Math for fun thread

[tk102]

I have no idea if there'll be any takers to this, but let's a try it anyway.

 

Rules: Present a math problem, or better yet, a story problem that you have already worked out yourself and see if others can solve it. Show your work when presenting an answer. Black magic earns no honor.

 

 

I'll start with an easy one to get the ball rolling.

 

Three guys order a pizza and they want to split it fairly 3 ways. One guy however doesn't want any crust, he only wants to eat the middle. So they devise a plan to cut a circle out of the middle of the pizza and give it to one guy and split the rest so they all have the same amount. What is the ratio of the diameter of cut-out portion to the diameter of the original pizza?

[Det. Bart Lasiter]

The circumference of the pizza equal C.

Let C = 16.

Since C = 16, the area of the pizza (A) = (8^2)*pi = 64*pi = 201.06 (rounded).

Therefore, (1/3)A = 67.02.

Accordingly, the circular center piece has an area of 67.02.

Therefore, the radius of the center piece = 67.02 = pi*(r^2) <=> 21.33 = r^2 <=> 4.62 = r.

Therefore, the center piece has a radius of 4.62 and a diameter equal to 2(4.62) = 9.24.

As such the ratio of the center piece's diameter to the original pizza is 9.24:16.00

 

 

Or not.

[Boba Rhett]

1/3 Pi

[tk102]

9.24:16.00

Yes that is approximately correct but not absolutely correct. The answer is an irrational number.

 

1/3 Pi

More lulz for 1/3 Pizza Pi. But nope.

[Det. Bart Lasiter]

I hate that smilie. I remember when I first joined there was someone with like nine of those in his sig and I hated him for it.

 

Please quantify my anger and calculate the limit of my anger as the number of 's approaches infinite.

 

 

 

Yes that is approximately correct but not absolutely correct. The answer is an irrational number.

Uhh. I was accounting for inaccurate cutting.

[Ray Jones]

a = radius of pizza

b = radius of desired piece (with one third of the area of the pizza)

 

3*b²*pi = a²*pi

 

3*b² = a²

 

sqrt(3)*b = a

 

b = a/sqrt(3)

 

the ratio is 1 : 1/sqrt(3)

[tk102]

the ratio is 1 : 1/sqrt(3)

*bling bling* you are correct sir. You could also just say sqrt(3).

 

 

Next question:

 

You want to build a cube from bricks. The bricks are 2x3x5 in dimension. Assuming you cannot break the bricks, what is the minimum number of bricks it will take to build this cube? What will be the size of the cube?

[Rev7]

ok, I think I will give this one a try:

Surface Area of Rectangle= 2wl+2wh+2hl

Volume of a cube= S to the third power (S3)

 

Height=2 Width=3 Lenght=5

SA=30x12x20

times

The volume of the cube S3 =810,000x104,976x160,000

Volume divided by 6 (sides of the cube) = 135,000x22,500x3,750

add this number togather=161,250 total bricks (without breaking the bricks)

 

Answer= minimum amount of bricks= 161,250 bricks (w/o breaking them)

Answer=size of cube= 135,000x 22,500 x 3,750

 

**** I bet ya I screwed something up!!!!! Hey, at least I tryed, eh?****

[Sabretooth]

I used a different tactic. Here's my analysis (haphazardly, I did this in my head):

 

LCM = 30, which is the length, breadth and height of the cube. Therefore, 5*6, 3*10 and 2*15.

Therefore, 30's cube is the size of the cube = 900 units cube.

The number of bricks required = 6*10*15 = 900 little brickie wickies.

 

A strong feeling I'm wrong, but it's a try. I'm working on verifying it.

[tk102]

You are correct Sabretooth. 900 bricks will create a cube of 30units length.

 

Another way to solve:

(2x3x5) * (2x3x5) * (2x3x5) = 27000 cu. units per cube

 

(2x3x5) = 30 cu. units per brick

 

27000/30 = 900 bricks/cube

[Sabretooth]

Damn, I feel so proud now. Next question, please.

[tk102]

Next problem.

 

You need to drive to somewhere that's 10 miles away and you have 20 minutes to get there. Unfortunately for the first 10 minutes you're stuck in traffic and only move at 5 miles per hour. How fast do you now have to go to make it there just in time?

