Chapter 7: Right Triangle Relationships and
Trigonometry
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The variable associated with the interior and exterior angles in the triangle is equal to 56.

The value of the exterior angle is equal to 116°.

This question presents the case of a geometric system formed by a triangle and a semirray that includes two interior angles and a exterior angle. Please notice that the sum of the measures of interior angles in a triangle equals 180° and the sum of two supplementary angles equals 180°.

First, derive the equation of the missing interior angles in triangle PQR:

m ∠ R = 180° - 60° - x

m ∠ R = 120° - x

Second, obtain the equation for the two supplementary angles:

(120° - x) + (2 · x + 4°) = 180°

124° + x = 180°

x = 56

Third, determine the value of the exterior angle:

θ = 2 · x + 4

θ = 2 · 56 + 4

θ = 116°

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To find the interquartile range of the amounts of radiation received at a greenhouse, we need to first calculate the first quartile (Q1) and the third quartile (Q3).

The explained solution to the problem is mentioned below.

First is to sort the data in ascending order:
6.1, 6.8, 8.7, 9.0, 10.2, 10.3, 10.6, 10.8, 10.9, 11.0, 11.1, 11.5, 11.7, 12.2, 13.4

The second is to find the median (Q2) of the sorted data. Since we have an odd number of data points, Q2 is the middle value, which is 10.8.

The third one is to find the median (Q1) of the lower half of the sorted data. In this case, the lower half consists of the values up to and including Q2. So, the lower half is 6.1, 6.8, 8.7, 9.0, 10.2, and 10.3. The median of this lower half is (8.7 + 9.0) / 2 = 8.85.

Next is to find the median (Q3) of the upper half of the sorted data. In this case, the upper half consists of the values from Q2 onwards. So, the upper half is 10.9, 11.0, 11.1, 11.5, 11.7, 12.2, and 13.4. The median of this upper half is (11.5 + 11.7) / 2 = 11.6.

Last but not least, calculate the interquartile range (IQR) by subtracting Q1 from Q3: IQR = Q3 - Q1 = 11.6 - 8.85 = 2.75.

Therefore, the interquartile range of the amounts of radiation received at the greenhouse is 2.75. The interquartile range represents the range of the middle 50% of the data, indicating the spread of values within that range.

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Answer:

Therefore any set of data that satisfies the 5-Number summary: 16,18,22,26 and 28 can be represented with the box plot.

Step-by-step explanation:

Interpreting Box Plots

We use a box plot to present the 5-Number summary of a set of data.

The values contained in the 5-Number summary are:

In the box plot, the following rules apply:

Using these, we interpret the given box plot

A left whisker extends from 16 to 18, therefore:

The box extends from 18 to 26 and is divided into 2 parts by a vertical line segment at 22.

The right whisker extends from 26 to 28.

Therefore any set of data that satisfies the 5-Number summary: 16,18,22,26 and 28 can be represented with the box plot.

Box plot has boxes with some lines ( called whiskers ).

Box plot on the graphical level tells about 5 points:

There is starting whisker or left whisker denoting point minimum (Q0) and extends from there the line to Q1 ( first quartile).

Then the box starts which starts from Q1 and touched Q3. In the middle it passes through Q2 ( The median of the data ), and thus has a line in it.

At last, there is end whisker or right whisker denoting end quantile or Q4 (the maximum point).

Given data:

Left whisker extends from 16 to 18.

     Thus Q0 = 16

              Q1 = 18

Box plot starts from 18 and extends to 26, and is split by a vertical line segment at 22.

    Thus Q2 = 22

             Q3 = 26

The right whisker or end whisker extends from 26 to 28

    Thus Q4 = 28.

Thus the description of data is given as:

\(Q_0 = Minimum = 16\\Q_1 = First\: quartile = 18\\Q_2 = Second \:quartile = Median= 22\\Q_3 = Third\: quartile = 26\\Q_4 = Maximum = 28\\\)

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Answer: 729 cubic yards

Step-by-step explanation:

To calculate the volume of a cube,

we need to multiply its side 3 times, so

9×9×9=729.

Step 1

Add the length of the 2 ropes of Jen.

