A mirror is placed 45 feet from the base of a waterfall by a hiker. The hiker walks backwards until they are 7.5 feet from the mirror. Determine how tall the waterfall is if the hiker is 6 feet tall.

36 ft
39.5 ft
56.25 ft
72 ft

We can start by isolating the secant term and then taking the inverse cosine of both sides. Remember that the inverse cosine function only gives results on the interval [0,π], so we will need to adjust our solutions accordingly:

2sec(θ) + 3 = 0

2sec(θ) = -3

sec(θ) = -3/2

cos(θ) = -2/3

θ = ±cos⁻¹(-2/3)

Using a calculator, we find:

θ ≈ 2.302, 4.981 radians

However, we need to adjust these solutions to be within the interval [0,2π). Since cos(x) = cos(2π - x), we can add 2π to the negative solution to get an equivalent positive solution:

θ ≈ 2.302, 7.438 radians

Rounding to the nearest thousandth, we get:

θ ≈ 2.302, 7.438 radians

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On sοlving the prοvided questiοn we can say that equatiοn Miles already ran -target distance = Miles still tο run 60 - 24 = 36 miles left tο run

A mathematical equatiοn is a fοrmula that links twο statements and uses the equals sign (=) tο indicate equality. In algebra, an equatiοn is a statement that demοnstrates the equality οf twο mathematical expressiοns.

We may begin by subtracting the miles Mοna has already ran frοm her οbjective distance tο get the number οf miles she must run each day tο accοmplish her gοal:

Miles already ran - target distance = Miles still tο run

60 - 24 = 36 miles left tο run

Since there are three days left in the week, Mοna must run:

36 miles ÷ 3 days = 12 miles per day

On a number line, we may begin with 24 and add 12 fοr each remaining day:

24 ------ 36 ------ 48 ------ 60

Tο represent this οn a number line, we can start at 24 and add 12 fοr each remaining day:

|---|---|---|---|---|---|---|

0 5 10 15 20 25 30 35 40

^ ^ ^

Day 5 6 7

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Answer:

25.12 m (or 25.13 m)

Step-by-step explanation:

Recall the formula for the circumference of a circle:

[tex]2\pi r[/tex]

where r is the radius.

We are given that the radius is 4 m, so let's plug that into our formula:

[tex]2\pi r=\\2\pi *4=\\8\pi[/tex]

[tex]\pi[/tex] is approximately 3.14, so let's multiply 8 m by 3.14. We get:

25.12

So, the circumference is about 25.12 m.

(if you multiply by pi itself using a calculator, the answer is about 25.13 m)

Answer: $10.92

Step-by-step explanation:

Since there are 16 ounces in a pound, 8 ounces would be equivalent to half a pound. Thus, 3 pounds and 8 ounces = 3.5 pounds.

Since apples cost $3.12 per pound and we have 3.5 pounds, the total cost would be $3.12*3.5 pounds = $10.92

A square root function that is represented by the graph include the following: A. f(x) = √(4x + 8).

In Mathematics and Geometry, a square root function can be defined as a type of function that typically has this form f(x) = √x, which basically represent the parent square root function i.e f(x) = √x.

In Mathematics and Geometry, a horizontal translation to the left is modeled by this mathematical equation g(x) = f(x + N) while a vertical translation to the positive y-direction (downward) is modeled by this mathematical equation g(x) = f(x) - N.

Where:

Therefore, the required square root function is given by;

f(x) = √(4x + 8)

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The correct option are - A. x²  + (y – 3)²  = 36 and E. x²  + (y + 8)²  = 36 , for the equation of circle.

The standard form equation for just a circle, which is (x - s)² + (y - t)² = r², is used to formulate our equation if we assume a generic centre point like (s,t)and r is the radius of circle.

By simply expanding basic binomial squares in their regular form and merging like words, circles can likewise be presented in expanded form.

Now for the given data:

Put the values in the standard form of the circle:

(x - s)² + (y - t)² = r²,

(x - 0)² + (y - t)² = 6²,

x² + (y - t)² = 36

Where 't' can be any integer.

Thus, the options that satisfy the stated conditions are-

A. x²  + (y – 3)²  = 36 : here t = 3

and E. x²  + (y + 8)²  = 36 ; here t = - 8.

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Correct question:

Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis?

Select two options.

