The solution to the original system is the pair and Explain why it makes sense that this pair of values is also the solution to the equation

If Jason starts building 5 feet out from the existing structure, and wants it to have a 60° angle, then the wall should be 10 feet long.

For given question,

Jason starts building 5 feet out from the existing structure, and wants it to have a 60°C angle.

The ground and the side of the play structure make a 90° angle.

So, the given triangle would be a - 30°- 60° - 90° - triangle.

The short side is 5 feet. The climbing wall will be the hypotenuse.

The angle opposite to the short side is 5 feet.

Now we need to find the hypotenuse.

Consider the sine of angle 30°

sin(30°) = 5/hypotenuse

1/2 = 5/hypotenuse

hypotenuse = 5 × 2

hypotenuse = 10

Therefore, if Jason starts building 5 feet out from the existing structure, and wants it to have a 60° angle, then the wall should be 10 feet long.

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The probability of hitting the area of the shaded region is 0.21 (in the form of two decimal places) when you throw a dar.

The probability is defined as the possibility of an event being equal to the ratio of the number of favorable outcomes and the total number of outcomes.

We have been given that the radius of the circle = 2cm

So one side of square = 2+2 = 4cm

Area of circle = πr² = (3.14) ×(2)²= (3.14)×4 = 12.57cm²

Area of square = (a)² = (4)² =16cm²

Then the area of the shaded region = 16−12.57 = 3.43cm²

So the probability of hits in the shaded region = 3.43/16 = 0.2144

Round to the nearest Hundredth (two decimal places).

The probability of hits in the shaded region = 0.21

Hence, the probability of hitting the area of the shaded region is 0.21 (in the form of two decimal places) when you throw a dar.

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A. The percentage of standardized test scores is between 210 and 840 is 99.7%

B. The percentage of standardized test scores that is less than 210 or greater than 840 is 0.3%.

C. The percentage of standardized test scores that is greater than 735 is 2.2%.

a) The z score for 210 will be:

= (210 - 525)/105

= -3

The score for 840 = (840 - 525)/105 = 3

Hence,

Percentage between 210 and 840 will be:

= P(-3 < z < 3)

= 99.7%

b) The percentage of standardized test scores is less than 210 or greater than 840 will be:

= 100% - P(Between 210 and 840)

= 100% - 99.7%

= 0.03%

c) The z score for 735 = (735 - 525)/105

= 2

Hence, the percentage greater than 735

= P(z > 2)

= 2.2%

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The diameter and radius of circle is 24.84 feet and 12.42 feet.

We are given the circumference of circle. So, using the formula of circumference, we will find the diameter. As we know, the radius of circle is half of its diameter.

Performing calculation now -

Circumference = π×d, where d stands for diameter of circle.

Taking the value of π as 3.14.

Keep the values in formula to find the diameter of circle

Diameter = 78÷3.14

Performing division

Diameter = 24.840

Rounding to nearest hundredth -

Diameter = 24.84 feet

Now, calculating radius by the formula -

Radius = diameter÷2

Radius = 24.84÷2

Performing division to find the radius of circle

Radius = 12.42 feet

Hence, the diameter and radius of circle is 24.84 feet and 12.42 feet.

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The approximate value of negative seventeen plus square root of sixty to the nearest tenth is -9. 3 . Option A

From the information given, we have;

Substitute the values into the expression, we have;

- 17 + √60

Find the square root of 60 and add to the expression;

-1 7 + 7. 7459

Find the sum of the values

-9. 3

Thus, the approximate value of negative seventeen plus square root of sixty to the nearest tenth is -9. 3 . Option A

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The simpler form of the complex fraction [tex]\frac{x-2}{\frac{x+2}{\frac{x+1}{\frac{3}{\frac{x-1-1}{x+1} } } } }[/tex] is equal to

3(x-2)² / (x+2)(x+1)².

