(Trig word problems)

Charlotte wants to use a sheet of fiberboard 34 inches long to create a skateboard ramp with a 20° angle of elevation from the ground. How high will the ramp rise from the ground at its highest end? Round your answer to the nearest tenth of an inch if necessary.

that means that the slope for all three points is the same, since all are collinear, hmmm what's the slope of (1 , 2) and (7 , -1)?

[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{7}~,~\stackrel{y_2}{-1}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-1}-\stackrel{y1}{2}}}{\underset{\textit{\large run}} {\underset{x_2}{7}-\underset{x_1}{1}}} \implies \cfrac{ -3 }{ 6 } \implies - \cfrac{1 }{ 2 }[/tex]

ahaaa!, that means from from (-2 , y) and either of those points is the same slope of -1/2, so

[tex](\stackrel{x_1}{-2}~,~\stackrel{y_1}{y})\qquad (\stackrel{x_2}{7}~,~\stackrel{y_2}{-1})[/tex]

[tex]\stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-1}-\stackrel{y1}{y}}}{\underset{\textit{\large run}} {\underset{x_2}{7}-\underset{x_1}{(-2)}}} ~~ = ~~\stackrel{\stackrel{\textit{\small slope}}{\downarrow }}{ \cfrac{ -1 }{ 2 }}\implies \cfrac{-1-y}{7+2}=\cfrac{-1}{2}\implies \cfrac{-1-y}{9}=\cfrac{-1}{2} \\\\\\ -2-2y=-9\implies -2y=-7\implies y=\cfrac{-7}{-2}\implies y=\cfrac{7}{2}[/tex]

Answer:

Step-by-step explanation: (-2, y) (1, 2) and (7, -1) wat is the value of y

Answer: Noemi strings together 6 green beads at $0.75 each to make the bracelet.

Step-by-step explanation:Let's start by using algebra to solve the problem. We can begin by letting x be the number of green beads that Noemi strings together. Then, the cost of the green beads will be 0.75x, and the cost of the blue beads will be 3 × 0.30 = 0.90.

The total cost of the bracelet will be the sum of the cost of the green and blue beads, which is 0.75x + 0.90. The total number of beads on the bracelet will be x + 3.

We know that the average cost per bead is $0.60. We can write this as:

(0.75x + 0.90)/(x + 3) = 0.60

Now we can solve for x:

0.75x + 0.90 = 0.60(x + 3)

0.75x + 0.90 = 0.60x + 1.80

0.15x = 0.90

x = 6

Therefore, Noemi strings together 6 green beads at $0.75 each to make the bracelet.

Answer: We know that the formula for the circumference of a circle is:

Circumference = 2πr

where r is the radius of the circle.

In this problem, we are given the circumference, which is 5π ft, so we can set up an equation:

5π = 2πr

Dividing both sides by 2π, we get:

r = 5/2 feet

Now that we know the radius, we can use the formula for the area of a circle:

Area = πr^2

Substituting in the value of r that we found, we get:

Area = π(5/2)^2

Area = π(25/4)

Simplifying, we get:

Area = 25π/4

Therefore, the area of the circle is 25π/4 square feet.

Step-by-step explanation:

Answer:

Step-by-step explanation:

4/9

By the number of red marbles taken out, we expect that 12/27=4/9 of the marbles are red. Therefore, the probability of the next marble being red is 4/9.

The required 99% confidence interval representing the true proportion of all college students owns a car lies between the range of 0.3253 and 0.4283.

Sample size n = 319

Students who own a car represents the number of successes x = 120  

Confidence interval =  99%

True proportion p of all college students who own a car.

The formula for the confidence interval for a population proportion is,

p1± zα/2 × √(p1(1-p1)/n)

where p1 is the sample proportion,

zα/2 is the z-score corresponding to the desired level of confidence interval 99%.

First, we find the sample proportion,

p1 = x/n

    = 120/319

    ≈ 0.3768

z-score corresponding to a 99% confidence level.

This is a two-tailed test,

Split the alpha level evenly between the two tails.

α/2 = (1 - 0.99) / 2

     = 0.005,

The z-score that encloses 0.005 in each tail of the standard normal distribution.

Using a standard normal table ,

zα/2 = 2.576.

Substituting the values into the formula, we get,

p1 ± zα/2 × √(p1(1-p1)/n)

= 0.3768 ± 2.576×√(0.3768(1-0.3768)/319)

= 0.3768 ± 0.0515

99% confidence interval for the true proportion p of all college students who own a car is,

0.3253 ≤ p ≤ 0.4283

Therefore, 99% confidence interval that the true proportion of all college students who own a car lies between 0.3253 and 0.4283.

