part 2 of important test, all solutions need to be the same

helppp

Answer:

  16.65 ft

Step-by-step explanation:

You want the length of the shortest ladder that will reach from the ground to a building over an 8 ft fence at a distance of 4 ft from the building.

We recall that the trig relations in a right triangle are ...

  Sin = Opposite/Hypotenuse   ⇒   Hypotenuse = Opposite/Sin

  Cos = Adjacent/Hypotenuse   ⇒   Hypotenuse = Adjacent/Cos

In the model attached, the ladder (GH) makes angle α with the ground. The above relations tell us the ladder length is ...

  GH = GC +CH

  GH = 8/sin(α) +4/cos(α)

Using the reciprocal trig relations, this becomes ...

  GH = 8·csc(α) +4·sec(α)

The minimum length of the ladder corresponds to the value of α that makes the derivative of GH with respect to α be zero.

  GH' = -8·cos(α)csc(α)² +4·sin(α)sec(α)² = 0

Dividing by 4·cos(α)csc(α)², we get

  tan(α)³ -2 = 0

  α = arctan(∛2) ≈ 51.56095°

The length of the shortest ladder is then ...

  GH = 8·csc(51.56095°) +4·sec(51.56095°) ≈ 16.65 . . . . feet

The shortest ladder that will reach is 16.65 feet long.

__

Additional comments

In the second attachment, the horizontal axis is degrees of angle α, and the vertical axis is feet of length GH. The calculator is set to degrees mode.

The derivatives of the trig functions can be found from a suitable table, or by using the power rule with the derivatives of the primary sin and cos functions.

We can divide by cos(α)csc(α)² because we know it is not zero for the angle of interest. The value 2 in the tangent formula is h/d, where the fence is h units high and d units from the building. You will notice the length expression is h·csc(α)+d·sec(α). This is a generic solution for this sort of problem.

The problem can be worked using similar triangles and the Pythagorean theorem. This seems easier. The solution in most cases will be irrational, involving cube roots at some point.

g(x) intercepts the x-axis at x = 3, then the correct option is (3, 0)

For a function y = f(x), we define the x-intercept as the value of x for which:

f(x)=  0

The x-intercepts are also called zeros or roots of the function, and are the points where the graph of the function intercepts the x-axis.

In this case, the function is g(x) = -3x + 9, so here we need write and solve the equation:

g(x) = 0

-3x + 9 = 0

9 = 3x

9/3 = x

3 = x

This means that the x-intercept is x = 3, or (3, 0) written in point form, when we evaluate g(x) in x = 3 we get:

g(3) = -3*3 + 9

g(3) = -9 + 9

g(3) = 0

As expected.

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Answer: Using the triangle theorem, the angle (64 + x)° and 110° are alternate angles, therefore the angle marked, x is 46°

Step-by-step explanation: Alternate angles are equals :

(64 + x)° = 110°

We can solve for x thus ;

x + 64 = 110

x + 64 - 64 = 110 - 64

x = 46°

Hence, the value of x in the triangle is 46°

Answer:

(-1/10)

Step-by-step explanation:

I'll assume 1/4xy is (1/4)xy and not 1/(4xy).

(1/4)xy

(1/4)(-2/3)(3/5)

Multiple the numerators and denominators separately:

Numerator + 1*(-2)*3 = -6

Deniominator 4*3*5 = 60

Put them back together:  (-6/60)  This reduces to (-1/10)

answer is a. this is the long method but there is a short method — angles on a "z" since there are 2 parallel lines and a straight line across

Answer:

A. 47

Step-by-step explanation:

they are two parallel lines where by a transversal has passed throw them ; but first of all you have to look at the corresponding angles followed by alternative angles and so on;as angles on a str

By using the concept of simple interest, it can be calculated that

Brionny has invested £8400.

What is simple interest?