A. x²  + (y – 3)²  = 36

B. x²  + (y – 5)²  = 6

C. (x – 4)² + y² = 36

D. (x + 6)² + y² = 144

E. x²  + (y + 8)²  = 36

The cost of an individual ticket is contingent upon the overall expenditure of the performance and the quantity of tickets purchased.

Ticket sales refer to the purchase of one or multiple admission passes to an occasion, which may be facilitated either through the Event Management System, online, mobile app, third-party voucher providers (such as Groupon), or physically, among other options.

Based on the fact that there is no much details on the ticket sales, it present ]an obstacle in knowing the number cost of individual tickets. To determine the cost of each ticket, we need to have the total expenditure or the quantity of tickets sold.

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Step-by-step explanation:

Steve spends 20% of £200 = 0.2 * £200 = £40 on his football magazine subscription.

The remaining amount he spends on football stickers, sweets, and fizzy drinks in the ratio of 5:2:1, which means that for every 5 parts he spends on football stickers, he spends 2 parts on sweets and 1 part on fizzy drinks.

So, let's call the amount he spends on football stickers "5x". Then, the amount he spends on sweets is "2x" and the amount he spends on fizzy drinks is "x".

The total amount he spends on these items is:

5x + 2x + x = 8x

We know that he has £200 - £40 = £160 left after paying for his magazine subscription. So, we can set up an equation:

8x = £160

Solving for x, we get:

x = £20

Therefore, Steve spends:

5x = 5 * £20 = £100

on football stickers.

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Answer:

5 : 2 : 1 = 100 : 40 : 20

Step-by-step explanation:

10% = 20 (200÷10=20)

20% = 40 (double both sides)

take away £40 from £200 = £160

£160 is left over after buying the magazine subscription

______________________________

add the radio all together = 5:2:1 = 8

divide 8 by 160 = £20

That now means 1 = 20

5 : 2 : 1

we already found 1, now to find 2 and 5

you times 2

1 = 20

2 = 40

and then do the same for 5

1 = 20

5 = 100

To know this is right, 100 + 40 + 20 = 160 the original number for the money left over.

Answer:

To determine the classification of a triangle based on its side lengths, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the legs (the shorter sides) is equal to the square of the length of the hypotenuse (the longest side).

For this particular triangle with side lengths 6 cm, 10 cm, and 12 cm, we can apply the Pythagorean theorem to determine whether the triangle is acute (all angles are less than 90 degrees), obtuse (one angle is greater than 90 degrees), or right (one angle is equal to 90 degrees).

Using the Pythagorean theorem, we find:

6^2 + 10^2 = 136

12^2 = 144

Since 136 is less than 144, we know that 6 cm, 10 cm, and 12 cm are the side lengths of a triangle that is obtuse (one angle is greater than 90 degrees), because the longest side (the hypotenuse) is greater than the sum of the squares of the lengths of the other two sides.

Therefore, the correct answer is: obtuse, because 6^2 + 10^2 < 12^2.

Answer:

y = -3x + 1

Step-by-step explanation:

We are given a couple ordered pairs from this table. Lets use the slope formula so we can find the slope of this line:

[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Y2, y1, x2, and x1 are constants (points on the line). To make it easy, lets take the coordinates (0,1) and (1,-2). Then we just plug these points in and evaluate.

[tex]\frac{-2-1}{1-0}[/tex]

Evaluate both sides of the fraction.

[tex]\frac{-3}{1} =-3[/tex]

Now that we have the slope, we are given the y intercept coordinate, (0,1) which means the y-intercept is 1. To write it in y = mx + b form, we just plug in our numbers and were good to go.

[tex]y=-3x+1[/tex]

Answer: The first choice.

Step-by-step explanation:

As you can see in the diagram, the line passes through the points[tex](x, y) = (4, -4)[/tex] and [tex](x, y) = (-4, 8)[/tex] . Thus, the slope of the line is [tex]\frac{8-(-4)}{-4-(4)} = \frac{12}{-8} = -\frac{3}{2}.[/tex] Thus, the correct answer is the first choice, as that line has a slope of [tex]-\frac64=-\frac32.[/tex]

Note that the probability of selling at least 30 widgets is 0.0468.

Given the information provided, the problem requires us to use a binomial distribution to estimate the probability of selling at least 30 widgets.