[Ray Jones]

okay lets see that is 5 miles per 60 minutes for 10 minutes makes 5/6 miles in 10 minutes

so you have another 10 minutes for the rest of the total distance of 10 miles

that would be 10 - 5/6 miles or 60/6 - 5/6 miles which equals 55/6 miles in 10 minutes or 55 miles in 1 hour

 

dude 55 mph

[tk102]

Bravo Ray. Your turn.

 

>_>

<_<

or anyone else?

[Det. Bart Lasiter]

or anyone else?

 

I hate that smilie. I remember when I first joined there was someone with like nine of those in his sig and I hated him for it.

 

Please quantify my anger and calculate the limit of my anger as the number of 's approaches infinite.

 

 

ore da

[tk102]

Based on the statement, I'd say your anger is proportional the number of smilies. Therefore as smiles -> infinity, jmac's anger -> infinity.

 

I'll step it up.

 

A batter hits a baseball up in the air at a 55° angle. It takes 5 seconds to land in the glove of an outfielder. How far did the batter hit the ball?

 

 

Bonus points: How far would the ball have traveled if it left the bat at a 45° angle with the same velocity?

[Det. Bart Lasiter]

Based on the statement, I'd say your anger is proportional the number of smilies. Therefore as smiles -> infinity, jmac's anger -> infinity.

You forgot to take into account that as anger -> ∞, blood pressure -> ∞, as such, the limit of my anger is capped at whatever it is when my heart explodes.

[stingerhs]

A batter hits a baseball up in the air at a 55° angle. It takes 5 seconds to land in the glove of an outfielder. How far did the batter hit the ball?

well, first you have to recognize that any object thrown into the air will move in a parobolic arc. at time = 0, the ball is moving upwards at a 55 degree angle while at time = 5, the ball is moving downwards at a 55 degree angle.

 

using your basic calculus, you can calculate the deflection of the curve by first coming up with the tangent line of the arc. using time = 0 as one tanget point and time = 5 as another, you can then create a triangle from the beginning point to the end point as both lines will intersect at the peak of the parabola. dividing the triangle in half at time = 2.5, you now have two right triangles to derive the formula for the two tangent lines to the parabolic curve.

 

remember, your basic parabolic arc will use the forumla y = ax^2 + bx + c. taking the derivative, you will then receive the tangent line at (x,y) with dx/dy = 2ax + b. these formulas will provide us with the basis of our calculations.

 

i used the triangle from time = 0 to time = 2.5 as it will produce a positive slope for the tangent line, and i find positive numbers easier to work with. since we're working with the positive side, we'll work with the points (0,0) and (1,y) on the tangent line. to find y, you have to remember that y is equal to the opposite side of the triangle from the 55 degree angle. 1 is equal to the adjacent side of the triangle. as such, we'll use the tangent function to deturmine the slope.

 

tan(55) = opposite/adjacent

tan(55) = y/1

tan(55) = y

tan(55) = 1.4281

 

to get the slope, we use the slope formula:

 

a = (y2 - y1) / (x2 - x1)

 

plugging it in using the points (0,0) and (1, 1.4281), we get:

 

a = (1.4281 - 0) / (1 - 0)

a = 1.4281 / 1

a = 1.4281

 

going back to our derived formula for the tangent line, we know that the slope is equal to 2a. therefore, the slope will then equal 1.4281/2 which is approximately equal to 0.564.

 

knowing that b from our derived function is equal to the y-intercept, we know that b equals 0. as such, we can derive our final formula for the tangent line:

 

dx/dy = 0.564x

 

now, knowing that the slope for the tangent line at time = 5 is a negative slope, we can then derive that the slope for that tangent line is:

 

dx/dy = -0.564x + b

 

now, at the point time = 5, we know the ball is at the point (d,0). as such, b in the previous problem will be equal to the distance traveled by ball. so, by integrating the problem, we will then be able to solve for the distance in terms of x.

 

dx/dy = -0.564x + d

y = (-0.564x^2)/2 + dx + C

 

from there, you should now be able to solve for d. just don't ask me because i'm out of time where i'm at.

[Rev7]

I think that I will try this one:

 

Given that the field is 375 feet long, I cross multiplied.

 

55 degrees -- N

_____________

180 degrees-- 375 feet

55*375=20,625 feet _

20,625 ft. / 114.583

 

D=rt

(t)= 5 seconds or 5/60

(D)=114.58

®= 95.4 mph

 

Therefore, the distance traveled is 114.58 feet out of a 375 foot field.