Step 2

Find the difference between the rope of Jen (279/8)ft and Kelsey(71/2)ft

Hence, Kelsey has 5/8 feet of more rope. Option C

Answer:

C

Step-by-step explanation:

The given double integral is ∫ ∫ (x² + y²)dx dy, and we need to evaluate it over the region in the positive quadrant where x+y≤1.

To evaluate this double integral, we can first determine the limits of integration for both x and y based on the given region. In the positive quadrant, x and y both range from 0 to 1.

Now, integrating the inner integral with respect to x, we get:

∫ (x² + y²)dx = (1/3)x³ + y²x + C1,

where C1 is the constant of integration.

Next, we integrate the resulting expression with respect to y:

∫ [(1/3)x³ + y²x + C1] dy = (1/3)x³y + (1/3)y³x + C1y + C2,

where C2 is another constant of integration.

Finally, we evaluate this double integral over the given region by substituting the limits of integration:

∫∫ (x² + y²)dx dy = ∫[0 to 1] ∫[0 to 1-x] (x² + y²)dy dx.

Performing the integration, we can find the numerical value of the double integral within the given region.

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Answer:

Answer is 69%

Step-by-step explanation:

Because 69% of 42 is 29

Answer:

29/42*100=69%

Step-by-step explanation:

To draw the largest possible circle on the triangular piece of wood, the furniture maker can follow these steps:

By following these steps, the furniture maker can ensure that the circle is as large as possible and fits within the given triangular piece of wood.

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Answer:

Because there are fewer than 30 (not no more than 30) one of the answer i x+y < 30

Because he needs both small and large another answer is x>0 and not x  0

Finally because he need at least dozen large cones the last answer is y

Step-by-step explanation:

The area of the shaded region is 36.3  to 3  significant figures

What is the area?

A two-dimensional figure's area is the amount of space it takes up. In other terms, it is the amount that counts the number of unit squares that span a closed figure's surface.

Step one: find the two diagonals of the kite.

The Horizontal diagonal can be obtained using the cosine rule:

AC² = OA²- OC² - 2 *OA*OC* cosθ

       = 10²+ 10² - 2* 10 *10 * cos(120)

AC² = 200

=> AC=√200 = 14.1

The vertical diagonal of the kite can be obtained by Pythagoras' Theorem:

Please note the law in circle geometry which states that a radius and a tangent always meet at right angles.

This implies that triangle OBC is a right-angled triangle, with angle OCB being 90 degrees, and COB being 60 degrees. This is because the diagonal divides the 120-degree angle into half.

cos(60)= 10/OB

=> OB= 10/ cos(60) = 20 m

Step two: Use the dimensions of the two diagonals of the Kite to find the area:

The area of a Kite is obtained using this formula:

area = pq/2, , where p and q are the two diagonals.

area =( 14.1*20)/2 = 141 m²

Step three: Calculate the area of the sector of the circle.

Area of the sector is obtained using this formula

Area =θ/360 * πr²  = 120/360 * 3.14 * 10²  = 104.66 m²

Step Four: Subtract the area of the sector from the area of the kite:

Area of the shaded region will be  141 m²  -  104.66 m²  = 36.34 m²

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Answer:

SA = 94 ft²

Step-by-step explanation:

To find the surface area of a rectangular prism, you can use the equation:

SA = 2 ( wl + hl + hw )

SA = surface area of rectangular prism

l = length

w = width

h = height

In the image, we are given the following information:

l = 4

w = 5

h = 3

Now, let's plug in the information given to us to solve for surface area:

SA = 2 ( wl + hl + hw)

SA = 2 ( 5(4) + 3(4) + 3(5) )

SA = 2 ( 20 + 12 + 15 )

SA = 2 ( 47 )

SA = 94 ft²

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Answer: True.

Step-by-step explanation:

The statement is:

"A linear equation has ONE solution when the variable equals zero."

First, a generic linear equation can be written as:

y = a*x + b

Where a and b are known values.

If the variable equals zero, we have:

y = a*0 + b = b

y = b.

Then it has only one solution if x = 0.

So the statement is true, but it is also true for any value in the variable.

So if we rewrite it as (for example):

"A linear equation has one solution when the variable equals 4"

y = a*4 + b.

Because a and b are known, this equation also does have only one solution.

