# Some Medium to Hard Math Questions

<p>Thanks in advance for reading what is sure to be a long post.</p>

<p>Volume of a Solid</p>

<p>If the 3 visible sides of a rectangular shown above (too lazy to draw out, but basically front, top, and right side) have areas 10,36, and 10 respectively, what is the volume of the solid. The answer is 60, but alas, I have no idea why.</p>

<p>A Lot of Factors</p>

<p>X=(1/3 *3)(1/3^2 *3) (1/3^3 *3)...(1/3^50 *3)
The three dots (...) in the product above represent 46 missing factors of the form (1/3^n *3), where the integer n represents all of the consecutive integers from 4 to 49, inclusive. Which of the following is equal to X? The answer is 1/3^1225.</p>

<p>Deceptively Simple Work-Rate Problem</p>

<p>If a certain job can be performed by 20 clerks in 30 days, the number of clerks, working at the same rate, needed to perform the same job in 14 days is? The answer is 43, but I don't know how to get that.</p>

<p>Pendulum</p>

<p>The pendulum of a certain clock travels 30 degrees per swing. If the length of the pendulum is 3/pie meters and the pendulum swings back and forth once every second, what is the distance in meters covered by the lower end of the pendulum in one hour. The answer is 3600, but I got 1800. This is what I did. 30/360 times 6pie/6 and got .5 meters per second, then multiplied by 3600.</p>

<p>Searching for a Quicker Way</p>

<p>3, 2, 0, 6, 0, 8
If a and b represent any two unequal numbers from the list above, how many different values are possible for ab?</p>

<p>Obviously, anything times 0 is 0, and 3<em>2 is the same as 2</em>3, and there are few enough possibilities that I can make a chart, but is there a quicker way to do this?</p>

<p>For the Searching for a Quicker Way problem, is the answer 10? If so, I might be able to tell you how.</p>

<p>I took a brief glance on the first problem you offered with the 10,36,10 </p>

<p>Here's the way it works:</p>

<p>If you name the length of the "Front Face" = x
Name the length of the "Right Face" = z
They should share a common width of "y"</p>

<p>This would make the following equations</p>

<p>xy=10
yz = 10
and
xz = 36
(XZ is the "Top Face")</p>

<p>Here z = 10/y from the above equation
substitute this into XZ = 36 we get
(x)(10/y) = 36
Furthermore, we know x = 10/y </p>

<p>Substitute:
(10/y)(10/y) =36
100/y^2 = 36
y^2 = 100/36
y=5/3</p>

<p>We're not done yet...since XZ = 36 and y = 5/3, the volume would be 36 x 5/3 which is 60.</p>

<p>As for your second problem, the way I did it:</p>

<p>Let 1/3 = x; let 3 = y</p>

<p>So accordingly series would be:</p>

<p>xy, x^2y, x^3y...x^50y</p>

<p>Using the Sum Formula S=(n(a1+an))/2 I got 1275= 50(1+50)/2
Plugging into the thing..</p>

<p>x^1275y^50 </p>

<p>Since 3 x 1/3 = 1 we can remove 3 50s from the x as (1/3)^50 would negate 3^50. You can test this by putting into your calculator (0.5)^4 x 2^4.</p>

<p>PS:::: Are these MATHIIC probs? They seem like it.</p>

<p>The quickest way in solving the first problem is just using to visualize the rectangular solid. Volume is lxwxh and since thwey gave you the area of the faces, just take the square root of 26 and you have 1 sie, and the other sides must be 2 and 5. 2x5x6= 60. bam, 10 seconds.</p>

<p>^ If you see that XY and YZ both = 10, you can assume XY = YZ and therefore X = Z . Since XZ = 36, X = 6, Z = 6 </p>

<p>Then plug in YZ so 6Y = 10; Y=5/3</p>

<p>However, Quix, I'm not sure what you mean by the other sides must be 2 and 5. The correct increments are Width = 6 Length = 6 Height = 5/3.</p>

<p>No, these are actually questions from a SAT I practice test, so that's why I'm so confused! Some of these questions seem too intense for SAT I.</p>

<p>Anyway, thanks for the help guys! Any thoughts on the work/rate or the pendulum problems?</p>

<p>Here is how to do the Pendulum Problem</p>

<p>We know that 3/pi is the radius and the angle 30 is between the radius. I drew a circle with two radius separated by 30 degrees. </p>

<p>Since we want to know what is covered by a swing, we want to know the arc length opposite of the 30 degrees. This can be found by finding 30/360 of the Circumference. </p>

<p>2(pi)(3/pi) = 6<br>
30/360 x 6 = 1/2</p>

<p>Now, we know it says 1/2 m per a swing back and forth so this # is doubled</p>

<p>1 m per 1 second
60 m per 60 second (1 min)
3600 m per 60 min ( 1 hr)</p>

<p>There you have it: total time - about 30 seconds</p>

<p>

</p>

<p>A very useful trick:
multiplying all three equations
(xy)(yz)(xz) = 10<em>10</em>36
(xyz)^2 = 3600
xyz = 60</p>

<p>^ Wow, I never thought of that! Isn't it very situational, though? How do you know when to use it? Any certain types of probs?</p>

<p>I did question 1 the way gcf did it, but he beat me to the post :).</p>

<p>Now, the work-rate problem.</p>

<p>Use some common sense here. If you needed to get something done quicker, you'd need more helping hands right? Therefore, the less the number of days, the greater the number of clerks. So the number of clerks and number of days are inversely proportional.</p>

<p>c1/c2=d2/d1 (c=no. of clerks, d=no. of days)</p>

<p>Setting up an equation, you get
20/c2=14/30
c2=20*30/14
c2=42.85</p>

<p>You obviously can't hire 0.85 of a clerk :D.</p>

<p>Some Medium to Hard Math Questions
Thanks in advance for reading what is sure to be a long post.</p>

<p>Volume of a Solid</p>

<p>If the 3 visible sides of a rectangular shown above (too lazy to draw out, but basically front, top, and right side) have areas 10,36, and 10 respectively, what is the volume of the solid. The answer is 60, but alas, I have no idea why.</p>

<p>Searching for a Quicker Way</p>

<p>Here are your numbers - 3, 2, 0, 6, 0, 8.</p>

<p>Start from the front.
3x2
3x0
3x6
3x8</p>

<p>2x6
2x8
(we leave out zero as we got an answer of 0 already(</p>

<p>6x8</p>

<p>In post #10

This was a true SAT question. In questions about the value of an expression quite often there is no need to solve for individual variable(s).
You play with given equations trying to construct that expression in some form.
With more practice you start seeing the path right away.</p>

<p>Important formulas:
a^2 - b^2 = (a-b)(a+b)
(a+-b)^2 = a^2 + b^2 +-2ab
(a +- 1/a)^2 = a^2 + 1/a^2 +-2</p>

<p>Pendulum</p>

<h1>The pendulum of a certain clock travels 30 degrees per swing. If the length of the pendulum is 3/pie meters and the pendulum swings back and forth once every second, what is the distance in meters covered by the lower end of the pendulum in one hour.</h1>

<p>Very similar to the lolilaughed's solution (post #8), just without fractions.
Working with degrees:
2x30=60deg in one sec.
60x60deg in one min.
60x60x60deg in one hour.
(60x60x60)/360 = 600 full circles

600x6m = 3600m</p>

<p>Here is how i did it
1)I used the same technique as gcf101.It's probably the shortest method.This trick comes in handy in many questions.</p>

<p>2)This could take some time</p>

<p>X=(1/3 <em>3) (1/3^2 *3) (1/3^3 *3) (1/3^4</em>3)......(1/3^50 *3)
or,X=(1/1) (1/3^1) (1/3^2) (1/3^3)......(1/3^49)
X=1/(3 * 3^2 * 3^3......3^49)
X=1/3^(1+2+3+4.......49)
Using the sum formula to find the sum of numbers from 1-49
sum = (n/2) (a+l)=(49/2) (1+49)=1225 </p>

<p>X=1/3^1225</p>

<p>3)20 men can do a work in 30 days
1 man can do the work in 30*20 days (Since less men require more time)
14 men can do the work in 600/14 days
=42.86
then looking at the options go for the closest answer tht u have</p>

<p>4)Pendulum
Similar to lolilaughed
The pendulum acts a radius to a circle and if the pendulum is rotated 360deg. it covers the length of the circumference
r=3/pie metres
circumference=2<em>pie</em>r = 6 metres
360 deg. = 6 metres
1 deg. = 6/360 metres
30 deg. = 0.5 metres
Its .5 metre per swing So in 1sec. it swings back and forth i.e.0.5*2=1
In 1 sec. it moves 1 metre
So, in 3600 sec. it moves 3600 metres</p>

<p>5)did the same way as tetrisfan did</p>

<p>anything left? :)</p>