As given in the question,

Given complex fraction is [tex]\frac{x-2}{\frac{x+2}{\frac{x+1}{\frac{3}{\frac{x-1-1}{x+1} } } } }[/tex]

Simplify the given complex form :

[tex]\frac{x-2}{\frac{x+2}{\frac{x+1}{\frac{3}{\frac{x-1-1}{x+1} } } } }\\\\= \frac{x-2}{\frac{x+2}{\frac{x+1}{\frac{3}{\frac{x-2}{x+1} } } } }\\\\=\frac{x-2}{\frac{x+2}{\frac{x+1}{\frac{3}{\frac{x-2}{x+1} } } } } \\\\=\frac{x-2}{\frac{x+2}{\frac{(x+1)^{2}}{3(x-2)} } } \\\\=\frac{3(x-2)^{2}}{(x+2)(x+1)^{2}}[/tex]

Therefore, the simpler form of the complex fraction [tex]\frac{x-2}{\frac{x+2}{\frac{x+1}{\frac{3}{\frac{x-1-1}{x+1} } } } }[/tex] is equal to 3(x-2)² / (x+2)(x+1)².

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Parenthesis

Exponents

Multiplication

Division

Addition

Subtraction

Multiplication and Division - whatever comes first from left to right (i.e. 6*7*2, do 6*7 first)

Addition and Subtraction - same thing as multiplication and division, whatever comes first

x=567

Step-by-step explanation:

The equation in slope-intercept form is y = -2/3x + 6

Given,

Points (x, y) = (6, 2)

Parallel to y = -2/3x + 1

Now,

Recall that a parallel line to the one given has to have the same slope. The slope of the line given is "-2/3" (the coefficient that accompanies "x").

Therefore the equation of a parallel line has to be of the form:

y = -2/3x + b

2 = -2/3(6) + b

2 = -12/3 + b

2 = -4 + b

b = 6

So the equation now becomes:

y = -2/3x + 6

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Using translation concepts, the new coordinates of the point are given as follows:

(-2.5, 9.8).

A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s range(involving values of y) or in it’s domain(involving values of x). Examples are shift left, shift right, shift down, shift up, vertical stretching, horizontal stretching, vertical compression, horizontal compression, reflection over the x-axis, reflection over the y-axis, and rotations by a determined amount(in degrees). All these translations have predetermined rules that we apply to the figures.

For this problem, the translation rule for each point is given as follows:

(x, y) → (x - 7.5, y + 6.8)

This translation rule means that the figure is shifted:

The original coordinate is given by:

(5,3), meaning that x = 5 and y = 3.

Applying the rule to the original coordinates, we have tat:

Hence the new coordinates, after the rule is applied to the original coordinates, are given as follows:

(-2.5, 9.8).

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Answer:

x = 24

Step-by-step explanation:

[tex]\frac{6+x}{3}[/tex]  = [tex]\frac{x}{4}[/tex] + 4  Multiply though by 12 to clear the fraction

4(6 + x) = 3x + 48  Distribute the 4)

24 + 4x = 3x + 48  Subtract 3x from both sides of the equation

24 + x = 48  Subtract 24 from both sides.

x = 24

Answer: 100 $

Step-by-step explanation:

300 divided by 3 is 100

The distance between the points R and S is 18.03 units.

We know that for points P(x1, y1) and Q(x2, y2) the distance between points is given as:

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

In this question, we have been given two points R(8,0), S(-9,6)

We need to find the the distance between the points R(8,0) and S(-9,6)

Let R(8,0) = (x1, y1) and S(-9,6) = (x2, y2)

Using distance formula the distance between the points R and S is:

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\\\d=\sqrt{(-9-8)^2+(6-0)^2} \\\\d=\sqrt{(-17)^2+6^2}\\\\ d=\sqrt{289+36}\\\\ d=\sqrt{325}\\\\d=18.03~~units[/tex]

Therefore, the distance between the points R and S is 18.03 units.

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Answer:

16red 24yellow in total  40

Step-by-step explanation:

8 x 2 =16

12 x 2 = 24

Answer:

Let P(h,k) be the point which is equidistant from the point (2,4) and the y-axis. The distance of point P(h,k) from the y-axis is h. ∴h=√(h−2)2+(k−4)2⇒h2−4h−4+k2−8k+16=h2⇒k2−4h−8k+20=0.

Hence,the locus of (h,k)is y2−4x−8y+20=0.

Step-by-step explanation:

The simplication of the given expression is (3x - 8) /4x² and the restriction on the variable in the denominator of the given expression is x ≠ 0.

According to the given question.

We have an expression 3/4x - 2/x^2

Since, we have to simplify the above expression.