Simple interest is the interest earned on a certain principal at a certain rate over a certain period of time

For the first year-

Principal = £P

Rate = 2%

Time = 1 year

Interest = [tex]\frac{P \times 2 \times 1}{100}[/tex] =  £[tex]\frac{P}{50}[/tex]

Amount = £[tex](P + \frac{P}{50})[/tex]

             = £[tex](\frac{50P + P}{50})[/tex]

             = £[tex]\frac{51P}{50}[/tex]

For the Second year:

Principal =  £[tex]\frac{51P}{50}[/tex]

Rate = 1%

Time = 1 year

Interest = [tex]\frac{51P \times 1 \times 1}{50 \times 100}[/tex] =  £[tex]\frac{51P}{5000}[/tex]

Amount = £[tex](\frac{51P}{50} + \frac{51P}{5000})[/tex]

             = £[tex](\frac{5100P + 51P}{5000})[/tex]

             = £[tex]\frac{5151P}{5000}[/tex]

By the problem,

[tex]\frac{5151P}{5000} = 8653.68\\P = \frac{8653.68 \times 5000}{5100}\\[/tex]

P =  £8400

Brionny has invested £8400.

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Answer:

[tex]\frac{5}{7}[/tex] x + 2

Step-by-step explanation:

([tex]\frac{2}{7}[/tex] x - 6) + ([tex]\frac{3}{7}[/tex] x + 8) ← remove parenthesis

= [tex]\frac{2}{7}[/tex] x - 6 + [tex]\frac{3}{7}[/tex] x + 8 ← collect like terms

= ( [tex]\frac{2}{7}[/tex] x + [tex]\frac{3}{7}[/tex] x ) + ( - 6 + 8 )

= [tex]\frac{2+3}{7}[/tex] x + 2

= [tex]\frac{5}{7}[/tex] x + 2

Predicted amount of time to travel 30 kilometers is 7 hours and 40 minutes. If A train travels from City A to City B.

The pace at which an object's position changes in any direction is referred to as speed. The distance travelled in relation to the time it took to travel that distance is how speed is defined.

Given,

Average speed

Speed = Distance/Time

Speed = 9/3

Speed = 3 km/h

When distance is 12 and time is 4

Speed = 12/4

Speed = 3 km/h

When distance is 18 and time is 9

Speed = 18/9

Speed = 2 km/h

Average speed = 3 + 3 + 2/3

Average speed = 8/3

Average speed = 2.6

Predicted amount of time to travel 30 kilometers

Time = Distance/Speed

Time - 30/2.7

Time = 7.4

Time = 7 hours and 40 minutes

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The tax paid by Kerry is $2.89

As per the given data in question:

Kerry bought a bag of dog food for $37. 99.

Total cost = $37. 99

Rate of sales tax is 7.6 %

A sales tax is a consumption tax paid to a government on the sale of certain goods and services.

Sales tax: = Sales tax percent / 100.

[tex]=\frac{7.6}{100}[/tex]

= $0.076

Tax paid by Kerry = Total cost × Sales tax

= 37. 99 × 0.076

= $2.89

Therefore the tax paid by Kerry is $2.89

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The smaller number is 25 and the larger number is 30.

Given that, the sum of two numbers is 55.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Let the smaller number be x.

The larger number is 5 more than the smaller number.

Then, the larger number be x+5

The sum of two numbers is 55.

x+x+5=55

⇒ 2x+5=55

The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true.

Now, 2x+5=55

⇒ 2x=50

⇒ x=25

So, x+5=30

Therefore, the smaller number is 25 and the larger number is 30.

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Answer:

1080cm²

Step-by-step explanation:

Area of a quadrilateral = 1/2 x length of diagonal (length of perpendicular)

[tex] \frac{1}{2} \times 10(18 \times 12) \\ \frac{1}{2} \times 10(216) \\ \frac{1}{2} \times 2160 \\ {1080cm}^{2} [/tex]

Answer:

ok so I am also a high school student so my answer may not be correct but here are the answers.

1. 18500

2. 16000

3. 68.3%

4. 47.7%

5. 2.3%

Step-by-step explanation:

To find the answers to questions 1, 2, just add 500 to mean for above and substract 500 for below. I just used a graphic calculator for 3, 4 and 5.

For a casio calculator, here are the steps

menu, 2, dist, norm, ncd,

data: variable

lower: 3. 16500, 4. 17000, 5. 0

upper: 3. 17500, 4. 18000, 5. 16000

deviation: 500

Mean: 17000

Hope this helps.

The variance of the number of heads which come up when a fair coin is flipped 9 times is 2.25.