Answer:true

Step-by-step explanation:

Answer:

23 - 24 = -1

Step-by-step explanation:

since the are all multiplied by square root of 2, we can set this as a variable

x = square root 2

8x + 15x -24x

this equals -1x = -x = -square root 2

Answer:

LM = 18

Step-by-step explanation:

since the triangles are similar then the ratios of corresponding sides are in proportion, that is

\(\frac{LM}{QM}\) = \(\frac{LN}{QP}\) ( substitute values )

\(\frac{LM}{12}\) = \(\frac{15}{10}\) ( cross- multiply )

10 LM = 180 ( divide both sides by 10 )

LM = 18

Answer:

(x+1)(x+5)(5x−1)

x= 1/5, -5, -1

Step-by-step explanation:

5x^3+29x^2+19x-5

factor

(x+1)(x+5)(5x−1)

the zeros are

x+1=0

x=-1

x+5=0

x=-5

5x-1=0

5x=1

x=1/5

maybe this helps

Answer:

The answer is: From any point on the graph, he counts down 2 then right 3 to end up on the line again.

Step-by-step explanation:

To count the slope of the graph from any point on the graph; He counts down 2 then right 3 to end up on the line again.

The slope of a linear graph refers to the change behaviour of its dependent variable (usually y-axis) with respect to its Independent variable (usually x-axis).

In essence;

slope, m = (y2-y1)/(x2-x1)

Consequently, From any point on the graph, he counts down 2 then right 3 to end up on the line again.

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9514 1404 393

Answer:

  10°, 80°

Step-by-step explanation:

Setup

Let x and y represent the measures of the two acute angles in the right triangle.

From your knowledge of right triangles, you know the two acute angles are complementary:

  x + y = 90

The problem statement gives you another relation:

  x = 2(y + 30) . . . . twice the sum ...

__

Solution

The second equation gives you an expression for x that can be substituted into the first equation.

  (2(y +30)) + y = 90

  3y +60 = 90 . . . . . . . . collect terms

  3y = 30 . . . . . . . . . subtract 60

  y = 10 . . . . . . . . divide by 3

  x = 2(y +30) = 2(40) = 80

The two angles are 10° and 80°.

Answer:

12

Step-by-step explanation:

So Josh is 6 years older than kims age and the sum of the ages is less than 31, the greatest number that is less than 31 is 30 so we can use that.

We can make an equation out of this and solve it.

n+(n+6)=30

2n+6=30

2n=24

n=12

12 is the oldest Kim could be.

12+(12+6)=12+18=30.

Answer:

none of the above is best andwer

Answer:

rotation

Step-by-step explanation:

Answer:

Reflection

Step-by-step explanation:

I think its reflection because if you reflect them across the y axis then the x axis I believe they will be in those positions.

Answer:

the coner

Step-by-step explanation:

Answer:

the area of a triangle is A =BxH

2

Answer:

2

Step-by-step explanation:

Answer:

Step-by-step explanation:

A=1/4πd2

A = 1/4(22/7)(18)^2

= 254.47

Answer:

Irrational

Step-by-step explanation:

\(\sqrt{2}\) produces 1.41421356237, so it is irrational.

To show that U = V, we need to demonstrate that U is a subset of V and V is a subset of U. This can be done by expressing each vector in U as a linear combination of vectors in V and vice versa.

Let's start by showing that U is a subset of V. For any vector u in U, we can express u as a linear combination of vectors v1 and v2 in V:

u = a1v1 + a2v2,

where a1 and a2 are scalars. Let's substitute the given values of u1, u2, and u3 into this equation:

u1 = 1v1 + 0v2,

u2 = 0v1 + 1v2,

u3 = -1v1 + 3v2.

This shows that u1, u2, and u3 can be expressed as linear combinations of v1 and v2, which implies that U is a subset of V.

Next, we need to demonstrate that V is a subset of U. For any vector v in V, we can express v as a linear combination of vectors u1, u2, and u3 in U:

v = b1u1 + b2u2 + b3u3,

where b1, b2, and b3 are scalars. Substituting the given values of v1 and v2 into this equation:

v1 = 1u1 - u3,

v2 = 2u1 + 3u2 + 2u3.

This shows that v1 and v2 can be expressed as linear combinations of u1, u2, and u3, indicating that V is a subset of U.

Since U is a subset of V and V is a subset of U, we can conclude that U = V, i.e., the spans of the given vectors are equal.

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