7.3 right triangle and trigonometry homework 

Therefore, the formula for T1 is T1(x1,x2) =[tex](24x1/5 - 18x2/5, -36x1/5 + 6x2/5).[/tex]

Given T is a linear transformation from R2 into R2, we need to show that T is invertible and find a formula for T1.T(x1,x2) = (3x1-9x2, -3x1+5x2)Let A be the standard matrix for T, then we have to find the determinant of A as it will help us determine whether T is invertible or not.The standard matrix for T is given as :A = [3 -9-3 5]Now, to calculate the determinant of A, we have to use the formula as:det(A) = ad - bcwhere a = 3, b = -9, c = -3, d = 5Now, substituting the values of a, b, c, d, we get,det(A) = (3 × 5) - (-9 × -3) = 15 - 27= -12Since the determinant of A is not equal to zero, therefore, T is invertible.

For finding the formula for T1, we need to calculate the inverse of T. We can find the inverse of T as:[tex][A|I] → [I|A-1][/tex]Now, appending the identity matrix to A, we get the following matrix.[A|I] = [tex][3 -9|1 0-3 5|0 1]→ [I|A-1] = [1 3/5|-9/5-3/5][/tex]

Therefore, the inverse of T is T-1(x1,x2) = (3x1/5 + 9x2/5, -9x1/5 - 3x2/5)Now, we can find T1 by substituting T-1 in T, we get,T1(x1,x2) = T(T-1(x1,x2)) =[tex]T(3x1/5 + 9x2/5, -9x1/5 - 3x2/5) = (3((3x1/5 + 9x2/5)) - 9(-9x1/5 - 3x2/5), -3((3x1/5 + 9x2/5)) + 5(-9x1/5 - 3x2/5)) = (24x1/5 - 18x2/5, -36x1/5 + 6x2/5)[/tex]

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Answer:

x=5

Step-by-step explanation:

[tex] log_{2x - 7}(81) = 4[/tex]

Raise both sides with a base of 2x-7

[tex]81 = (2x - 7) {}^{4} [/tex]

We get two solutions

[tex]3 = (2x - 7)[/tex]

[tex]10 = 2x[/tex]

[tex]x = 5[/tex]

The other solutions is

[tex] - 3 = 2x - 7[/tex]

[tex]4 = 2x[/tex]

[tex]x = 2[/tex]

The solution x=2 wont work since it will make our base negative.

So the answer is 5

The light bulb should be placed 1 inch from the vertex so that the rays will be reflected parallel to the axis.

The reflector of a flashlight is in the shape of a paraboloid of revolution.

Its diameter is 4 inches and its depth is 1 inch.

To find the distance from the vertex to the light bulb, so that the rays will be reflected parallel to the axis, we will use the following steps:

Step 1: Write the equation of the paraboloid.

For the given paraboloid, the vertex is at the origin (0,0,0) and the axis of symmetry is the z-axis.

Since the paraboloid is symmetric with respect to the z-axis, the equation of the paraboloid can be written as:

[tex]$$z=\frac {x^2+y^2}{4p}$$[/tex]

where p is the distance from the vertex to the focus.

Here, we have to find p.

Step 2: Find the value of p.

We know that the diameter of the reflector is 4 inches.

Since the paraboloid is symmetric with respect to the z-axis, the diameter of the paraboloid is also 4 inches.

Therefore, the radius of the base of the paraboloid is 2 inches.

Since the depth of the paraboloid is 1 inch, the vertex is 1 inch away from the base of the paraboloid.

Hence, the value of p can be found as follows: p = distance from vertex to focus

= [tex]$\frac{1}{4}$[/tex]

distance from vertex to the base

[tex]= $\frac {1}{4}$ (2)[/tex]

[tex]= $\frac{1}{2}$[/tex] inches

Step 3: Find the distance from the vertex to the light bulb.

Since the rays should be reflected parallel to the axis, the light bulb should be placed at a distance of

[tex]2p = 2 $\times$ $\frac{1}{2}$ = 1 inch[/tex] from the vertex.

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To find the fraction of the square quilt block that is shaded, we need to count the number of shaded unit squares and divide it by the total number of unit squares in the quilt block. Let us begin by counting the number of shaded unit squares.

