Question One
Consider the following information that relates to a certain
economy
C= 40 +0.35
I = 70- 2r
G=400
T= 15+0.2Y
X=240
M=45+0.4Y
r = 0.5
Required:
i). Compute the autonomous investment, government
expenditure, export, and tax multipliers
ii). Find the equilibrium income

Reason of congruency for each of the triangle are :

(a) SAS rule

(b) Triangles are not congruent.

Given are some triangles where some of the angles and sides are given to be equal.

We have to find the reason of congruency.

(a) Given two sides are equal.

One of the triangle has an included angle of 110°.

Find the included angle of the other one.

Included angle = 180 - (40 + 30) = 110°

So two sides and included angle of one is equal to the corresponding sides and angle.

So congruency is SAS rule.

(b) Length of the hypotenuse of first triangle = √(2² + 4²) = √20 cm

Length of hypotenuse of other triangle = 4 cm

The lengths are not equal.

So they are not congruent.

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The inequality states that Jayda's age (j) is greater than 12 i.e., j > 12.

Let j represent Jayda's age.

Since Jayda is older than her 12-year-old sister, we can write the inequality:

j > 12

Thus, the inequality states that Jayda's age (j) is greater than 12.

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The values of x that satisfy the equation are x = (-4 + 2√6) / 5 and x = (-4 - 2√6) / 5

The values of x that satisfy the equation 5x² + 4x - 4 = 0 can be determined using the quadratic formula given below:

x = (-b ± √(b² - 4ac)) / (2a)

From the equation:

a = 5, b = 4, and c = -4.

Substituting these values into the quadratic formula:

x = (-(4) ± √((4)² - 4(5)(-4))) / (2(5))

Simplifying to solve for values of x

x = (-4 ± √(16 + 80)) / 10

x = (-4 ± √96) / 10

x = (-4 ± 4√6) / 10

x = (-4 ± 2√6) / 5

Hence, x = (-4 + 2√6) / 5 or x = (-4 - 2√6)

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Answer:

(i) Attached below.  You can copy what I drew

(ii) (a) 35, (b) 60, (c) 5

Step-by-step explanation:

(a) Thus, 35 students study French only.

(c) Thus, 5 students' study neither Arabic nor French

the number of students who study French only + the number of students who study both French and Arabic + the number of students who study Arabic only + the number of students who study neither Arabic nor French equals 75:

35 + 10 + 25 + 5 = 75

45 + 30 = 75

75 = 75

Thus, our numbers are correct.

(b) Also, the number of students who study only subject is 60, as 35 (number of students studying French only) + 25 (number of students studying Arabic only) = 60

For positive acute angles A and B, if it is known that SinA= 11/61 and tan B=4/3, cos(A-B) = 224/305.

We can use the trigonometric identity cos(A-B) = cosA cosB + sinA sinB to find the value of cos(A-B).

First, we need to find the value of cosA and sinB:

Since sinA = opposite/hypotenuse, we can draw a right triangle with opposite side 11 and hypotenuse 61, and use the Pythagorean theorem to find the adjacent side:

cosA = adjacent/hypotenuse = √(61² - 11²)/61 = 60/61

Since tanB = opposite/adjacent, we can draw another right triangle with opposite side 4 and adjacent side 3, and use the Pythagorean theorem to find the hypotenuse:

hypotenuse = √(4² + 3²) = 5

sinB = opposite/hypotenuse = 4/5

Now we can substitute these values into the formula:

cos(A-B) = cosA cosB + sinA sinB

= (60/61)(3/5) + (11/61)(4/5)

= 180/305 + 44/305

= 224/305

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Answer:

The distance between Garland and Mansfield on the map is 15 inches.

Step-by-step explanation:

On Giana's map of Texas, the scale is 1 inch = 3.5 miles. To determine the distance between Garland and Mansfield on the map, we can use the scale to find the corresponding measurement. Given that the actual distance between these two cities is 52.5 miles, we can set up a proportion:

1 inch / 3.5 miles = x inches / 52.5 miles

Cross-multiplying, we have:

3.5 miles * x inches = 1 inch * 52.5 miles

3.5x = 52.5

Dividing both sides by 3.5, we find:

x = 15

Therefore, the distance between Garland and Mansfield on the map is 15 inches.

