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Korolek [52]
3 years ago
11

Need help with this problem:

Mathematics
1 answer:
Nitella [24]3 years ago
8 0

Answer:

<em>Henson: 3x + y = 163</em>

<em>Garcia: 2x + 3y = 174</em>

<em>adult ticket price: $45</em>

<em>child ticket price: $28</em>

Step-by-step explanation:

Henson Family:

3 adults + 1 child; total $163

3x + y = 163

Garcia Family:

2 adults + 3 children; total $174

2x + 3y = 174

Now we solve the system of equations.

Solve the first equation (Henson Family) for y.

y = 163 - 3x

Substitute 163 - 3x for y in the second equation (Garcia Family).

2x + 3<em>y</em> = 174

2x + 3(<em>163 - 3x</em>) = 174

2x + 489 - 9x = 174

-7x + 489 = 174

-7x = -315

x = 45

Now substitute 45 for x in the first original equation and solve for y.

3x + y = 163

3(45) + y = 163

135 + y = 163

y = 28

adult ticket price: $45

child ticket price: $28

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saw5 [17]

Answer: we reject the initial claim

(50. 3802, 51.6198)

Step-by-step explanation:

A)

The initial statement is that the mean temperature throughout history is 50°

This is the null hypothesis and it is denoted as H'

H': u = 50.

After taking a sample of 40 years, we realized that the mean temperature is 51°.

This is the alternative hypothesis, in contradiction to the alternative.

The alternative hypothesis is denoted as H1

H1: u > 50 ( upper tailed).

Sample size (n) = 40

Sample mean (x) = 51

Population standard deviation (σ) = 2

We use a z test to get the value of the test statistics.

We are using a z test because sample size is greater than 30 ( n = 40) and population standard deviation is given.

Z score = x - u/ (σ/√n)

Z score = 51 - 50 / (2/√40)

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Z score = 3.16

Our level of significance is 5%, and the critical value at this level of significance is 1.645.

By comparing the z score relative to the critical value, since our z score is greater than 1.645, it implies that we are in the rejection region hence we reject the initial claim.

B)

We are to construct a 95% confidence interval for population mean temperature.

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Where Zα/2 = 1.96 which is the critical value for a two tailed test at 5% level of significance.

For upper limit, we have that

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u = 51 + 1.96 (0.3162)

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u = 51.6198.

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Answer:

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General Formulas and Concepts:

<u>Alg I</u>

  • Terms/Coefficients
  • Factor
  • Exponential Rule [Dividing]: \displaystyle \frac{b^m}{b^n} = b^{m - n}

<u>Pre-Calc</u>

[Right Triangle Only] Pythagorean Theorem: a² + b² = c²

  • a is a leg
  • b is a leg
  • c is hypotenuse

Trigonometric Ratio: \displaystyle sec(\theta) = \frac{1}{cos(\theta)}

Trigonometric Identity: \displaystyle tan^2\theta + 1 = sec^2\theta

TI: \displaystyle sin(2x) = 2sin(x)cos(x)

TI: \displaystyle cos^2(\theta) = \frac{cos(2x) + 1}{2}

<u>Calc</u>

Integration Rule [Reverse Power Rule]:                                                                \displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C

Integration Property [Multiplied Constant]:                                                         \displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx

IP [Addition/Subtraction]:                                                             \displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx

U-Substitution

U-Trig Substitution: x² + a² → x = atanθ

Step-by-step explanation:

<u>Step 1: Define</u>

\displaystyle \int {\frac{dx}{(81x^2 + 4)^2}}

<u>Step 2: Identify Sub Variables Pt.1</u>

Rewrite integral [factor expression]:

\displaystyle \int {\frac{dx}{[(9x)^2 + 4]^2}}

Identify u-trig sub:

\displaystyle x = atan\theta\\9x = 2tan\theta \rightarrow x = \frac{2}{9}tan\theta\\dx = \frac{2}{9}sec^2\theta d\theta

Later, back-sub θ (integrate w/ respect to <em>x</em>):

\displaystyle tan\theta = \frac{9x}{2}  \rightarrow \theta = arctan(\frac{9x}{2})