We notice that there are a total of 6 unit squares that are shaded. The unit squares that are shaded are the 2 squares that are completely shaded and the 4 squares that are half shaded due to the presence of triangles.

Next, we need to count the total number of unit squares in the quilt block. We notice that the quilt block is made up of 9 unit squares, each of which can be divided into 4 smaller unit squares. Thus, the total number of unit squares in the quilt block is 9 x 4 = 36.

Therefore, the fraction of the square quilt block that is shaded is 6/36 or 1/6.

To summarize, the shaded portion of the quilt block consists of 6 unit squares out of a total of 36 unit squares. Thus, the fraction of the square quilt block that is shaded is 1/6.

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The factors that affect the width of a confidence interval are a. population variation, c. sample size, and d. confidence level

Confidence interval refers to a statistical measure that attempts to estimate the range of values that could plausibly contain the true value of a population parameter. A confidence interval, in other words, is a statistical range that tells us how sure we are that a statistical estimate we're interested in lies between two values.

There are several factors that can affect the width of a confidence interval. Here are some of them:

The Question was Incomplete, Find the full content below :

what factors of the following affect the width of a confidence interval?

a. population variation

b. none of them

c. sample size

d. confidence level

e. sample mean

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Answer:B

Step-by-step explanation:

Adjacent side to the given angle = 7.072

Hypotenuse = opposite side to given angle ^2 + adjacent side to given angle ^2

Hypotenuse = 8.27666502886

The value of the expression 29.2 + 4.81/√97.2​ when each value is estimated to 1 significant figure is 30.5

Given that we have the following expression

29.2 + 4.81/√97.2​

Using the order of operations, we need to first take the square root of 97.2, and then divide 4.81 by the result, before finally adding 29.2 (all estimated)

To 1 significant figure, the square root of 97.2 is 10,

Mathematically, we have

29.2 + 4.81/√97.2 ≈ 29.2 + 4.81/10

Next, we estimate others

So, we have

29.2 + 4.81/√97.2 ≈ 30 +  5/10

Divide

29.2 + 4.81/√97.2 ≈ 30 +  0.5

Evaluate the sum

29.2 + 4.81/√97.2 ≈ 30.5

Hence, the estimate is 30.5

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The volume for each cylinder, given the dimensions, is as follows:

1. r = 4 units, h = 6 units, V = 96π units².

2. r = 8 units , h = 3 units , V = 192π units².

3. r = 1 unit(half of the diameter), h = 5 units, V = 5 units².

4. r = 5 units, h = 7 units, V = 175π units².

The volume of a cylinder of radius r and height h is given by the equation presented as follows, in which the square of the radius is multiplied by π and the height, hence:

V = πr²h.

Then the equation is applied to each option in this problem with the given dimensions to obtain the volumes.

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The best-fitting regression line can be used to estimate or predict the value of y for a given value of x. Thus, option b is correct.The best-fitting regression line is a statistical model that summarizes the relationship between two variables, x and y. It allows us to estimate or predict the value of y for a given value of x based on the observed data.

Regression is a statistical method that helps in the understanding and quantification of the relationship between two or more variables. It includes a set of techniques to build models that explain how the value of one variable depends on one or more other variables. The best-fitting regression line is a line that gives the best possible fit to the data on a scatter plot of points.

.The equation of the line is given as

y = mx + c, where

y is the dependent variable,

x is the independent variable,

m is the slope of the line and

c is the y-intercept.

The value of m and c is calculated using the given data points.The equation of the best-fitting regression line gives us a way to estimate or predict the value of y for a given value of x. Thus, the correct option is b.

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The answer of the given question based on the pentagon is , the apothem of the pentagon is approximately 13.76 meters, and the area of the pentagon is approximately 688 meter².

An apothem is line segment that is drawn from center of regular polygon to midpoint of one of its sides. The apothem is always perpendicular to the side it intersects and is typically used to calculate the area of the polygon. In a regular polygon, all sides are congruent and all interior angles are also congruent. The apothem, therefore, is the radius of the inscribed circle of the polygon.

Apothem can be found using the formula:

a = (s/2) * (1/tan(π/n))

where s is the length of one side of the pentagon, and n is the number of sides (in this case, n=5 for a pentagon).

Plugging in s=20 and n=5, we get:

a = (20/2) * (1/tan(π/5))

a ≈ 13.76m

So the apothem of the pentagon is approximately 13.76 meters.