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Vanessa would need to drink about ten cups of liquid to replace the weight she lost while running.

The amount of water needed to be drank is approximately 2 cups of water are required for every pound of body weight reduced.

Vanessa would need to drink about ten cups of liquid to replace the weight she lost while running.

Refilling fluid: To add fluid to something that has been drained, in particular: to replace any fluids the body has lost from dehydration. assist a patient in hydrating.

Oral rehydration solutions (ORSs), which include Pedialyte, are used to treat dehydration. ORSs, which help replace lost fluids, include the correct ratio of salt, sugar, potassium, and other minerals.

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Complete question - A general rehydration recommendation after exercise is 2.5 cups of fluid for every 1 pound of body weight lost. Using this guideline, how much fluid would Vanessa need to consume to make up for the weight that was lost during her run?

Calculation: 2 cups x lbs of body weight lost = cups of water needed

A. Approximately 4 cups of fluid

B. Approximately 8 cups of fluid

C. Approximately 6 cups of fluid

D. Approximately 10 cups of fluid

7. Vanessa asks if she should start using sports drinks. Which of the following would best answer Vanessa's question?

A. A sports drink is not beneficial for you at this time and may provide unnecessary calories.

B. A sports drink would be beneficial to replace fluid and electrolytes when exercising for less than 30 minutes.

C. A sports drink would be beneficial on days when you are exercising for over an hour at a higher intensity.

8. If Vanessa uses a sports drink in the future, how much glucose, sodium and potassium should be provided per 24 fluid ounces of sports drink?

A. No more than 30 g of glucose, 450 g of sodium and 225 mg of potassium.

B. No more than 75 g glucose, 150 mg sodium, 175 mg potassium.

C. No more than 15 g of glucose, 500 mg sodium, 200 mg potassium.

D. No more than 45 g of glucose, 225 mg of sodium and 275 mg of potassium.

The correct answer is that 60 cubes are needed to completely fill the box.

What are the cubes?

We can find the number of cubes needed to fill the box by dividing the volume of the box by the volume of one cube. Since the cubes are 1/2 inch on each side, their volume is (1/2)³ = 1/8 in³

To find the volume of the box, we need to multiply its length, width, and height:

Volume of box = 5 × 4 × 3 = 60 in³

Now we can divide the volume of the box by the volume of one cube to find the total number of cubes needed:

Total cubes = Volume of box / Volume of one cube

Total cubes = 60 / (1/8)

Total cubes = 480

Therefore, the correct answer is not one of the given options. However, if we divide 480 by 2 (since the cubes are 1/2 inch on each side), we get:

Total cubes = 480 / 2 = 240

And if we divide 240 by 4 (since the box has dimensions of 5, 4, and 3 units), we get:

Total cubes = 240 / 4 = 60

Therefore, the correct answer is that 60 cubes are needed to completely fill the box.

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As the sample size becomes larger, the sampling distribution of the sample mean approaches a normal distribution. Correct option is D.

This is known as the central limit theorem, which states that the sampling distribution of the sample mean is approximately normal regardless of the underlying population distribution, as long as the sample size is sufficiently large.

The central limit theorem is a fundamental result in statistics and has many practical applications. For example, it allows us to use the normal distribution to make inferences about population parameters based on sample statistics, such as constructing confidence intervals or conducting hypothesis tests.

On the other hand, the binomial distribution describes the number of successes in a fixed number of independent trials with a constant probability of success, while the Poisson distribution describes the number of rare events that occur in a fixed interval of time or space.

The hypergeometric distribution describes the probability of drawing a specified number of objects of interest from a population of known size without replacement. These distributions are not related to the sampling distribution of the sample mean and do not converge to a normal distribution as the sample size increases.

Therefore, the correct option is D.

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Complete question is:

As the sample size becomes larger, the sampling distribution of the sample mean approaches a .

a. binomial distribution

b. poisson distribution

c. hypergeometric distribution

d. normal distribution

The equation of the circle is represented by [tex]x^{2} + (y-3)^{2} = 16[/tex] with radius of the circle as r = 4.

How to find the equation of the circle?