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The domain and the range of the graph seen in the figure are X = {- 5, - 4, - 1, 3} (or x ∈ [- 5, 3]) and Y = {- 2, 0, 2, 4} (or y ∈ [- 2, 4]).

Herein we find a graph generated by four points of the form (x, y) set on Cartesian plane. The domain of the graph consists in the set of all x-coordinates of the graph, whereas range is for the set of all y-coordinates.

Domain

X = {- 5, - 4, - 1, 3}

(Interval notation)

x ∈ [- 5, 3]

Range

Y = {- 2, 0, 2, 4}

(Interval notation)

y ∈ [- 2, 4]

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Answer:

2x and 180 - 5x are complementary angles.

2x + 180 - 5x = 90

-3x + 180 = 90

-3x = -90

x = 30

Answer:

  x ≥ 5

Step-by-step explanation:

You want the domain of the equation y = √(x -5) -1.

The domain of the function is the set of x-values for which it is defined. The square root is not defined for negative values, so the domain is ...

  x -5 ≥ 0

  x ≥ 5 . . . . . . the domain of this function

__

Additional comment

The range is the set of y-values the function may produce. Here, that set of values is y ≥ -1.

<95141404393>

The volume of the storage nook is 24 cubic feet.

To find the area of 8 puzzles, we multiply the area of one puzzle by 8:

Area of 8 puzzles = 1 square foot * 8 = 8 square feet

Now, to find the volume of the storage nook, we multiply the area of the floor by its height:

Volume of the nook = Area of the floor * Height

= 8 square feet * 3 feet

= 24 cubic feet

Hence, the volume of the storage nook is 24 cubic feet.

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The required sample size for the confidence interval and given margin of error is given as follows:

n = 1037.

A confidence interval of proportions has the bounds given by the rule presented as follows:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which the variables used to calculated these bounds are listed as follows:

The margin of error is given as follows:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

The confidence level is of 99%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.99}{2} = 0.995[/tex], so the critical value is z = 2.575.

As we have no estimate, the parameter is given as follows:

[tex]\pi = 0.5[/tex]

Then the sample size for M = 0.04 is obtained as follows:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

[tex]0.04 = 2.575\sqrt{\frac{0.5(0.5)}{n}}[/tex]

[tex]0.04\sqrt{n} = 2.575 \times 0.5[/tex]

[tex]\sqrt{n} = \frac{2.575 \times 0.5}{0.04}[/tex]

[tex](\sqrt{n})^2 = \left(\frac{2.575 \times 0.5}{0.04}\right)^2[/tex]

n = 1037.

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Approximately 5048 sports cars will have less than 150,000 miles on the odometer

Mean = 120,000

Standard deviation = 30,000

Total group = 6000

Total Miles = 150,0000

An odometer is a device used to gauge how far a wheeled vehicle has driven. The tool might be mechanical, electrical, or a hybrid of the two.

Calculating the Z score -

z = x - u/a

= 15,0000 - 12,0000/30000

= 30000/30000

= 1

Using the standard normal table, to determine the value -

P ( Z<1) = 0.8413

Calculating the total number of cars -

= P value x total group of cars

= 6000 x 0.8413

= 5047.7 or 5048

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Answer:

7

Step-by-step explanation:

Since one card was removed:

6 = x/4

x = 6 X 4

x = 24

Then you sum the numbers on the five cards and subtract 24 from it

(2 + 5 + 7 + 8 + 9) - 24

= 31 - 24

= 7

Therefore card with number 7 was removed.

Answer:

  (e)  $13,480

Step-by-step explanation:

You want the estimated profit from selling 60 computers, if the profit from selling x computers is P = 4x² -15x -20.

Put 60 where x is in the profit equation and do the arithmetic.

The result is an estimated profit of $13,480.

__

Additional comment

To simplify evaluation, the polynomial can be written in "Horner form:"

  P = (4x -15)x -20

Evaluating it this way takes fewer math operations and generally involves smaller numbers.