<u>Step 3: Integrate Pt.1</u>

  1. [Int] Sub u-trig variables:                                                                                 \displaystyle \int {\frac{\frac{2}{9}sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta
  2. [Int] Rewrite [Int Prop - MC]:                                                                           \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta
  3. [Int] Evaluate exponents:                                                                                \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4tan^2\theta + 4]^2}} \ d\theta
  4. [Int] Factor:                                                                                                      \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4(tan^2\theta + 1)]^2}} \ d\theta
  5. [Int] Rewrite [TI]:                                                                                              \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4sec^2\theta]^2}} \ d\theta
  6. [Int] Evaluate exponents:                                                                                \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{16sec^4\theta} \ d\theta
  7. [Int] Rewrite [Int Prop - MC]:                                                                          \displaystyle \frac{1}{72} \int {\frac{sec^2\theta}{sec^4\theta} \ d\theta
  8. [Int] Divide [ER - D]:                                                                                         \displaystyle \frac{1}{72} \int {\frac{1}{sec^2\theta} \ d\theta
  9. [Int] Rewrite [TR]:                                                                                            \displaystyle \frac{1}{72} \int {cos^2\theta} \ d\theta
  10. [Int] Rewrite [TI]:                                                                                              \displaystyle \frac{1}{72} \int {\frac{cos(2\theta) + 1}{2}} \ d\theta
  11. [Int] Rewrite [Int Prop - MC]:                                                                          \displaystyle \frac{1}{144} \int {cos(2\theta) + 1} \ d\theta
  12. [Int] Rewrite [Int Prop - A/S]:                                                                          \displaystyle \frac{1}{144} [\int {cos(2\theta) \ d\theta + \int {1} \ d\theta]  

<u>Step 4: Identify Sub Variables Pt.2</u>

Determine u-sub for trig int:

u = 2θ

du = 2dθ

<u>Step 5: Integrate Pt.2</u>

  1. [Ints] Rewrite [Int Prop - MC]:                                                                       \displaystyle \frac{1}{144} [\frac{1}{2} \int {2cos(2\theta) \ d\theta + \int {1 \theta ^0} \ d\theta]
  2. [Int] U-Sub:                                                                                                     \displaystyle \frac{1}{144} [\frac{1}{2} \int {cos(u) \ du + \int {1 \theta ^0} \ d\theta]
  3. [Ints] Integrate [Trig/Int Rule - RPR]:                                                             \displaystyle \frac{1}{144} [\frac{1}{2} sin(u) + \theta + C]
  4. [Expression] Back Sub:                                                                                 \displaystyle \frac{1}{144} [\frac{1}{2} sin(2 \theta) + arctan(\frac{9x}{2}) + C]
  5. [Exp] Rewrite [TI]:                                                                                           \displaystyle \frac{1}{144} [\frac{1}{2}(2sin(\theta)cos(\theta)) + arctan(\frac{9x}{2}) + C]
  6. [Exp] Multiply:                                                                                                 \displaystyle \frac{1}{144} [sin(\theta)cos(\theta) + arctan(\frac{9x}{2}) + C]
  7. [Exp] Back Sub:                                                                                             \displaystyle \frac{1}{144} [sin(arctan(\frac{9x}{2}))cos(arctan(\frac{9x}{2})) + arctan(\frac{9x}{2}) + C]

<u>Step 6: Triangle</u>

Find trig values:

\displaystyle tan\theta = \frac{9x}{2}

\displaystyle \theta = arctan(\frac{9x}{2})

tanθ = opposite / adjacent; solve hypotenuse of right triangle, determine trig ratios:

sinθ = opposite / hypotenuse

cosθ = adjacent / hypotenuse

Leg <em>a</em> = 2

Leg <em>b</em> = 9x

Leg <em>c</em> = ?

  1. Sub variables [PT]:                                                                                         \displaystyle 2^2 + (9x)^2 = c^2
  2. Evaluate exponents:                                                                                      \displaystyle 4 + 81x^2 = c^2
  3. [Equality Property] Square root both sides:                                                  \displaystyle \sqrt{4 + 81x^2} = c
  4. Rewrite:                                                                                                           c = \sqrt{81x^2 + 4}

Substitute into trig ratios:

\displaystyle sin\theta = \frac{9x}{\sqrt{81x^2 + 4}}

\displaystyle cos\theta = \frac{2}{\sqrt{81x^2 + 4}}

<u>Step 7: Integrate Pt.3</u>

  1. [Exp] Sub variables [TR]:                                                                               \displaystyle \frac{1}{144} [\frac{9x}{\sqrt{81x^2 + 4}} \cdot \frac{2}{\sqrt{81x^2 + 4}} + arctan(\frac{9x}{2}) + C]
  2. [Exp] Multiply:                                                                                                 \displaystyle \frac{1}{144} [\frac{18x}{81x^2 + 4} + arctan(\frac{9x}{2}) + C]
  3. [Exp] Distribute:                                                                                             \displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C
3 0
2 years ago
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