Therefore,

3/4x - 2/x²

⇒ (3x - 8) /4x²

Hence, the simplication of the given expression is (3x - 8) /4x².

Also, we have to find the restriction on variable.

As we know that, we never divide by 0, a restriction on the variable of a rational function with a linear denominator would be any value of the variable that makes the linear denominator equals to 0.

So, for the restriction on the variable.

Set the denominator equals to 0.

⇒ 4x² = 0

Solve for x

x = 0 ( because 4 ≠ 0)

The denominator is 0 when x = 0. This is the only value that makes the given expression undefined.

Therefore, the restriction on the variable in the denominator of the given expression is x ≠ 0.

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The following set of numbers, 20, 21 and 31, can be the measures of the sides of an obtuse triangle.

A triangle is a polygon with three sides and three vertices.

To know if a set of number is a valid measure of the sides of a triangle, it must follow the Triangle Inequality Theorem which states that the sum of any two sides must be greater than the third side, such that a + b > c, a + c > b, and b + c > a.

let a = 20, b = 21, and c = 31

a + b > c     20 + 21 > 31     41 > 31     True

a + c > b     20 + 31 > 21     51 > 21     True

b + c > a     21 + 31 > 21     52 > 20     True

To know what type of triangle based on the length of its side:

1. Take the sum of the squares of the two smaller sides.

20^2 + 21^2 = 841

2. Compare it to the square of the largest side.

31^2 = 961

961 > 841

If the sum of the squares of the 2 is larger than the square of the 3rd, it is an acute triangle.

if they are equal, it is a right triangle

if they are smaller, then it is an obtuse triangle.

Hence, 20, 21 and 31 can be the measure of the sides of an obtuse triangle.

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Chords within a circle can be related in a variety of ways. Parallel chords cut congruent arcs in the same circle. That is, the arcs whose endpoints include one of each chord's endpoints have equal measures.

Chords within a circle can be related in a variety of ways. Parallel chords cut congruent arcs in the same circle. That is, the arcs whose endpoints include one of each chord's endpoints have equal measures.

The lengths of the arcs connecting two parallel chords of a circle are equal. Similarly, if the lengths of the two arcs connecting two distinct chords are the same, the chords are parallel. If two chords have the same length, the arcs between their endpoints will have the same measure.

The term "parallel" refers to how each note in the chord rises or falls by the same interval.

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The local gym holds three 45-minute workout sessions and two 30-minute sessions each week. Then the, total number of minutes Judy worked out for the week was 155 minutes.

We are to determine the total number of minutes Judy worked out for the week.

The gym holds three 45-minute workout sessions and two 30-minute sessions each week

So,

We can write,

The total number of minutes the gym holds workout sessions is

3 × 45 + 2 × 30

= 135 + 60

= 195 minutes

Also, from the information,

Judy left 5 minutes early during the 30-minute sessions and 10 minutes early during the 45-minute sessions.

The total number of minutes Judy didn't attend is

= 3 × 10 + 2 × 5

= 30 + 10

= 40 minutes

Then,

The total number of minutes Judy worked out for the week was,

= 195 minutes - 40 minutes

= 155 minutes

Therefore,

The total number of minutes Judy worked out for the week was 155 minutes.

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Answer:

The rabbit will get hit because the car takes too long to break

Step-by-step explanation:

The relationship between the ∠2 and ∠8 is that both are alternate exterior angles

Given,

The angles =∠2 and ∠8

Here,

∠2 and ∠8  are both lies on opposite sides of the transversal and they are nonadjacent exterior angles.

So, ∠2 and ∠8  are alternate exterior angles

Hence, the relationship between the ∠2 and ∠8 is, both are alternate exterior angles

The complete question is :

Classify the relationship between the pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

∠2 and ∠8

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Answer:

If two variables have a cause-and-effect relationship, then the variables will have a correlation.

Step-by-step explanation:

Using elimination method, the solution of the given system of equations is (-1, -2)

A system of equations can be solved using:

The given system of equations is:

8y + 10 = 6x   ..... (1)

8y - 4x = -12   ...... (2)

Re-arrange equation (1) to

8y - 6x = -10   ........ (3)

To eliminate variable y, subtract equation (3) from equation (2):

8y - 4x = -12

8y  - 6x = -10 –

      2x  = -2

         x = -1

Substitute  x = -1 into equation (2)

8y - 4x = -12

8y - 4. (-1) = -12

8y + 4 = -12

Add (-4) to both sides:

8y = -16

 y = -2

Hence, the solution is (-1, -2)

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Answer:

KN = 45, NM= 31, m<K = 61° m<L = 119°, m<M = 61°

Step-by-step explanation:

Given: KL = 31 LM = 45 N = 119°

This is a parallelogram so sides will equal.