To find the area of the pentagon, we can divide it into five congruent triangles, each with base equal to the length of one side (20m) and height equal to the apothem (13.76m). The area of each triangle is:

(1/2) * base * height = (1/2) * 20m * 13.76m ≈ 137.6m²

So the area of the pentagon is the sum of the areas of the five triangles:

A = 5 * (1/2) * base * height = 5 * 137.6m² ≈ 688m²

Therefore, the apothem of the pentagon is approximately 13.76 meters, and the area of the pentagon is approximately 688 meter².

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If the average GPA of juniors, seniors, and graduate students in your class differ, you should use a one-way ANOVA test. This is the proper statistical test used to compare the average GPAs of different groups.

ANOVA is used to test whether there are any significant differences between the means of three or more independent groups. In this case, we have three independent groups (juniors, seniors, and graduate students) and we want to determine whether their average GPAs differ significantly by  proper statistical test.

If the ANOVA test indicates that there are significant differences between the means, we can use post-hoc tests (such as Tukey's test or Bonferroni's test) to determine which specific group means differ significantly.

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The answer this riddle is time as it is always in a hurry, constantly moves forward and waits for no one. And time is often represented as being in a high position, such as on a clock or a calendar.

A riddle is a type of puzzle or question that is designed to challenge and engage the mind. It usually presents a seemingly paradoxical or enigmatic situation or description, and the goal is to figure out the hidden meaning or answer through careful thought and analysis.

Riddles can take many different forms, but they often involve wordplay, metaphor, or clever misdirection to lead the solver towards the solution. Riddles have been used for entertainment, education, and even as a means of passing on cultural knowledge and wisdom from one generation to the next.

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Answer:

13xy^2 - 7x^2y

Step-by-step explanation:

1) Collect like terms.

13xy^2+(-12x^2y+5x^2y)

2) Simplify.

13xy^2 - 7x^2y

Answer:

the area ot the triangle is half the area of the rectangle, so the area of the triangle is what sqauare units

Answer: It's  much easier to form a rectangle and multiply the width and the length and then divide it by 2  

Step-by-step explanation:

One-hundred-two-and-eighty-one-hundredths can be written as 102.81 in standard form.

About 68% percent of light bulbs can last within one standard deviation of the mean, between 57.6 hours and 66.4 hours.

To determine the percentage of light bulbs that can last within one standard deviation of the mean, between 57.6 hours and 66.4 hours, you can follow these steps:

1. Identify the range of values:

In this case, the range is between 57.6 hours and 66.4 hours.

2. Recognize that this range represents one standard deviation from the mean.

In a normal distribution, approximately 68% of values fall within one standard deviation of the mean.

So, about 68% of light bulbs can last within one standard deviation of the mean, between 57.6 hours and 66.4 hours.

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the formula for y in terms of x is y = 12/x²3/4).

Why it is?

We have two statements about the relationship between the variables y, t, and x:

"y is directly proportional to the √ of t" means that y = k√t for some constant of proportionality k.

"T is inversely proportional to the cube of x" means that T = k'/x²3 for some constant of proportionality k'.

We are given that y = 12 when t = 16 and x = 2. Using these values, we can solve for the constants k and k':

y = k√t

12 = k√16

12 = 4k

k = 3

T = k'/x²3

16 = k'/2²3

k' = 128

Now we can substitute these values for k and k' into the expression for y in terms of t and simplify:

y = k√t

y = 3√t

We can also substitute the expression for t in terms of x:

T = k'/x²3

T = 128/x²3

And substitute the value of T when x = 2:

T = 128/2²3

T = 16

Now we can substitute this value for t in the expression for y:

y = 3√t

y = 3√16/x²(3/2)

Simplifying the radical and exponents, we get:

y = 12/x²(3/4)

Therefore, the formula for y in terms of x is y = 12/x²(3/4).

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Answer:

x > 8

Step-by-step explanation:

-6x + 18 < 2 - 4x

-2x + 18 < 2

-2x < -16

x > 8

Answer:

The two angles are 57° and 33°.

Step-by-step explanation:

The three angles of a triangle add up to 180°. The angle at the top of this triangle has a little tiny square in it to show that its a right angle. So it is 90°

That means that the other two angles have to add up to 90° We can use this to write an equation.