A binomial distribution is a probability distribution that describes the number of successes in a set number of trials.

Where n = 200 (the number of people that visit the store and buy a widget each day) and

p = 0.1667 (likelihood of a client purchasing a widget)

We are looking for the probability of selling at least 30 widgets, which is equivalent to finding the probability of selling 30 or more widgets.

This can be calculated using the cumulative probability formula for a binomial distribution:

P(X >= 30) = 1 - P(X < 30) = 1 - Σ (nCx)(p^x)(1-p)^(n-x) from x=0 to 29

We can use a calculator or a software to calculate this probability. The result is 0.0468 rounded to 4 decimals.

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The woman invested $4,000 at 5% and $3,000 at 12%.

Let x be the amount invested at 5% and y be the amount invested at 12%. Then we have:

x + y = 7,000 (equation 1)

0.05x + 0.12y = 560 (equation 2)

To solve for x and y, we can use substitution method :

From equation 1, we have y = 7,000 - x. Substituting this into equation 2, we get:

0.05x + 0.12(7,000 - x) = 560

Simplifying and solving for x, we get:

0.05x + 840 - 0.12x = 560

-0.07x = -280

x = 4,000

Substituting x = 4,000 into equation 1, we get:

4,000 + y = 7,000

y = 3,000

Therefore, the woman invested $4,000 at 5% and $3,000 at 12%.

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Answer:

Yes it is a parallelogram because opposite sides of this quadrilateral are equal.

The regular price of the running shoes is $80.

Let's denote the regular price of the running shoes as P. We know that during the sale, the price is reduced by 20%, which means the sales price is 80% (100% - 20%) of the regular price.

Given that the sales price is $64, we can set up the equation:

0.8P = $64

To find the regular price P, we divide both sides of the equation by 0.8:

P = $64 / 0.8

P = $80

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Answer:

3.15%

Step-by-step explanation:

To find the expected return and standard deviation of a portfolio that is invested in two assets A and B, we need to know the expected returns and standard deviations of those assets, as well as the correlation between them.

Let's assume that the expected return and standard deviation of asset A are 10% and 5%, respectively, and that the expected return and standard deviation of asset B are 8% and 3%, respectively. Let's also assume that the correlation between the two assets is 0.5.

To find the expected return of the portfolio, we can use the following formula:

Expected return of portfolio = weight of A * expected return of A + weight of B * expected return of B

where the weights are the proportions of the portfolio invested in each asset:

weight of A = 40% = 0.4

weight of B = 60% = 0.6

Expected return of portfolio = 0.4 * 10% + 0.6 * 8%

Expected return of portfolio = 9.2%

Therefore, the expected return of the portfolio is 9.2%.

To find the standard deviation of the portfolio, we can use the following formula:

Standard deviation of portfolio = sqrt(weight of A^2 * variance of A + weight of B^2 * variance of B + 2 * weight of A * weight of B * correlation * standard deviation of A * standard deviation of B)

where the variances are the square of the standard deviations of each asset:

variance of A = (5%)^2 = 0.0025

variance of B = (3%)^2 = 0.0009

Standard deviation of portfolio = sqrt(0.4^2 * 0.0025 + 0.6^2 * 0.0009 + 2 * 0.4 * 0.6 * 0.5 * 0.05 * 0.03)

Standard deviation of portfolio = 0.0315 or 3.15% (rounded to two decimal places)

Therefore, the standard deviation of the portfolio is 3.15%.

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It shtbe noted that 68% of the data should fall in the interval from 616 to 756.

z score for values:

z = (X-µ)/σ = (616-686)/70 = -1

z = (X-µ)/σ = (756-686)/70 = 1

Approximately 68% of the data should fall in the interval from 616 to 756.

b) B. 71% of the data falls in the interval from 616 to 756. The estimate is very close to the value predicted by the Empirical Rule.

c) (µ - 2*σ, µ + 2*σ)

(686 - 2*70, 686 + 2*70)

(546, 826)

You expect to find about 95% of the teams between the two values 546 and 826.

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Bank B would be the best option since it offers a higher interest rate than Bank A and does not have any fees.

To compare different bank accounts, we need to know the interest rates offered by each bank. Let's assume that we have three local banks, each with a different interest rate: Bank A offers 2% interest, Bank B offers 2.5% interest, and Bank C offers 3% interest.