<95141404393>

Step-by-step explanation:

3775  +  2890-1395 = total diving meters = 5270 m

Total movement = 3775  + ( 3775 - 1395 )   +  (2890 - 1395 ) = 7650 m

The point (4, - 1) will not be a solution to the system of linear inequalities shown.

The solution of the  linear inequalities is calculated by simplifying the  linear inequalities as follows;

The given linear inequalities;

y ≥ -x/4  + 7

y > 4x + 4

Solve the linear inequalities as follows;

4x + 4 ≥ -x/4 + 7

Collect similar terms together;

4x + x/4 ≥ 7 - 4

4x + x/4 ≥ 3

Multiply through by 4;

16x + x ≥ 12

17x ≥ 12

divide through by 17;

x ≥ 12/17

x ≥ 0.7

The value of y is calculated as;

y ≥ -x/4  + 7

y ≥ - (0.25 x 0.7) + 7

y ≥ 6.8

Since y is greater than or equal to 6.8, the point (4, - 1) will not be a solution to the system of linear inequalities shown.

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16800 cubic inches of concrete is needed to make the stairs

Rectangular prism is a three dimensional shape which consist of six faces. The two faces are at the top and bottom and the other faces are lateral faces.

The given stairs can be divided into 3 rectangular prisms as shown in the image attached.

Volume of a concrete needed = Volume of A + Volume of B + Volume of C

Volume of a rectangular prism = l × w × h

where l is the length, w is the width and h is the height

For A:

l = 40, w = 10 in and h = 7 + 7 + 7 = 21 in  

For B:

l = 40, w = 10 in and h = 7 + 7 = 14 in

For C:

l = 40, w = 10 in and h = 7 in

Volume of a concrete needed = (40 * 10 * 21) * (40 * 10 * 14) + (40 * 10 * 7)

Volume of a concrete needed = 8400 + 5600 + 2800

Volume of a concrete needed = 16800 in³

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The distance of the perpendicular from Q to line PR is D = 2.2135 units

Given data ,

Let the triangle be represented as ΔPQR

Now , the coordinates are P(1, 3), Q(5, 4) and R(5, 15)

And , the equation of line of PR is given by

Slope m = ( 15 - 3 ) / ( 5 - 1 )

m = 3

y - 3 = 3 ( x - 1 )

Adding 3 on both sides , we get

y = 3x

y - 3x = 0

And , the point is Q(5, 4)

Now , distance of a point to line D = | Ax₀ + By₀ + C | / √ ( A² + B² )

D = | ( 1 ) ( 5 ) + ( -3 ) ( 4 ) + 0 | / √ ( 1 )² + ( 3 )²

D = | 5 - 12 | / √10

D = 7/√10

D = 2.2135 units

Hence , the distance is 2.2135 units

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Answer:

500 and 4.3

Step-by-step explanation:

A decigram is one-tenth of a gram and a kilometer is one-thousandth of a meter. Using this information, we must multiply the given amounts by the values.

5000 × 0.1 = 500

4300 × 0.001 = 4.3

Thus, the answers [in order] are 500 and 4.3

Hope this helps and feel free to ask any questions.

Answer:

[tex]9^{\frac{1}{8}x }[/tex]

Step-by-step explanation:

The function in the question

[tex]\sqrt[4]{9} ^{\frac{1}{2}x }[/tex]

is equivalent to

[tex]9^{(\frac{1}{4})(\frac{1}{2} x) }[/tex]

where you multiply the powers and get the final answer

[tex]9^{\frac{1}{8}x }[/tex]

Answer:

Step-by-step explanation:

5/12 pounds more

Step-by-step explanation:

the square root of 5 can be written as 5 ^ (1/2).

We can then multiply the exponents, so it's 9/2.

So the answer could be written as 5 ^ (9/2).

Safiya can use either ∛12 or ∛(1/6) to find a rational product when multiplied by ∛18. So, correct options are e and f.