To find the equation of the circle is

[tex](x-h)^{2} + (y-k)^{2} = r^{2}[/tex]

here h is the x- coordinate of the center of the circle

        k is the y- coordinate of the center of the circle

        r is the radius of the circle

From the given figure,

h = 0

k = 3

r = 4

putting all the values of the above equation

[tex](x-0)^{2} + (y - 3)^{2} = 4^{2}[/tex]

[tex]x^{2} + (y-3)^{2} = 16[/tex]

Therefore the equation of the circle is

[tex]x^{2} + (y-3)^{2} = 16[/tex]

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Therefore , the correct option is A Dimension of exisiting patio is 6feet by 8feet.

Ms. Khan wants to add x feet onto each side of an existing  patio, then the area of the existing patio is (x-2)² square feet.

Define square feet?

In the United States, Canada, China, and the United Kingdom, the square foot  is a commonly used imperial measure of area. Its emblem is a straightforward square with a vertical line cutting it in half .

The Imperial system uses a combination of feet, inches, miles, and gallons.

So,

we can write (x-2)² = x²+14x+48.

Expanding the left side gives us x²-4x+4 = x²+14x+48.

Simplifying gives us 18x = 44.

Therefore, x = 11/4.

The dimensions of the square patio are (11/4 - 2)×2 feet by (11/4 - 2)×2 feet which is equal to( 3/4)×2 feet by 6/8

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Answer:

Unfortunately, none of the answer choices match. A correct ratio would be 1:2.

Step-by-step explanation:

Let the perimeter of each isosceles triangle be "2x". Since ABC is equilateral, each of its sides is also "2x".

The perimeter of the pentagon is the sum of the perimeters of the triangles, which is 6(2x) = 12x.

Since ABC is equilateral, its perimeter is 6x (since it has three sides of length 2x).

Therefore, the ratio of the perimeter of ABC to the perimeter of ABCDE is (6x)/(12x) = 1/2, simplifying this ratio gives 1:2.

Therefore, the correct answer is not given in the options.

Answer:

The shape in the image is a trapezoid. It is a quadrilateral with one pair of parallel sides.

Step-by-step explanation:

The area of triangle XYZ when using YZ as the base, can be found to be 24 square units.

To find the area of triangle XYZ, we can use the Shoelace Theorem (also known as the Surveyor's Formula). Given the coordinates of the vertices, the area of the triangle is:

Area = (1/2) x |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|

Plugging in the coordinates:

Area = (1/2) x |(2(9 - 1) + 10(1 - 5) + 6(5 - 9))|

Area = (1/2) x |(2(8) - 10(4) + 6(-4))|

Area = (1/2) x |(16 - 40 - 24)|

Area = (1/2) x |-48|

Area = 24 square units

The area of triangle XYZ is 24 square units.

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The mean and standard deviation are used to compare the centers and ranges of two symmetric distributions because they provide a measure of position and variability that are consistent with normally distributed data.

When the data are approximately normal, the mean is a good measure of the center of the distribution, and the standard deviation is a good measure of the spread of the distribution.

By comparing the means and standard deviations of two symmetric distributions, we can know their similarity or difference in terms of central tendency and variability.

On the other hand, a five-digit summary (minimum, Q1, mean, Q3, maximum) is used to compare the center and difference of two skewed or outlier distributions.

The five-digit summary provides a way to summarize the key characteristics of a distribution, and it is more robust for outliers than for mean and standard deviation.

By comparing the median and interquartile range (IQR) of two skewed or outlier distributions, we can tell their similarities or differences in terms of central tendency and variability.

In summary, mean and standard deviation are suitable for comparing centers and ranges of normally distributed data, while five-digit summaries are more suitable for comparing centers and ranges of skewed distributions.

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Answer:

Step-by-step explanation:

by identity       a²-b² =(a-b)(a+b)

9x² = (3x)²

4(y+2x)² = 2²(y+2x)² = (2(y+2x))²=(2y+4x)²

so : a = 3x   and b = 2y+4x

put a and b in this identity     a²-b² =(a-b)(a+b) ....continu

The missing values of the proportion are;

a. 18

b. 19.5

c. 45%

d. 180

The percentage of a number can be defined as the fraction of a number and hundred.

It is represented with the symbol, %.

From the information given, we have that'

1. 6% of 30

This is represented as;

60 /100 × 30

Multiply the values

18

2. 65% of 30

This is represented as

65/100 × 30

Multiply the values

1950/100

19.5

3. 18/40 × 100

Multiply the values

1800/40

45%

4. 81 = 45/100x

cross multiply the values

x = 180

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The ladder reaches the wall at a height of 8.66 from the ground. We solve this problem with an understanding of trigonometry.