Using the formula for simple interest, we can calculate the amount of interest earned on the initial deposit of $2000 over four years for each bank:

Bank A: 2% interest per year

After four years, the interest earned is:

$2000 x 2% x 4 = $160

Bank B: 2.5% interest per year

After four years, the interest earned is:

$2000 x 2.5% x 4 = $200

Bank C: 3% interest per year

After four years, the interest earned is:

$2000 x 3% x 4 = $240

So, Bank C offers the highest interest rate and would earn the most interest over the four-year period. However, we also need to consider any fees associated with each account. Let's assume that Bank A and Bank B have no fees, but Bank C charges a $50 annual fee.

Bank A: Total amount after four years = $2000 + $160 = $2160

Bank B: Total amount after four years = $2000 + $200 = $2200

Bank C: Total amount after four years = $2000 + $240 - $50 x 4 = $1960

Even though Bank C offers the highest interest rate, the annual fee significantly reduces the total amount earned over four years. Therefore, Bank B would be the best option since it offers a higher interest rate than Bank A and does not have any fees.

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The length of the tank should be approximately 150.4 inches in order to hold 450 gallons of water, with a width of 21 inches and a height of 33 inches.

To find the length of the tank, we first need to convert the volume of water from gallons to cubic inches, using the given conversion factor:

1 gallon = 231 cubic inches

Therefore, the volume of water in the tank is:

450 gallons * 231 cubic inches/gallon = 103950 cubic inches

We can then use the formula for the volume of a rectangular prism to find the length of the tank:

Volume = length * width * height

Substituting the given dimensions, and solving for the length, we get:

103950 cubic inches = length * 21 inches * 33 inches

length = 103950 cubic inches / (21 inches * 33 inches)

length ≈ 150.4 inches

Therefore, the length of the tank should be approximately 150.4 inches in order to hold 450 gallons of water, with a width of 21 inches and a height of 33 inches.

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Answer:

y^2 - 4

Step-by-step explanation:

We are given the statement,'The square of y decreased by the quotient of 28 and 7'. i.e. Subtracting the quotient 28/7 = 4 from y^2 . i.e. y^2 - 4

Hence, the expression which represents the given statement is y^2 - 4 .

The model predicts that the population of the inner-city area will be 107,079 people in 7 years

The number of inhabitants in a given region is characterized as the quantity of individuals normally living around there, estimated on 1 January in a given year.

The given model for the number of inhabitants in the rough part of town region is:

P(t) = 108,000 - 0.011t

where t is the number of years from the present and P(t) is the estimated population at time t.

To find the predicted population in 7 years, we can substitute t=7 into the given model and evaluate:

P(7) = 108,000 - 0.011(7)

P(7) = 108,000 - 0.077

P(7) = 107,923

As a result, the model projects that the inner-city area will have a population of 107,079 people in seven years, rounded to the nearest person.

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The senior math contest ended at 10:40 pm on Sunday

If the senior math contest began at 6:00 am on a Sunday and ended 1000 minutes later, we can add 1000 minutes to the starting time to find the ending time.

To do this, we can first convert the starting time of 6:00 am into minutes. There are 60 minutes in one hour, so 6:00 am is equivalent to:

6 hours × 60 minutes/hour = 360 minutes

Now, we can add 1000 minutes to this starting time:

360 minutes + 1000 minutes = 1360 minutes

Finally, we can convert this total number of minutes back into hours and minutes.

Since there are 60 minutes in one hour, we can divide 1360 minutes by 60 to get:

1360 minutes ÷ 60 = 22 hours and 40 minutes

Therefore, the senior math contest ended at 10:40 pm on the day (Sunday).

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The estimated actual temperature (x) when the temperature adjusted for wind chill (y) is -35 degrees Fahrenheit =

approximately -6.92 degrees Fahrenheit.

To estimate the actual temperature (x) when the temperature adjusted for wind chill (y) is given as -35 degrees Fahrenheit using the formula y = -26 + 1.3x, we can substitute y with -35 and solve for x.

-35 = -26 + 1.3x

Adding 26 to both sides to isolate the x term:

-35 + 26 = 1.3x

-9 = 1.3x

Dividing both sides by 1.3 to solve for x:

x = -9 / 1.3

x ≈ -6.92

Thus, the estimated actual temperature (x) when the temperature adjusted for wind chill (y) is -35 degrees Fahrenheit would be approximately -6.92 degrees Fahrenheit.

