxed{2 \over3}\)