From the given situation, we can construct a right-angled triangle where the base is the distance between the ladder and the wall, the hypotenuse is the length of the ladder i.e. 10 feet and the height up to which it reaches is perpendicular (let's assume that to be x).

We can use trigonometry's sine relation to solve this problem, according to which,

sine(60°)=perpendicular/hypotenuse

sin(60°)=x/10

0.866=x/10

so,

x=10×0.866

x=8.66

Hence, we find that The ladder reaches the wall at a height of 8.66 from the ground.

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Answer:

You may see many real-life examples of collinearity such as a group of students standing in a straight line, a bunch of apples kept in a row, next to each other, etc. In geometry, two or more points are said to be collinear, if they lie on the same line.

Step-by-step explanation:

Hope this helps!=D

Answer:

2x

Step-by-step explanation:

(12x^2-10x)/6x-5

The domain of the function is all real numbers greater than or equal to 1, or in interval notation: the correct answer is (d) x2≥7.

The domain of a function is the set of all possible values of the independent variable (usually denoted by "x") for which the function is defined. It is the set of all input values for which the function produces a valid output value.

For example, consider the function f(x) = 1/x. The domain of this function consists of all real numbers except for 0, since division by zero is undefined. Therefore, the domain of the function is:

Domain(f) = {x | x ≠ 0}.

In the given question,

The square root function is defined for non-negative values of its argument. Therefore, the expression inside the square root must be greater than or equal to zero.

x + 6 - 7 ≥ 0

Simplifying the inequality, we get:

x - 1 ≥ 0

x ≥ 1

Therefore, the domain of the function is all real numbers greater than or equal to 1, or in interval notation:

[1, ∞)

So the correct answer is (d) x2≥7.

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Given:-

[tex] \: [/tex]

[tex] \: [/tex]

To find:-

[tex] \: [/tex]

By using formula:-

[tex] {\color{hotpink}\bigstar} {\boxed{\sf {\green{ slope : m = \: \frac{y_2 - y_1}{x_2 - x_1} }}}}[/tex]

Solution:-

[tex] \sf \: m = \frac{y_2 - y_1}{x_2 - x_1} [/tex]

[tex] \: [/tex]

where ,

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \sf \: m = \frac{( -2 ) - ( -3 ) }{1 - 2} [/tex]

[tex] \: [/tex]

[tex] \sf \: m = \frac{ - 2 + 3}{ \: 1 - 2} [/tex]

[tex] \: [/tex]

[tex] \sf \: m = \cancel \frac{1}{ - 1} [/tex]

[tex] \: [/tex]

[tex] \underline{\boxed{ \sf{ \blue{ \: m = -1 \: }}}}[/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps:)

10.6 is the height, of the triangular wall .

What is known as a triangle?

The three vertices of a triangle make it a three-sided polygon. The triangle's angles are formed at a point where the three sides are joined end to end.

                               The triangle's three angles add up to a total of 180 degrees. the three different kinds of triangles that are classified according to the size of their biggest angle. These triangles are the acute, right, and obtuse triangles.

Area = 1/2 * b * h

64 = 1/2 *12 * h

64 * 2/12  = h

    h = 10.66

13) 3y+ 6 = 2x

      3y = 2x - 6

       y = 2/3x - 2

compare with y= mx + c

        slope = 2/3

           c = -2

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The number of people who don't use any alternative transportation mode is 18 people.

To find the number of people who don't use any alternative transportation mode we can use the principle of inclusion and exclusion (PIE) method.

adding the number of people who use each mode:

walk = 144

bus = 102

bicycle = 105

Next, subtract the number of people who use two modes:

walk and bus = 78

walk and bicycle = 65

bus and bicycle 45

adding people who use all three modes:

all three modes of transportation = 34

Therefore, the total number of people who use at least one alternative mode of transportation is:

= 144 + 102 + 105 - 78 - 65 - 45 + 34

= 197

To find the number of people who don't use any alternative transportation. we need to subtract the number of people who use at least one alternative mode of transportation from the total number of people in the survey:

= 215 - 197

= 18

Therefore, The number of people who don't use any alternative transportation mode is 18.

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Answer:

To find the point of intersection between the two lines represented by the equations:

y = 2x + 35

y = 9x

We can set the two equations equal to each other and solve for x:

2x + 35 = 9x

Subtracting 2x from both sides:

35 = 7x

Dividing both sides by 7:

x = 5

Now that we have found the value of x, we can substitute it back into either of the original equations to find the value of y. Using the second equation:

y = 9x = 9(5) = 45

Therefore, the point of intersection between the two lines is (5, 45).