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The estimated actual temperature is -6.92 degrees Fahrenheit.

We can use the given formula to solve for x when y is known. Rearranging the formula, we get:

x = (y + 26) / 1.3

Substituting y = -35 into this formula, we get:

x = (-35 + 26) / 1.3

x = -6.92 (rounded to two decimal places)

Therefore, the estimated actual temperature is -6.92 degrees Fahrenheit.

The term actual temperature typically refers to the true or measured temperature of an object, substance, or environment. It is the temperature that is currently present in a given location or situation, as opposed to a predicted or estimated temperature. The actual temperature can be measured using a thermometer or other temperature sensing device.

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Here the correct statements can be

The y-intercept represents D) the starting amount in his savings account.

The slope represents C) the change in the dependent variable with respect to the independent variable.

What is meant by y-intercept?

The y-intercept is the point where a line intersects the y-axis on a graph. It represents the value of the dependent variable when the independent variable is zero.

What is meant by the term slope?

The slope is a measure of the steepness of a line on a graph. It is calculated by dividing the change in the y-coordinate by the change in the x-coordinate between two points on the line.

According to the information

The y-intercept of the linear model y = 200x + 500 represents the starting amount in Richard's savings account.

The slope of the linear model y = 200x + 500 represents the change in the dependent variable (y, the savings amount) with respect to the independent variable (x, the number of months). Specifically, the slope of 200 means that for every additional month, Richard's savings will increase by $200.

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The compositions are:

f * f(x) = x⁴ - 2x²

g * g(x) = 81/64x

To find the composition f * f, we start by substituting f(x) into f(x) wherever we see x:

f * f (x) = f(f(x)) = f(x² - 1) = (x² - 1)² - 1

Expanding the expression, we get:

f * f (x) = x⁴ - 2x² + 1 - 1 = x⁴ - 2x²

So, f * f(x) = x⁴ - 2x².

To find the composition g * g, we substitute g(x) into g(x) wherever we see x:

g * g (x) = g(g(x)) = g(9/8x) = 9/8(9/8x) = 81/64x

So, g * g(x) = 81/64x.

Therefore, the compositions are:

f * f(x) = x⁴ - 2x²

g * g(x) = 81/64x

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The text format of the question is:

Suppose that the functions f and g are defined as follows.

f(x) = x² - 1

g(x) = 9/8x, x ≠

Find the compositions f * f and g * g

Simplify your answers as much as possible.

(Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain)

Answer:

The salary of a paramedic can vary depending on several factors such as location, experience, and employer. According to the Bureau of Labor Statistics, the median annual wage for paramedics and EMTs was $36,650 in May 2020.

Assuming a constant annual income of $36,650 over 30 years, the estimated total income for a paramedic would be:

$36,650 x 30 = $1,099,500

However, it is important to note that this is just an estimate and does not take into account potential salary increases, promotions, or changes in the job market.

The description describes the solutions to this inequality in all the positive numbers that are greater than ten. The correct option is B.

Relationships between two expressions that aren't equal to one another are known as inequalities. Social justice theories are fundamentally based on the idea of inequality, which is the condition of not being equal, especially in terms of status, rights, and opportunities.

The given equation is 9 > 52.

everything that is positive and bigger than 10

Therefore, the correct option is B. All the positive numbers are greater than ten.

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The number of more points that Owen must earn to meet his goal would be = 31 points.

The total number of point expected to be scored by Owen = 50 points

The total amount of points that Owen has earned so far = 19 point.

The amount of points that Owen should earn more to meet up with the points expected of him = 50-19= 31 points.

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The equilibrium tuition price p and the consumers' and producers' surpluses at this tuition level is 11400

From the question, we have the following parameters that can be used in our computation:

q = 20,000 − 2p

q = 8,500 + 0.5p

The equilibrium tuition price p and the consumers' and producers' surpluses at this tuition level is when q = q

So, we have

20,000 − 2p = 8,500 + 0.5p

Evaluate the like terms

2.5p = 28500

Divide by 2.5

p = 28500/2.5

Evaluate

p = 11400

Hence, the equilibrium tuition price p is 11400

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