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marishachu [46]
3 years ago
13

The shop sold 12,789 chocolate and 9,324 cookie dough cones. It sold 1078 more peanut butter cones than cookie dough cones. And

999 more vanilla than chocolate cones. What was the total number of ice cream cones sold?
Mathematics
1 answer:
Brut [27]3 years ago
5 0

Answer:

46,303

Step-by-step explanation:

12789+9324+9324+1078+12789+999= 46,303

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Solve these recurrence relations together with the initial conditions given. a) an= an−1+6an−2 for n ≥ 2, a0= 3, a1= 6 b) an= 7a
8_murik_8 [283]

Answer:

  • a) 3/5·((-2)^n + 4·3^n)
  • b) 3·2^n - 5^n
  • c) 3·2^n + 4^n
  • d) 4 - 3 n
  • e) 2 + 3·(-1)^n
  • f) (-3)^n·(3 - 2n)
  • g) ((-2 - √19)^n·(-6 + √19) + (-2 + √19)^n·(6 + √19))/√19

Step-by-step explanation:

These homogeneous recurrence relations of degree 2 have one of two solutions. Problems a, b, c, e, g have one solution; problems d and f have a slightly different solution. The solution method is similar, up to a point.

If there is a solution of the form a[n]=r^n, then it will satisfy ...

  r^n=c_1\cdot r^{n-1}+c_2\cdot r^{n-2}

Rearranging and dividing by r^{n-2}, we get the quadratic ...

  r^2-c_1r-c_2=0

The quadratic formula tells us values of r that satisfy this are ...

  r=\dfrac{c_1\pm\sqrt{c_1^2+4c_2}}{2}

We can call these values of r by the names r₁ and r₂.

Then, for some coefficients p and q, the solution to the recurrence relation is ...

  a[n]=pr_1^n+qr_2^n

We can find p and q by solving the initial condition equations:

\left[\begin{array}{cc}1&1\\r_1&r_2\end{array}\right] \left[\begin{array}{c}p\\q\end{array}\right] =\left[\begin{array}{c}a[0]\\a[1]\end{array}\right]

These have the solution ...

p=\dfrac{a[0]r_2-a[1]}{r_2-r_1}\\\\q=\dfrac{a[1]-a[0]r_1}{r_2-r_1}

_____

Using these formulas on the first recurrence relation, we get ...

a)

c_1=1,\ c_2=6,\ a[0]=3,\ a[1]=6\\\\r_1=\dfrac{1+\sqrt{1^2+4\cdot 6}}{2}=3,\ r_2=\dfrac{1-\sqrt{1^2+4\cdot 6}}{2}=-2\\\\p=\dfrac{3(-2)-6}{-5}=\dfrac{12}{5},\ q=\dfrac{6-3(3)}{-5}=\dfrac{3}{5}\\\\a[n]=\dfrac{3}{5}(-2)^n+\dfrac{12}{5}3^n

__

The rest of (b), (c), (e), (g) are solved in exactly the same way. A spreadsheet or graphing calculator can ease the process of finding the roots and coefficients for the given recurrence constants. (It's a matter of plugging in the numbers and doing the arithmetic.)

_____

For problems (d) and (f), the quadratic has one root with multiplicity 2. So, the formulas for p and q don't work and we must do something different. The generic solution in this case is ...

  a[n]=(p+qn)r^n

The initial condition equations are now ...

\left[\begin{array}{cc}1&0\\r&r\end{array}\right] \left[\begin{array}{c}p\\q\end{array}\right] =\left[\begin{array}{c}a[0]\\a[1]\end{array}\right]

and the solutions for p and q are ...

p=a[0]\\\\q=\dfrac{a[1]-a[0]r}{r}

__

Using these formulas on problem (d), we get ...

d)

c_1=2,\ c_2=-1,\ a[0]=4,\ a[1]=1\\\\r=\dfrac{2+\sqrt{2^2+4(-1)}}{2}=1\\\\p=4,\ q=\dfrac{1-4(1)}{1}=-3\\\\a[n]=4-3n

__

And for problem (f), we get ...

f)

c_1=-6,\ c_2=-9,\ a[0]=3,\ a[1]=-3\\\\r=\dfrac{-6+\sqrt{6^2+4(-9)}}{2}=-3\\\\p=3,\ q=\dfrac{-3-3(-3)}{-3}=-2\\\\a[n]=(3-2n)(-3)^n

_____

<em>Comment on problem g</em>

Yes, the bases of the exponential terms are conjugate irrational numbers. When the terms are evaluated, they do resolve to rational numbers.

6 0
2 years ago
Solve F=D+Drt for t.....
Anuta_ua [19.1K]

Answer:

t = \frac{-D+F}{Dr}

Step-by-step explanation:

Step 1: Flip the equation.

  • Drt + D = F

Step 2: Subtract D from both sides.

  • Drt + D - D = F - D
  • Drt = -D + F

Step 3: Divide both sides by Dr.

  • \frac{Drt}{Dr} = \frac{-D+F}{Dr}
  • t = \frac{-D+F}{Dr}

6 0
2 years ago
What is the value of m in the equation 5m − 7 = 6m 11? 18 1 −18 −1.
denpristay [2]

Answer:

-7 = m

Step-by-step explanation:

5m − 7 = 6m

Subtract 5m from each side

5m-5m − 7 = 6m-5m

-7 = m

6 0
2 years ago
HELPPPPPPPPPPPPPPPPP PLZZZZZZZZZZZZZZZZZZZZZZ
Amiraneli [1.4K]

Answer:

54$

Step-by-step explanation:

If 6% per year for 3 years, we can do 6 x 3 = 18 to find the total interest percent over all the years. Then we can do 18% of 300 = 54 Therefore, you will gain 54$ of simple in three years.                

P.S. Dont fall for those link scams! ;)

Hope this helped! If it did, please give me brainliest! It would help a lot! Thanks! :D

3 0
3 years ago
Estimate the perimeter and the area of the shaded figure to the nearest whole number
ollegr [7]

Answer:

See below ~

Step-by-step explanation:

<u>Perimeter</u>

  • 2(6 + 8)
  • 2(14)
  • <u>28</u> units

<u>Area</u>

  • 2(4 x 2) + 2(8)
  • 4(8)
  • <u>32</u> square units
7 0
1 year ago
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