In this case, number of trials n = 9

Note that X = X₁ + X₂ + X₃ + X₄ +X₅ + X₆ + X₇ + X₈ + X₉

Xₐ = 0 if the flip #i is tails, and

Xₐ = 1 if the flip #i is heads.

Since the Xi are independent, then we have:

Var (X) = Var (X₁ + … + X₉)

= Var (X₁) + Var (X₂) + … + Var (X₉)

Then,

Var (Xₐ) = E(Xₐ^2) – E(Xₐ)^2

= 1/2 – 1/4

= 1/4

So, the variance is:

Var (X) = Var (X₁) + Var (X₂) + … + Var (X₉)

= 9 * (1/4)

= 9/4

= 2.25

Hence, the variance of the number of heads which come up when a fair coin is flipped 9 times is 2.25.

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To eliminate factors not discussed, the answer would be 3/7 shed each, assuming everyone builds at the same rate and effort.

The unitary method is a technique that involves determining the value of a single unit and then calculating the value of the requisite number of units based on that value. The unitary method's formula is to get the value of a single unit and then multiply that value by the number of units to achieve the required value. The term unitary refers to a single or unique entity. As a result, the goal of this approach is to determine values in reference to a single unit. For example, if a car travels 44 kilometers on two litres of gasoline, we may use the unitary technique to calculate how far it will go on one litre of gasoline.

Here,

3 sheds built by 7 workers,

let x = amount of shed each built

7x = 3

x = 3/7

Assuming everyone built at equal speeds and effort, to remove variables not addressed, the answer would be 3/7 shed each.

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The value of I in the obtained inequality is the number of levels that can be played for free.

A difference between two values indicates whether one is smaller, larger, or basically not similar to the other.

In other words, inequality is just the opposite of equality for example 2 =2 then it is equal but if I say 3 =6 then it is wrong the correct expression is 3 < 6.

As per the given,

Set up time + time taken in complete levels ≤ 60

For I number of levels,

5(¹/₆) + 7(¹/₃)I ≤ 60

By comparing the given inequality,

I = number of levels that are played for free.

Hence "The number of levels that can be played for free is the value of I in the resulting inequality".

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Answer:

a) The cost of buying x bottles of water is represented by the equation v = 0.50x, where v is the cost in dollars and x is the number of bottles of water. The gradient of this equation, which is 0.50, represents the cost per bottle of water.

b) The total cost of buying one bottle of water per day for a year (365 days) is 365 * 0.50 = $182.50.

c) If a refillable water bottle costs $40, then a person would have to drink 40 / 0.50 = 80 bottles of water in order for the refillable water bottle to be more cost effective.

d) Some factors that might influence someone's decision to carry their own water bottle or to buy bottled water include convenience, availability of clean drinking water, personal preference, and environmental impact.

The area of the region bounded by the given curves. y = 6x2 ln(x), y = 24 ln(x) is 2.85 sq.units.

In this question we need to find the area of the region bounded by the given curves. y = 6x^2 ln(x), y = 24 ln(x)

Equating both the equations of the curve,

6x^2 ln(x) = 24 ln(x)

24 ln(x) - 6x^2 ln(x) = 0

x = 1, 2

This means,  the curves intersect at x = 1 and x = 2.

So, the required area would be,

A = ∫[1 to 2] [24 ln(x) - 6x^2 ln(x)] dx

First we find the indefinite integral ∫[24 ln(x) - 6x^2 ln(x)] dx

= -6  ∫[-4 ln(x) + x^2 ln(x)] dx

= -6 ∫ln(x) (x^2 - 4) dx

= -6 ln(x) (1/3 x^3 - 4x) + 2/3 x^3 - 24x

So,  ∫[1 to 2] [24 ln(x) - 6x^2 ln(x)] dx

= [-6 ln(x) (1/3 x^3 - 4x) + 2/3 x^3 - 24x] _(x = 1 to x = 2)

= 32 ln(2) - 58/3

= 22.18 - 19.33

= 2.85 sq.units.

Therefore, the area of the region is 2.85 sq.units.

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(i) Yes, it is even and divisible by 3 as 1 + 0 + 2 = 3

(ii) No, it does not end in 5 or 0

What are Divisibility Rules?