KL = NM (31) LM = KN (45)

The inside of a parallelogram equals 360°. Opposite angles are congruent.

L = N (119°) So, 119 + 119 = 238

360 - 238 = 122

122 ÷ 2 = 61°

So, K = 61° M = 61° L = 119°

The magnitude of the vector JK is estimated to be  2√13.

The given vector JK's direction is - Ф = 33.69° (angle lies in the fourth quadrant with having direction in clockwise)

The algebraic demonstration of the coordinate-based distances between two points (in coordinate system). Using rectangular coordinates, the Pythagorean theorem is employed to determine distances among points in 2- and 3-dimensional Euclidean space.

Now, in answer to your questioning;

J and K have coordinates of (8,1) & (2,5), respectively.

The following is the distance formula for the points J & K:

JK² = (x₂ - x₁)² + (y₂ - y₁)²

JK = √[(x₂ - x₁)² + (y₂ - y₁)²]

The given of coordinates of J and K's  are-

(x₁,y₁) = (8,1)   and (x₂,y₂) = (2,5).

Substituting the given values in the distance formula;

JK = √[(2 - 8)² + (5 - 1)²]

JK = √[(-6)² + (4)²]

JK = √[36 + 16] = √52

JK  = 2√13 (magnitude of the vector JK)

Now, estimate the vector JK's direction.

The angle formed by a vector with a horizontal line defines its direction.

Use one of the given equation to find the direction of a vector.

tan Ф = Δy/Δx

Where, x = horizontal change

and, y = vertical change

tan Ф = (y₂ - y₁)/(x₂ - x₁)

        = (5 - 1)/(2 - 8)

tan Ф = 4/(-6) = -2/3

Ф = 33.69° (angle lies in the fourth quadrant with having direction in clockwise)

As a result, the distance between J and K is found to be 2√13, that is the same as a magnitude of the given vector JK.

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Answer:

A) y=10x-4

Step-by-step explanation:

There are 10 full boxes which means 10x (since x is the boxes of pizza)

but there are 4 already eaten slices in one, which can be subtracted at the end, making the equation y=10x-4.

Hope this helped and feel better soon! :)

Answer: x = 32.0°

Step-by-step explanation:

When the sum of two angles is equal to 90 degrees, they are called complementary angles.

x + 58° = 90°

x = 90° - 58°

x = 32.0°

The average of the last three numbers is 58 if the average of the five numbers in a list is 54. the average of the first two numbers is 48.

It is defined as the single number that represents the mean value for the given set of data or the closed value for each entry given in the set of data.

It is given that:

The average of the five numbers in a list is 54. and the average of the first two numbers is 48.

54 = sum of the five numbers/5

54x5 = sum of the five numbers

270 = sum of the five numbers

48 = sum of the two numbers/2

48x2 = sum of the two numbers

96 = sum of the two numbers

Sum of three numbers = 270 - 96

Sum of three numbers = 174

The average of the last three numbers = 174/3 = 58

Thus, the average of the last three numbers is 58 if the average of the five numbers in a list is 54. the average of the first two numbers is 48.

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After evaluating the expression | - 3 d - 6 | + | - 5 - d² | for d = - 3, we get that the result is 17.

We are given the expression:

| - 3 d - 6 | + | - 5 - d² |

We need to evaluate the expression for d = - 3

Put the value of d as - 3 in the expression, we get that:

| - 3 ( - 3 ) - 6 | + | - 5 - ( - 3 )² |

Solving the expression further, we get that:

| 9 - 6 | + | - 5 - 9 |

| 3 | + | - 14 |

= 3 + 14

= 17

Therefore, after evaluating the expression | - 3 d - 6 | + | - 5 - d² | for d = - 3, we get that the result is 17.

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Answer:

x = - 10/3

Step-by-step explanation:

1. Add 4 to both sides

6x = -20

2. Simplify

x = - 10/3