(4x-7)° + (3x-15)° = 90°

Combine like terms.

7x - 22 = 90

add 22

7x = 112

divide by 7

x = 16

Then the question says to find the angle measures. Put the x = 16 back into the given expressions.

4x - 7

= 4(16) - 7

= 64 - 7

= 57

and

3x - 15

= 3(16) - 15

= 48 - 15

= 33

The two angles are 57° and 33°.

Answer:

[tex](x+2)^2-16[/tex]

Step-by-step explanation:

Given quadratic expression:

[tex]x^2+4x-12[/tex]

To complete the square, add and subtract the square of half the coefficient of the term in x after the term in x:

[tex]\implies x^2+4x+\left(\dfrac{4}{2}\right)^2-\left(\dfrac{4}{2}\right)^2-12[/tex]

Simplify:

[tex]\implies x^2+4x+\left(2\right)^2-\left(2\right)^2-12[/tex]

[tex]\implies x^2+4x+4-4-12[/tex]

The first three terms form a perfect square trinomial which can be factored:

[tex]\implies \boxed{x^2+4x+4}-4-12[/tex]

[tex]\implies \boxed{(x+2)^2}-4-12[/tex]

Finally, subtract the numbers:

[tex]\implies (x+2)^2-16[/tex]

The triangle shown in the image above that has sides 8, 24, and 25 is a right triangle because it satisfies the Pythagorean theorem.

A triangle with sides 8, 24, 25 is a right triangle.

This is because the sides satisfy the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (the legs) is equal to the square of the longest side (the hypotenuse).

In this case, we have:

8² + 24² = 64 + 576 = 640

25² = 625

Since 8² + 24² = 25², the triangle satisfies the Pythagorean theorem, and is therefore a right triangle.

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Answer:

Step-by-step explanation:

4 and a half

Answer: 6

Step-by-step explanation:

The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42

The factors of 78 are: 1, 2, 3, 6, 13, 26, 39, 78

Then the greatest common factor is 6.

Heres something you need to learn about the greatest common factor (gcf)

What is the Greatest Common Factor?

The largest number, which is the factor of two or more numbers is called the Greatest Common Factor (GCF). It is the largest number (factor) that divide them resulting in a Natural number. Once all the factors of the number are found, there are few factors that are common in both. The largest number that is found in the common factors is called the greatest common factor. The GCF is also known as the Highest Common Factor (HCF)

Let us consider the example given below:

Greatest Common Factor (GCF)

For example – The GCF of 18, 21 is 3. Because the factors of the number 18 and 21 are:

Factors of 18 = 2×9 =2×3×3

Factors of 21 = 3×7

Here, the number 3 is common in both the factors of numbers. Hence, the greatest common factor of 18 and 21 is 3.

Similarly, the GCF of 10, 15 and 25 is 5.

How to Find the Greatest Common Factor?

If we have to find out the GCF of two numbers, we will first list the prime factors of each number. The multiple of common factors of both the numbers results in GCF. If there are no common prime factors, the greatest common factor is 1.

Finding the GCF of a given number set can be easy. However, there are several steps need to be followed to get the correct GCF. In order to find the greatest common factor of two given numbers, you need to find all the factors of both the numbers and then identify the common factors.

Find out the GCF of 18 and 24

Prime factors of 18 – 2×3×3

Prime factors of 24 –2×2×2×3

They have factors 2 and 3 in common so, thus G.C.F of 18 and 24 is 2×3 = 6

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GCF and LCM

Greatest Common Factor of two or more numbers is defined as the largest number that is a factor of all the numbers.

Least Common Multiple of two or more numbers is the smallest number (non-zero) that is a multiple of all the numbers.

Factoring Greatest Common Factor

Factor method is used to list out all the prime factors, and you can easily find out the LCM and GCF. Factors are usually the numbers that we multiply together to get another number.

Example- Factors of 12 are 1,2,3,4,6 and 12 because 2×6 =12, 4×3 = 12 or 1×12 = 12. After finding out the factors of two numbers, we need to circle all the numbers that appear in both the list.

Greatest Common Factor Examples

Example 1:

Find the greatest common factor of 18 and 24.

Solution:

First list all the factors of the given numbers.