Divisibility criteria can be used to determine whether or not to diminish a fraction. The rules are based on patterns seen while listing the multiples of any integer.

Solution:

(i) 102 is divisible by 6 - TRUE

It is because 102 is even number thus, it is divisible by 2 and since the sum of digits 1 + 0 + 2 is 3 therefore, it is divisible by 3

- Yes, it is even and divisible by 3 as 1 + 0 + 2 = 3

(ii) 102 is divisible by 5 - FALSE

It is because any number to be divisible by 5 it should always end with 5 or 0 at ones place.

- No, it does not end in 5 or 0

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Answer: 1. 108.     2. 1         mark brainlist if helpful

Step-by-step explanation: hope it helps

98 plus 10.   108 plus one 109.  

It would take about 18 days for a pair of chickens to lay three dozen of eggs.

The unitary method is a method for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value. Finding the value of just one unit is the key to answering such questions.

Given that 1.5 chickens can lay 1.5 eggs in a day.

The number of eggs 1 chicken can lay in a day = (1.5/1.5) eggs = 1 egg.

Then, two chickens can lay about 2 eggs in a day.

The number of days it takes for two chickens to lay 3 dozen (3×12=36) eggs is given by

[tex]\frac{\text{total number of eggs}}{\text{number of chicken}}=\frac{36}{2}=18\;\text{days}[/tex]

The answer is 18 days.

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Answer:

children = $6 ; adults = $13

Step-by-step explanation:

6c + 8a = 140

12c + 9a = 189

3c + 4a = 70

4c + 3a = 63

3c = 70 - 4a

c = 70/3 - 4/3 a

4(70/3 - 4/3 a) + 3a = 63

280/3 - 16/3 a + 3a = 63

(-16a + 9a)/3 = 63 - 280/3

-7/3 a = (189 -280)/3

-7/3 a = -91/3

a = - 91/3 · -3/7

a = 91/7

a = $13

c = 70/3 - 4/3 · 13

c = 70/3 - 52/3

c = (70-52)/3

c = 18/3

c = $6

Answer: 56%

Step-by-step explanation:

Pot A

75% = 0.75

12 times 0.75 = 9 seeds sprouted

Pot B

50% = 0.5

20 times 0.5 = 10 seeds sprouted

Pot C

50% = 0.5

18 times 0.5 = 9 seeds sprouted

Add total; we get 28 seed sprouted

Then we take 28 divided by 50, times 100% = 56%

So, 56% of the seeds from the packet sprouted!

Answer:

The answer C) 5 * 106

Step by step explanation:

Answer:

The area of the rectangular lawn is 100m long and 45m wide, so it has an area of 100 * 45 = <<10045=4500>>4500m².

The three circular ponds have diameters of 20m, 10m, and 5m, so their areas are 314.16, 78.54, and 19.63 square meters respectively.

The total area of the ponds is 314.16 + 78.54 + 19.63 = <<314.16+78.54+19.63=412.33>>412.33 square meters.

The total area of the lawn that needs to be covered with grass seed is 4500 - 412.33 = <<4500-412.33=4087.67>>4087.67 square meters.

Each packet of grass seed covers 50 square meters, so Mrs Jones will need 4087.67 / 50 = <<4087.67/50=81.75>>81.75 packets of grass seed.

Each packet costs £1.49, so the total cost to Mrs Jones will be 81.75 * 1.49 = £<<81.751.49=121.78>>121.78. Answer: {121.78}.

Step-by-step explanation:

The empirical probability of rolling a 3 is 467% more than its theoretical probability

Its fundamental concept is that someone will nearly surely occur. The proportion of positive events in comparison to the total of occurrences.

Then the probability is given as,

P = (Favorable event) / (Total event)

A six-sided pass-on from obscure predisposition is moved multiple times, and the number 3 comes up multiple times. In the following three adjustments (the bite of the dust is moved multiple times in each round), the number 3 comes up 6 times, 5 times, and 7 times.

Then empirical probability is given as,

P = (6/20) x (5/20) x (7/20)

P = 21/800

P = 0.02625

The theoretical probability is given as,

P = (1/6) x (1/6) x (1/6)

P = 1 / 216

P = 0.004629

Then the percentage is given as,

Percentage = [(0.02625 - 0.004629) / 0.004629] x 100

Percentage = 467%

The empirical probability of rolling a 3 is 467% more than its theoretical probability

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Answer:

Day 7

Step-by-step explanation:

Given information:

Therefore, each day Celeste has three times as much as she had the previous day.

