proyaop

I can give a more detailed explanation.

Because BE is an angle bisector, we can use the angle bisector theorem. AB / BC = 9 / 12.

If AB = 15, so 15 / BC = 9 / 12

Thus, BC = 20.

\(\sqrt{2\sqrt{t-2}} = \sqrt[4]{7 - t}\)

First I multiplied both sides of the equation by ^4.

4(t-2) = 7 - t

4t - 8 = 7 - t

t = 3.

Our two equations are:

3a + 4b = 7

9a + 12b = k

If (a,b) have infinite many solutions, then those two equations above should be redundant.

If they are redundant, then the second equation is equal to the first equation by a factor of x3.

Thus, k = 7 x 3 = 21. 

ABC has lengths 6, 8, 12. To find angle B, we can use law of cosines.

12^2 = 8^2 + 6^2 - 2*8*6*cos(angB)

144 = 64 + 36 - 96cosB.

-11/24 = cosB.

Now that we have cosB, we can use the Angle Bisector theorem to get length of BD. 8/12 = BD/DC, so BD is 12/5. 

Then we can apply the law of cosines again. 

AD^2 = 8^2 + (12/5)^2 - 2*12/5*8*cos(angB)

AD^2 = 64 + 144/25 - (-88/5)

AD^2 = 1600/25 + 144/25 + 440/25

Thus, AD^2 is:

\({AD}^2 = {2184\over25} \) 

\(AD^2 = 87.36\)

First I drew out all the instructions. AME is a right triangle because it intersects AD with a perpendicular bisector. Then I drew a line down from E to AC that was perpendicular to AC. That length is the height of triangle ACE, and AC is the base, which is 30. So we have to find that height. Let us call the intersection of the orthogonal line segment from E to AC, P.

ME is 20, and AME is a right triangle. AD is a perpendicular bisector, so angle CAE is congruent to angle MAE. Angle AME is congruent to angle APE. AE is congruent to AE using reflexive property, so triangle APE is congruent to triangle AME. Thus PE, the height of triangle ACE with base AC, is 20 (because ME is 20). 

Now since we know the base of triangle ACE is 30, and the height is 20, then we can use (Base x Height)/2 to get the area.

Thus, the area of triangle ACE is 300 units squared. 

The circle has radius 25, so the circumference is 50pi. Since the sector is 1/3 of the area, then the circumference is 1/3 of the total circumference. Thus the circumference of the sector is 50/3pi. Then the sector is rolled into a cone so the circumference of the base of the cone is 50/3pi. The radius of the cone is 25/3. The height of the cone is 25. Thus, the volume of the cone is 15625/27pi units^3. 

If 6 were red than 4 were orange, then the fraction of orange lamps to total lamps is 4/10, which simplifies to 2/5.

0.806 :)

If 30 minutes costs $8.50, then we divide $25.50 by $8.50 getting 3.

Thus, Leon and his friends can ride a total of 1.5 hours or 90 minutes. 

Since AB and BC form a right angle, they are perpendicular. If the equation of AB is 4x + 8y = 16 (or x + 2y = 4), then the equation of point c has an opposite reciprocal slope of AB and it passes through point (1,-3). The slope of AB is -1/2, so the slope of BC is 2. Then we can use point slope form.

Thus, the equation of line representing side BC is \(y = 2x - 5\).