Factors of 18 = 1, 2, 3, 6, 9 and 18

Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24

The largest common factor of 18 and 24 is 6.

Thus G.C.F. is 6.

Example 2:

Find the GCF of 8, 18, 28 and 48.

Solution:

Factors are as follows-

Factors of 8 = 1, 2, 4, 8

Factors of 18 = 1, 2, 3, 6, 9, 18

Factors of 28 = 1, 2, 4, 7, 14, 28

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The largest common factor of 8, 18, 28, 48 is 2. Because the factors 1 and 2 are found all the factors of numbers. Among these two numbers, the number 2 is the largest numbers. Hence, the GCF of these numbers is 2.

If you know it, dont read it

Answer:

Step-by-step explanation:

Supplementary Angles and Adjacent Angles

Answer: It would be both supplementary and adjacent angles

Step-by-step explanation: both of the angles add up to 180 degrees

The total number of combinations of sandwiches possible is 360.

A local sub shop offers five different breads, four different meats, three different cheeses, and six different vegetables. You can choose one bread and any number of the other items.

To find the total number of combinations of sandwiches possible, we can use the multiplication rule.The multiplication rule states that if there are m ways to do one task and n ways to do another task, then there are m × n ways to do both tasks.

Similarly, if there are m ways to do one task, n ways to do another task, and p ways to do a third task, then there are m × n × p ways to do all three tasks.

To apply the multiplication rule to this problem, we can multiply the number of options for each component:Breads: 5 optionsMeats: 4 optionsCheeses: 3 optionsVegetables: 6 options

To find the total number of combinations, we can multiply the number of options for each component:5 × 4 × 3 × 6 = 360Therefore, there are 360 possible combinations of sandwiches.

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Answer:

first box: 4

second box: 1

Step-by-step explanation:

f(x) is a pretty standard curve, its a parabola, opens up, vertex at (0,0)

Its the most basic version of a parabola (a kind of U-ish, V-ish smooth curve shape)

But if you mess around with the x...that "- 4" that's in close to the x, it will slide the whole graph over to the right 4 units. Left and right shifts (shifts, slide, translate...all the same thing--it means what you think it should mean) anyway, left and right shifts are a little backwards from what you think it should be:

- anumber shifts right

+ anumber shifts left

There are math reasons why this happens, but just memorize it!

Up and down shifts go the way you think they should. If you tack a number on the end of an equation:

- anumber slides down

+ anumber slides up

So the graph of h is a translation (shift or slide) 4 units right and 1 unit down of f.

We have so established that m and m are equal as As congruent angles also referred to as vertical angles are formed when two lines connect.

The degree of rotation between two lines, rays, or segments that have a common endpoint is measured in mathematics as an angle. Angles are categorised in geometry according on how much they are rotated. Angles that are less than 90 degrees are referred to as acute angles, those that are precisely 90 degrees are referred to as right angles, those that are between 90 and 180 degrees are referred to as obtuse angles, and those that are exactly 180 degrees are referred to as straight angles.

given

We can write: since mXOY = mWOV.

mWOV - m = mXOY - m

This can be stated simply as:

OXY = OWV

As congruent angles—also referred to as vertical angles—are formed when two lines connect, we can state:

OXY = MOVY.

Therefore:

mOVW mOVY

And:

mOYX = mOXY

Therefore:

mOYX = mOVY

Finally:

OYX = OWV

We thus have:

m = m

We have so established that m and m are equal as As congruent angles also referred to as vertical angles are formed when two lines connect.

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The simple random sample of size 2 that will be selected is {b, c}.

To create a simple random sample of size 2 using the random digits 9, 0, 3, 7, 3, 2, 4, follow these steps:
1. Assign each letter a random number: 1 corresponds to a, 2 corresponds to b, and so on.
2. Go through the given random digits and find the first two that correspond to a valid letter.

The random digits are 9, 0, 3, 7, 3, 2, 4.
9 and 0 do not correspond to any letters (since we only have numbers 1, 2, 3, and so on). The next digit is 3, which corresponds to the letter c. The following digit, 7, does not correspond to a letter, so we move on to the next digit. The next digit is 3 again, which we have already selected. The next valid digit is 2, which corresponds to the letter b.
Thus, the simple random sample of size 2 that will be selected using these random digits is the set {b, c}.

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