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That is correct. Thank you.

A transit train from Boston to New York and a passenger train from New York to Boston both departed at 3:00. The distance between the New York station and the Boston station is 280 miles. The speed of the transit train was 80 miles per hour. The speed of the passenger train was 60 miles per hour. At what time will the trains pass each other?    

The rate of closure is (80 mi/hr + 60 mi/hr) = 140 mi/hr   

At 140 mi/hr, it will take 2 hours to travel 280 miles

The engines of the trains will meet in 2 hours past 3:00  

Answer  5:00 o'clock  

_.

It says that that is incorrect.

Thank you for trying.

1 2/3 is not correct...

Thank you though.

Determine the number of ways to arrange the letters of the word PARABOLA, where the As are all adjacent.    

Since the A's all have to be adjacent, you can consider them as a single character. 

Then you just have to calculate the permutations of 6 items taken 6 at a time.  example:  PRBOL(AAA)  

6P6  =  720 ways  

_.

Two trains on opposite tracks leave the same station at the same time. One train travels at an average speed of 80 km/hr and the other travels at an average speed of 70 km/hr. How long after they leave the station will they be 50 km apart?    

The combined rate of separation is (80 km/hr + 70 km/hr) = 150 km/hr,

therefore they will be 1/3 of 150 km apart after traveling 1/3 of an hour. 

The answer is 1/3 hour, or 20 minutes  

_.

From logarithm properties, a/b = 3.

because that line are parallel, then :

111 + b = 180 => b = 69.

a + 39 = 180 => a = 141.

So, the answer is trapezoid; a = 141, b = 69

This doesn't make much sense.

Perhaps this is part of a larger question?

Hey there!

Since \(BD=DC\) we can do some algebra to find \(x\):

\(5x-10=3x+10\)

\(2x=20\)

\(\therefore x=10\)

Hope this helped :)

 2log_2(x+5) = log_2(x-9) + log_2(x+53)+1

Replace 1 by log_2(2)

2log_2(x+5) = log_2(x-9) + log_2(x+53) + log_2(2)

loga + logb = logab, so

2log_2(x+5) = log_2( (x-9)*(x+53)*2)

alogb = log(b^a), so

log_2( (x+5)² ) = log_2( (x-9)*(x+53)*2)

Hence

(x+5)² = 2(x-9)(x+53)

Can you take it from here?

16 = (2y - 2) + (y + 3)

16 = 3y + 1

15 = 3y

y = 5

You can double check by plugging 5 in. BC is equal to CD, because it is a perpendicular bisector.

Actually

-b/(2a)=4

6x + 8 = 3x + 38

3x = 30

x = ?

We will find y first:

y / 21 = (10 + y) / 36

y = 14

Using the angle bisector theorem we have:

DE / DG = x / x + 6

DG * x = DE (x + 6)

36x = 24(x + 6)

36x = 24x + 144

x = 12

EG = x + (x + 6) = 84

x/y = 100/9.

By the quadratic axis of symmetry, b/a = -2.

9 / 12 = 15 / x

9x = 12 * 15

x = 20

Both of these problems require the use of constant ratios. 

8. 

LM/QR = NK/SP

(4x+4)/48 = (7x-9)/60

(x+1)/12 = (7x-9)/60

12(7x--9) = 60(x+1)

84x - 108 = 60x + 60

24x = 168

x = 7

9.

YZ/SZ = RT/ST

(2x+2)/35 = (3x-7)/40

35(3x-7) = 40(2x+2)

105x - 245 = 80x + 80

25x = 325

x = 13

A parallelogram with a right angle is simply a rectangle.

Therefore, RT and SW are equivalent diagonals and Z, their intersection, is also their midpoint.

Therefore SW is twice the length of RZ.

Now we can use this information to solve the respective equation:

2(3x+5) = 7x-12

6x+10 = 7x-12

x = 22

What you should do for any composite figure is divide it up into smaller parts. We can divide this figure into two parts, the semicircle at the top, and the rectangle at the bottom. 

We can find the area of the rectangle easily, it is just 8*10 = 80.

Now to find the area of the semicircle. In case you didn't know, the formula for the area of a circle is π r^2. The r in this case will be 4, because it is half of the diameter, 8. Plug this into the formula to get:

= π (4)^2

= 16π

Dont forget to divide by 2, because it is a semicircle. 

= 8π

Now, we add the rectangle and the semicircle's area together to get 80 + 8π.

If you want it in decimal form, it is 105.1327412

Notice that you can form a square in the left corner with side lengths of 5, as shown below. Lines that are tangent to a circle and meet at a point are congruent, so we can label the triangle as shown above. Then, it's just the Pythagorean theorem:

\(18^2+(5+x)^2=(13+x)^2\)

I think you can take it from here.

Since the angles in a triangle add up to 180 degrees, angle ADC is 180-30-90=60. Since BDC = ADC - ADB, BDC = 60-25=35

The way that I remember how logarithms work is to move the base of the log so that is it "under" the number on the right.

So in this case it becomes:

81 = b ^ (2/3)

81^(3/2) = b

I think you can find b now. Hope this helped!

The length of the shorest possible path is 2*pi + 4.