This can be expressed by the recursive rule:

[tex]\begin{cases}a_n=3a_{n-1}\\a_1=3\end{cases}[/tex]

Therefore:

[tex]\textsf{Day 2}: \quad a_2=3 \cdot a_{1}=3 \cdot 3=9[/tex]

[tex]\textsf{Day 3}: \quad a_3=3 \cdot a_{2}=3 \cdot9=27[/tex]

[tex]\textsf{Day 4}: \quad a_4=3 \cdot a_{3}=3 \cdot 27=81[/tex]

[tex]\textsf{Day 5}: \quad a_5=3 \cdot a_{4}=3 \cdot 81=243[/tex]

[tex]\textsf{Day 6}: \quad a_6=3 \cdot a_{5}=3 \cdot 243=729[/tex]

[tex]\textsf{Day 7}: \quad a_7=3 \cdot a_{6}=3 \cdot 729=2187[/tex]

So the day on which Celeste will have 2,187¢ is day 7.

The number 58 represents the Degree of freedom in the equation: t(58)=2.001, p<.05.

Option (c) is the correct choice.

The largest number of logically independent values—that is, values with the freedom to change—in the data sample is referred to as the degree of freedom. When calculating degrees of freedom, one is subtracted from the total number of items in the data sample.

A straightforward statistical method may be used to calculate degrees of freedom. It reads as follows: df = N-1 and indicates that degrees of freedom are equal to the number of values in a data collection minus 1. N is the number of values in the data collection, in this case (sample size).

The degrees of freedom, or the number of free data points available in a t-test to compare two groups, are shown in the above t-test by the number 58.

(A T-test is employed in statistics to compare the means of two groups. The t-value, p-value, and degrees of freedom are the values needed for a t test.)

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The complete question may be:

What does the number 58 represent in the following: t(58)=2.001, p<.05?

(a) Test statistic

(b) T-value

(c) Degrees of freedom

(d) Significance level

Answer:

Below

Step-by-step explanation:

Slope = rise/run = 2/3

Point (3,5)      y-5 = 2/3 (x-3)

                         y = 2/3x +3

The area of the region enclosed by one loop of the curve r = sin(12θ) is π/48.

We have to find the area of the region enclosed by one loop of the curve.

The given curve is:

r = sin(12θ)

Consider the region r = sin(12θ)

The area of region bounded by the curve r = f(θ) in the sector a ≤ θ ≤ b is

A =

Now to find the area of the region enclosed by one loop of the curve, we have to find the limit by setting r=0.

sin(12θ) = 0

sin(12θ) = sin0   or    sin(12θ) = sinπ

So θ = 0             or      θ = π/12

Hence, the limit of θ is 0 ≤ θ ≤ π/12.

Now the area of the required region is

A = [tex]\int ^{\pi/12}_{0} \frac{1}{2}(\sin12\theta)^2d\theta[/tex]

A = [tex]\frac{1}{2}\int ^{\pi/12}_{0}\sin^212\thetad\theta[/tex]

A = [tex]\frac{1}{2}\int ^{\pi/12}_{0}\frac{1-\cos24\theta}{2}d\theta[/tex]

A = [tex]\frac{1}{4}\int ^{\pi/12}_{0}(1-\cos24\theta)d\theta[/tex]

A = [tex]\frac{1}{4}\left[(\theta-\frac{1}{24}\sin24\theta)\right]^{\pi/12}_{0}[/tex]

A = [tex]\frac{1}{4}\left[(\frac{\pi}{12}-\frac{1}{24}\sin24\frac{\pi}{12})-(0-\frac{1}{24}\sin24\cdot 0)\right][/tex]

A = [tex]\frac{1}{4}\left[(\frac{\pi}{12}-\frac{1}{24}\sin2\pi)-(0-\frac{1}{24}\sin0)\right][/tex]

A = 1/4[(π/12-0)-(0-0)]

A = 1/4(π/12)

A = π/48

Hence, the area of the region enclosed by one loop of the curve r = sin(12θ) is π/48.

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