A train passes a man standing on a platform in 9 sec. and pa

If one of the angles of a parallelogram is 120 degrees, what is the ratio of the lengths of the diagonals inside the parallelogram?



Can anybody help me out with a step by step solution?

Thanks!!!

The question is not correctly worded.
See the attached drawing. \(x\) and \(y\) can be any positive numbers.
If one of the angles is 120, the acute angle then is 60. We have 30-60-90 right triangles.

The longer diagonal squared is \((2x+y)^2+3x^2\) and the shorter one squared is \(3x^2+y^2.\)
Unless you know the ratio \(x:y,\) you cannot find the ratio of the diagonals.
In case \(x = y\), the requested ratio is \(\sqrt{\frac{4}{12}}=\sqrt{\frac{1}{3}}=1:\sqrt{3},\) which seems to be the given answer.

What is the source of this question? BTW, a parallelogram doesn't have diagonals outside itself...

Working alone, Andy can do a job in 20 days and Beth can do the same job in 30 days. When the two work together Andy has to leave 5 days before the job is finished. If Beth finishes the rest of the job herself, how long does the entire job take?



Can anybody help me out! I must do something wrong. Here is how I did it:

Andy: 1/20
Beth: 1/30

1/20x + 1/30x = 1

X= 1/12 (Rate of Andy and Beth)

5 days before the job is finished means that Andy and Beth worked 7 days together which still leaves 5/12 of the job to do.


Beth is working alone now which means:

1/30x = 5/12

Result: Additional 12,5

Unfortunately the wrong answer. Could somebody please help me?

Thanks!!!!

Mike and Tina can complete a job in 5 days working together. If Mike worked twice as efficiently as he did and Tina worked one third as efficiently as she did, the job would have been completed in only three days. How long does Tina need to do the job alone?



Many thanks for your effort!

Your mistake is the assumption that the job took 12 days to finish.
If Andy left 5 days before the job was finished and Beth had to work alone until the end, it means that the project took more than 12 days.

If you denote by T the time you are looking for, the following equation can be written:
\(\frac{T-5}{12}+\frac{5}{30}=1,\) as \(T-5\) days they worked together at a constant common rate of 1/12, then Beth worked 5 more days to finish the job alone.

Solving, you get \(T=15.\)

If it takes Mike M days to finish the job alone and Tina T days to do the same job, then we can write:

\(\frac{1}{M}+\frac{1}{T}=\frac{1}{5}\) and
\(\frac{2}{M}+\frac{1}{3T}=\frac{1}{3}.\)

Multiply the first equation by 2, subtract from it the second equation, and solve for T.
You will get \(T=25.\)

4 men and 3 women can do a job in 8 days. If 6 men and 8 women work on the same job, they will complete the job in half the time. How long will it take if one man and one woman work on the job?



Thanks guys!

Denote by \(M\) the rate of a man and by \(W\) that of a woman.
We can write the following equations:

\(4M+3W=\frac{1}{8}\) and \(6M+8W=\frac{1}{4}.\)

Solving the above system we get \(M=W=\frac{1}{56}\) so \(M+W=\frac{1}{28}.\)

Answer 28 days.

Could you please explain how you get to this formula:
The longer diagonal squared is \((2x+y)^2+3x^2\) and the shorter one squared is \(3x^2+y^2.\)

I am not getting what the longer and what the shorter diagonal is.

Sorry!! Many thanks!

Refer to the drawing attached to my previous post.

The longer diagonal is the one slanted to the right having one leg \(x+y+x\) (last \(x\) is dashed), other leg is the height of the parallelogram and is \(x\sqrt{3}\). Use Pythagoras.
The other diagonal is slanted to the left, one leg, vertical, again the height of the parallelogram and the other leg is the side of the parallelogram \(y.\)
Use again Pythagoras.

If an = an + 1 + 1 and the sum of the first 99 numbers is 99 what does an and a1 equal?




Thanks in advance

Does the formula say \(a_n=a_{n+1}+1\) ?

Oh, I checked the answer and I think this is what the formula should be. But the question must be what does \(a_{99}\) and \(a_1\) equal?

The given formula can be written as \(a_{n+1}=a_{n}-1\). It means that we have an arithmetic progression with a constant difference between consecutive terms of \(-1\). Or forget about the mathematical jargon, we have a sequence of evenly spaced numbers, the difference between each term in the sequence and the previous one being \(-1\).
Then \(a_2=a_1-1, \, a_3=a_2-1=a_1-2\,\) ...\(a_{99}=a_1-98.\)
The sum of any number of consecutive terms in such a sequence equals \(the \, average \, of \, the \, terms * the \, number \, of \, terms \, in \, the \,sum.\)
And the average of the terms always equals the average of the first and the last term. Therefore, the sum being 99, we have \(\frac{a_1+a_{99}}{2}*99=\frac{a_1+a_1-98}{2}*99=99\) .
Solving we get \(a_1=50\) so \(a_{99}=a_1-98=50-98=-48.\)

use the equilateral triangle area formula to get the length of the sides sqrt(3)/4 * a^2 = 4sqrt(3)
we find that a=4
Now the next step I drew a line bisecting the triangle to form a right triangle and solve for the hypotenuse which is the diameter. Using the 30-60-90 triangle numbers i get 8/sqrt(3) as the diameter and then solve for the area of the circle which comes to approximately 15.9 or round up to 16.

In the figure to the right what is the length of the longest side, if all of the triangles are right triangles?



Looking forward to your answers.. Thanks!!!

If Radius BC equals 2, What is the area of the shaded region of the circle formed by the rectangle inscribed in the circle?



Can somebody please explain a step-by-step solution please.

Thanks!!!

Apply Pythagoras starting with the triangle with sides 1 and 1. The hypotenuse is \(\sqrt{2}.\)
In the next right triangle, the hypotenuse is \(\sqrt{3}\) ... the longest hypotenuse is \(\sqrt{12}=\sqrt{2^2\cdot3}=2\sqrt{3}.\)