Ask The Wizard #339

The answer is 27,981,391,815/6^15 = 0.059511.

Using a spreadsheet can simplify finding answers like this. For instance, let's consider a similar question: what are the odds of rolling a total of 20 with eight dice?

In the '1 Die' column, there is clearly one way to roll each total from 1 to 6.

For any cell depicting two dice or more, move one cell to the left and then sum the six cells located above it. The reasoning behind this is straightforward. Apply this formula to the cell corresponding to eight dice and a total of 20.

The resulting cell yields a total of 36,688. Since there are 8^{6} = 262,144 potential combinations to roll eight six-sided dice, the probability of achieving a total score of 20 with eight dice is calculated as 36688 / 262,144 = 0.139954.⁶Following a similar approach, the odds of rolling a total of 53 using 20 dice comes out to be 0.059511.

As a pyrotechnician responsible for the evening fireworks at a theme park, you've received a new type of rocket from Europe, and you are testing it to synchronize with the musical score of your show.

The answer is 483/49 = apx. 9.8571 seconds.

Let:
t = time since rocket fuel runs out.
r = time rocket fuel lasted.

Remember, the integral of acceleration gives us velocity, while the integral of velocity provides location. Let’s consider location relative to the ground.

Upon launching the rocket, we know its acceleration is 4.

Integrating this, the velocity of the rocket after r seconds can be expressed as 4r.

Integrating the velocity further, we determine the location of the rocket after r seconds to be 2r².

Next, let’s analyze what occurs once the rocket's fuel has burned out.².

We know the gravitational acceleration at that time will be -9.8.

The velocity imparted by gravity at time t is -9.8t, but there is also the upward velocity from the rocket, which is 4r.

The rocket will reach its maximum height when the velocity v(t) = 0. Let's find that point.

Let v(t) = velocity at time t

v(t) = -9.8t + 4r

In essence, whatever duration the rocket fuel lasted, the rocket will continue ascending for 20/49 of that time.

v(t) = 0 = -9.8t + 4r
4r = 9.8t
t = 40/98 r = 20r/49.

We also understand that the distance covered upon reaching maximum altitude is 138 meters.

Let's integrate v(t) to derive the expression for the distance traveled, which we will refer to as d(t).

+ 4rt + c, where c represents the constant of integration.

d(t) = -4.9t²As shown earlier, the distance traveled would be 2r² at the time the fuel runs out, which must equal the constant of integration, yielding:

We already established that the maximum height of 138 meters was attained at t = 20r/49, so let's insert t=20r/49 into the equation to solve for r:²Consequently, the rocket fuel lasted a total of seven seconds.

d(t) = -4.9t²+ 4rt + 2r²

We've established that the rocket continues its ascent for 20/49 of this total duration, equating to 140/49, approximately 2.8571 seconds.

d((20r/49) = -4.9((20r/49)²+ 4r(20r/49) + 2r²= 138

r²*(-1960/2401 + 80/49 + 2) = 138

r²= 49

r = 7

Therefore, the time elapsed from launch to the peak velocity reached is 7 + 140/49 = 483/49, approximately 9.8571 seconds.

Strategies and insights grounded in mathematics for playing various casino games such as blackjack, craps, roulette, and many others.

Kindly check your inbox for an email and follow the link provided to finalize your registration.