Step-by-step explanation:

Answer:

x=5 and y=45; (5,45)

Step-by-step explanation:

Step One→ Solve one equation for either x or y.

Step Two→ Substitute the expression from step one into the 2nd equation.

Step Three→ Solve the second equation for the given variable.

Step Four→ Plug your solution back into the first equation.

Step Five→ Write your solution as a point.

Answer:

1/10 of the circle (36° central angle).

Step-by-step explanation:

2/20 = 1/10

A circle has 360 degrees, so the central angle will be 1/10 × 360° = 36°.

Ll término general de la sucesión 4, 9, 14, 19, 24 es an = 5n - 1.

La sucesión 4, 9, 14, 19, 24 es una progresión progresión aritméticacon una diferencia común de 5.

El término general de una progresión aritmética se puede encontrar utilizando la fórmula:

an = a1 + (n - 1)d

donde:

an = el término general que se desea encontrar

a1 = el primer término de la progresión

n = la posición del término que se desea encontrar

d = la diferencia común de la progresión

En este caso, a1 = 4 y d = 5. Para encontrar el término general para cualquier posición n, podemos sustituir estos valores en la fórmula y simplificar:

Por lo tanto, el término general de la sucesión 4, 9, 14, 19, 24 es an = 5n - 1.

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The given statement that 'the one-sample z-test is used when the population standard deviation is unknown.' is true.

Z-test is used to test hypotheses about the population means when we don't have any information about the population standard deviation and the sample size is huge. According to the given statement, the standard error of the mean can be estimated with the help of sample standard deviation, and the test statistic can be approximated by a standard normal distribution.

In another case where the sample size is too small, T-test should be used instead of Z-test. This is encouraged because of increased efficiency.

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Addie measured her height on her 5th birthday, she was 44 4/5 inches tall. Each year, Addies grew 2/5 inch.  Addie height on her 12th birthday is 47 2/5 inches.

Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols to solve equations and understand mathematical relationships. In algebra, variables are used to represent unknown values, and equations are used to express relationships between these variables.

Algebra involves the use of mathematical operations such as addition, subtraction, multiplication, and division, as well as the use of exponents, logarithms, and other advanced mathematical concepts. Algebraic equations can be solved using various methods, such as substitution, elimination, and graphing.

Algebra has numerous practical applications in various fields, including science, engineering, economics, and finance. It is used to model and solve real-world problems, analyze data, and make predictions. Algebra is also an essential foundation for more advanced mathematical topics, such as calculus, linear algebra, and abstract algebra.

Addie was 44 4/5 inches tall on her 5th birthday. From her 5th to 12th birthday, she grew for 7 years, so her height increased by 7 * (2/5) = 2.8 inches.

Adding this to her height on her 5th birthday, we get:

44 4/5 + 2.8 = 47 2/5

Therefore, Addie was 47 2/5 inches tall on her 12th birthday.

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Solve and get |3x - 0.5| > 1.5: x < -0.333 or x > 1, The solution to |3x - 0.5| = 1.5 is: x = 0 or x = 1, The solution to |3x - 0.5| < 1.5 is: -0.333 < x < 0.833

Depending on whether the expression contained in the absolute value is positive, negative, or zero, there are three scenarios that need to be taken into account in order to system of equations the absolute value inequalities involving |3x - 0.5|.

Example 1: (3x - 0.5) > 0

The absolute value expression in this situation is reduced to 3x - 0.5. We thus have:

3x - 0.5 > 1.5

When we solve for x, we get x > 1.

Example 2: (3x - 0.5) < 0

The absolute value expression in this situation is -(3x - 0.5), which is equivalent to -3x + 0.5. We thus have:

-3x + 0.5 > 1.5

The result of solving for x is: x -0.333

Example 3: (3x - 0.5) = 0

The equation for the absolute value in this situation is |0|, which equals 0. The result is either 3x - 0.5 = 1.5 or 3x - 0.5 = -1.5.

The results of each equation's x term are: x = 1 or x = 0.

As a result, the answer to |3x - 0.5| > 1.5 is either x -0.333 or x > 1.

x = 0 or x = 1 is the answer to the equation |3x - 0.5| = 1.5.

The answer to the equation |3x - 0.5| 1.5 is: -0.333 x 0.833

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