Velocity= 95.4 mph ball (assumably fastball )

******bonus******

Using the same strategy, the distance traveled would be 93.75 feet.

 

I have confidence........(to say the least).........

[Commander Obi-Wan]

geez, what's with the Physics question? Physics is my bane.

 

If your tired of asking teh questions tee kay, I can certainly find a few.

[tk102]

I think you were on the right track stingerhs.

 

Here's the answer from a physics point of view:

 

 

**************

Acceleration is the change in velocity over time: a = dV/dt

 

Integrating acceleration through time gives us velocity. For a constant acceleration:

 

dV/dt = a

∫ dV/dt = ∫ a

∫ dV = ∫ a dt

V = at+ Vo

 

 

Now acceleration of gravity is 32.2 ft/sec² on earth.

 

The vertical velocity Vyof the baseball at time t then is given as:

Vy = -32.2t + Vyo

 

Since the curve is a parabola, we know that half way through the flight, the ball changes direction and comes back down. Therefore the vertical velocity, Vy, at t=2.5 is zero.

 

Solving for Vyo by setting Vy =0 at t=2.5:

 

0 = -32.2(2.5) + Vyo

 

Vyo = 80.5 ft/sec

This is the speed that ball is traveling vertically as it leave the bat.

 

 

To get the horizontal velocity, Vxo, we need to use some trigonometry, tan(55°) = Vyo /Vxo

 

Vxo = 80.5/tan(55°)

 

Vxo = 56.37 ft/sec

This is the speed the ball is traveling parallel to the ground when it leaves the bat.

 

 

Velocity is the change of distance over time: V = dX/dt

Integrating velocity through time gives us distance.

 

V = at + Vo

∫ V = ∫ ( at + Vo )

∫dX/dt = ∫ ( at + Vo )

∫dX = ∫ ( at + Vo ) dt

X = at²/2 + Vot + Xo

 

but in the horizonal direction, there is no acceleration (a=0) and we're starting the origin (Xo=0) is our origin so

 

X = 0 + (56.37)t + 0

 

t = 5

therefore

 

X = 281.8 feet

 

----------

Bonus

----------

For the bonus question, we need to know what the composite (both X & Y) velocity was when it left the bat. In the original scenario, Vyo was 80.5 and represented the 55° angle.

 

The composite velocity would be hypotenuse so

sin(angle) = Vyo/Vo

 

80.5/Vo = sin(55)

Vo = 98.27 ft/sec.

This is total speed the ball leaves the bat.

 

 

Now at a 45°,

Vyo = Vo*sin(45) = 69.48 ft/s vertically

Vxo = Vo*cos(45) = 69.48 ft/s horizontally

Note that the horizontal velocity is greater than in the original problem because it was hit at a flatter trajectory. So this makes sense.

 

We need to determine how long the ball remained in the air.

 

Vy = 0 at t/2 so

-32.2(t/2) + 69.48 = 0

-16.1t=-69.48

t = 4.31 seconds in the air

Note that it stayed less time in the air than in our original problem because it was hit at a flatter trajectory. So this makes sense.

 

 

X = Vxo * t

X = (69.48)(4.31)

X = 299.8 ft

 

Probably would've hit the fence in most parks.

 

***************

 

 

geez, what's with the Physics question? Physics is my bane.

Hope that helped then. :\

If your tired of asking teh questions tee kay, I can certainly find a few.

Go for it!

[Commander Obi-Wan]

Here is an easier question:

 

A car dealer claims that by buying a new car, Mike will pay 1/5 less for gas than he pays for the car he currently owns. If the car Mike currently drives costs 1/6 less to gas up than Dave's car, and Dave pays $700 per year, what will it cost Mike to put gas in a new car for 1 year? Assume all cars will be traveling the same distance.

[Rev7]

Interesting. Don't know anything about physics YET, but I WILL, so expect me back in a few years.

[Hallucination]

Here is an easier question:

 

A car dealer claims that by buying a new car, Mike will pay 1/5 less for gas than he pays for the car he currently owns. If the car Mike currently drives costs 1/6 less to gas up than Dave's car, and Dave pays $700 per year, what will it cost Mike to put gas in a new car for 1 year? Assume all cars will be traveling the same distance.

$486.12.

[Commander Obi-Wan]

$486.12.

 

Close, but that's not the answer. Remember to show your work.