The Balcony Scene
Math brain teasers require computations to solve.
You are a scenery creator for plays and you have just been asked to create the Balcony for Romeo and Juliet.
The stage is 7m from the floor to the spotlights.
The balcony has to be a full circle (the other half will act as a platform for the actors waiting on the other side of the wall).
The balcony floor is 30cm thick.
The balcony needs a 1m(height) x 10cm(thickness) solid fence surrounding the perimeter of the half that will be seen by the audience.
The floor of the balcony must be exactly 4m from the ground.
The balcony is like so in order for the spot light (that shines at a 45 degree angle to the roof) will cover the entire balcony precisely when it is positioned above the middle of the balcony.
Pine is cheap at exactly $1 per 5000cm^3, tax included.
Exactly how much money will you spend to build the balcony?
Conversions:
100cm = 1m
2.54cm = 1inch
HintIf the floor of the balcony is 4 meters from the ground, and the room is 7 meters, then the floor is 3 meters from the spotlight. If the spot light shines at a 45 degree angle and the spot light will cover the entire balcony precisely when it is positioned above the middle of the balcony, then the radius of the balcony floor will be 3 meters (if the hypotenuse of a triangle is at a 45 degree angle to the y axis (and the side x is 90 degrees to the side y), the side x will always equal the side y and the remaining side will be the hypotenuse of those [the rules of an isosceles triangle]).
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Answer
[SEE HINT]
Therefore, the stage will be solved by the formula for a cylinder, [pi(r)^2]h where r = 300cm and h = 30cm (rule 3).
The balcony now needs a 100 cm x 10 cm solid fence. This now defies the simple rules of perimeter, you must incorporate the loss of radius per added centimeter towards the middle. In other words, you solve for the perimeter 10 times with the thickness valued as 1, incorporating the new radius each time. Remember, since it is only on one half, each answer must be halved.
2pi(r) finds the perimeter. Then multiplied by the height will be formulated 2pi(r)h where r = 300cm and h = 100cm. Since we only need half of that, it will be [2pi(r)h]/2 = pi(r)h.
Then we need to find the 9 other rings incorporating the loss of radius.
pi(r1)h + pi(r2)h + pi(r3)h + pi(r4)h + pi(r5)h + pi(r6)h + pi(r7)h + pi(r8)h + pi(r9)h
This added to pi(r)h will yield the volume of the fence surrounding half of the stage.
Thus, the final equation will be:
let a represent the height of the room, then b represent the height of the balcony off the floor, then f represent the thickness of the balcony floor, then r represent ab or the radius of the balcony floor, then h represent the height of the fence.
V = {[pi(ab)^2]f + [pi(r)h + pi(r1)h + pi(r2)h + pi(r3)h + pi(r4)h + pi(r5)h + pi(r6)h + pi(r7)h + pi(r8)h + pi(r9)h]}
= {[pi(90000)30] + [pi(300)100 + pi(299)100 + pi(298)100 ... ... pi(291)100]}
= {(2 700 000pi) + [30 000pi + 29 900pi + 29 800pi + 29 700pi ... ... 29 100pi]}
~= 8482300.16469244 + 928340.629135783
~= 9410640.79382822 cm^3
If pine is cheap at exactly $1 per 5000cm^3, tax included,
the price is then:
P ~= (9410640.79382822 / 5000) * 1.00
~= $1882.13
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