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Fudgin [204]
1 year ago
14

6. Write the explicit formula for the geometric sequence of the height of the ball on the 10th bounce.First bounce: 54 inchescom

mon ratio: 0. 750
Mathematics
1 answer:
Nat2105 [25]1 year ago
5 0

The height of the ball initially is 54 inches. The common ratio has been provided which is 0.750.

The explicit formular for the geometric sequence is shown as;

\begin{gathered} ar^{n-1} \\ \text{Where a is the first term in the sequence,} \\ r\text{ is the common ratio betw}een\text{ every successive term} \\ \text{And n is the nth term} \\ \text{The explicit formular for the height of the ball on the 10th bounce is;} \\ ar^{n-1} \\ 54\times(0.750)^{10-1} \\ 54\times(0.750)^9 \\ 54\times0.07508468\ldots \\ 4.0545 \end{gathered}

The explicit formula is shown as;

54 x (0.750)^9

The rest is simply to solve for the height on the 10th bounce

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Find two power series solutions of the given differential equation about the ordinary point x = 0. compare the series solutions
monitta
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y''-y'=0\implies z'-z=0\implies z=C_1e^x\implies y=C_1e^x+C_2

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\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}-\sum_{n\ge1}na_nx^{n-1}=0

\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}-\sum_{n\ge2}(n-1)a_{n-1}x^{n-2}=0

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All the coefficients of the series vanish, and setting x=0 in the power series forms for y and y' tell us that y(0)=a_0 and y'(0)=a_1, so we get the recurrence

\begin{cases}a_0=a_0\\\\a_1=a_1\\\\a_n=\dfrac{a_{n-1}}n&\text{for }n\ge2\end{cases}

We can solve explicitly for a_n quite easily:

a_n=\dfrac{a_{n-1}}n\implies a_{n-1}=\dfrac{a_{n-2}}{n-1}\implies a_n=\dfrac{a_{n-2}}{n(n-1)}

and so on. Continuing in this way we end up with

a_n=\dfrac{a_1}{n!}

so that the solution to the ODE is

y(x)=\displaystyle\sum_{n\ge0}\dfrac{a_1}{n!}x^n=a_1+a_1x+\dfrac{a_1}2x^2+\cdots=a_1e^x

We also require the solution to satisfy y(0)=a_0, which we can do easily by adding and subtracting a constant as needed:

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4 0
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Step-by-step explanation:

Split the area into two sections by extending 6 in segment.

We get a rectangle of 5 in x 10 in and a trapezoid.

<u>Rectangle:</u>

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<u>Trapezoid:</u>

  • A = 1/2(b₁ + b₂)h
  • A = 1/2((15 - 5) + 7)*(10 - 6) = 1/2*17*4 = 34 in²

<u>Total area:</u>

  • 50 + 34 = 84 in²
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