The cube root of 18 is an irrational number because it cannot be expressed as a fraction of two integers. Safiya needs to find a rational number that, when multiplied by ∛18, results in a rational product. To do this, she needs to look for cube roots of numbers that are perfect cubes of integers.

Option a) ∛4 is a rational number, but it is not a perfect cube of any integer. Therefore, it is not a valid choice.

Option b) ∛18 is the given number and is not a perfect cube of any integer. Therefore, it is not a valid choice.

Option c) ∛3 is a rational number, but it is not a perfect cube of any integer. Therefore, it is not a valid choice.

Option d) ∛(1/18) is a rational number, but it is not a perfect cube of any integer. Therefore, it is not a valid choice.

Option e) ∛12 is a rational number and it can be expressed as ∛(2³ × 3). Therefore, it is a valid choice.

Option f) ∛(1/6) is a rational number and can be expressed as ∛(1/216). Therefore, it is a valid choice.

So, correct options are e and f.

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The equation of the Damari's investment is B(x) = 30000 * 1.03ˣ

Damari's investment

Given that

Initial value, a = 30000

B(3) = 32306.72

The function is calculated as

B(x) = a * bˣ

Using B(3), we have

30000 * b³ = 32306.72

So, we have

b³ = 1.077

Take the cube root of both sides

b = 1.03

So, we have

B(x) = 30000 * 1.03ˣ

So, the function is B(x) = 30000 * 1.03ˣ

The boat of Sky's family

Here, we have

Initial value = 6000

Rate of depreciation = 6%

So, the function is

f(x) = 6000 * (1 - 6%)ˣ

So, we have

f(x) = 6000 * (0.94)ˣ

In 2024, we have

x = 2024 - 2021

x = 3

So, we have

f(3) = 6000 * (0.94)³

Evaluate

f(3) = 4983.50

This value is less than the offered value of $5000

This means that Sky's family should take the offer

The rule of the function

Here, we have the graph

From the graph, we have

Initial value, a = 8

Rate, b = 4.8/8

So, we have

Rate, b = 0.6

So, the function is

f(x) = 8 * 0.6ˣ

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Answer:

The third option

Step-by-step explanation:

144.) The value of X if the two angles of the triangle are congruent would be = 4. That is option D.

145.) The equation that should be used to find the value of X would be = 13x -15 = 22x + 9. That is option H.

For question 144.)

Two triangles are congruent means that the angles are equal to each other. That is;

2.3x +25 = 5.8x + 11

Bring like terms together;

5.8x-2.3x = 25-11

3.5x = 14

X = 14/3.5

= 4.

For question 145.)

When two parallel lines are being cut across by a diagonal line a relationship exists between the angles that are formed.

The two given angles are congruent;

That is;

13x -15 = 22x + 9

Therefore the expression that can be used to calculate X would be = 13x -15 = 22x + 9

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The size of angle  can be expressed as 130 degrees

Two angles are said to be supplementary angles because they combine to generate a linear angle when their sum is 180 degrees.

Considering the given figure, if we look at the right top triangle  we can deduced that it is an isosceles, then the supplementary angle of angle α  can be calculated as

(180 - 80)/2

= 50 degrees.

Hence,  α = 180 - 50

= 130 degrees.

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The value of a3 in the Arithmetic sequence with a1 = -10 and d = -2 is -14.

The value of a3 in the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

an = a1 + (n - 1) * d

Where:

- an is the nth term in the sequence

- a1 is the first term of the sequence

- d is the common difference between consecutive terms

- n is the position of the term we want to find

In this case, we are given that a1 = -10 and d = -2. We want to find the value of a3, which corresponds to the third term in the sequence (n = 3).

Plugging in the values into the formula, we have:

a3 = a1 + (3 - 1) * d

  = -10 + 2 * (-2)

  = -10 - 4

  = -14

Therefore, the value of a3 in the arithmetic sequence with a1 = -10 and d = -2 is -14.

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Answer:

Step-by-step explanation:

Interior angle sum = 180

2x - 45  +  x  =  180

3x  -  45  =  180

x  -  15  =  60

x  =  75

Answer:

Step-by-step explanation: