Question 4 (Essay Worth 10 points)
(05.06 HC)
Michael has $16 and wants to buy a mixture of cupcakes and fudge to feed at least 4 siblings. Each cupcake costs $4, and each piece of fudge costs $2.
This system of equations models the scenario:

4x+2y = 16
x+y=4

Part A: Describe the graph of the system, including shading and the types of lines graphed. Provide a description of the solution set.

Part B: Is the point (2, 3) included in the solution area for the system? Justify your answer mathematically

Part C: Choose a different point in the solution set and interpret what it means in terms of the real-world context.

The system of equations is

15x + 10y = 800

x + y = 60

and the solution to these equations is

x = 40

y = 20

Given is that an office manager requests a calculator or calendar for each of the 60 office employees. Each calculator is $15 and each calendar is $10. The total order amounted to $800.

We can write the system of equations as -

15x + 10y = 800

x + y = 60

We can write -

y = 60 - x

So, we can write -

15x + 10(60 - x) = 800

15x - 10x + 600 = 800

5x = 200

x = 40

Then -

y = 20

Therefore, the system of equations is

15x + 10y = 800

x + y = 60

and the solution to these equations is

x = 40

y = 20

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{Question in english is -

An office manager requests a calculator or calendar for each of the 60 office employees. Each calculator is $15 and each calendar is $10. The total order amounted to $800. Part A: Write the system of equations that models this scenario. (5 points) Part B: Use the substitution method or the elimination method to determine the number of calculators and calendars ordered. Show all the necessary steps. (5 points)

}

The coordinates of point A'  and B' are A'(-12,8) and B'(20, 0)

A dilation is a transformation that changes the size of a figure but not its shape.

In a dilation with respect to the origin, a point (x, y) is transformed to a point (kx, ky) where k is a scale factor.

In this case, the point A (-3, 2) and B (5, 0) are transformed under a dilation of 4 with respect to the origin.

So the images are

A' = (kx, ky) = (-3*4, 2*4) = (-12, 8)

B' = (kx, ky) = (5*4, 0*4) = (20, 0)

So the image point of A(-3,2) under a dilation of 4 with respect to the origin is A'(-12,8)

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Complete question

The triangle ABC is dilated by a scale factor of 4 with the center of dilation at the origin.

a) What are the coordinates of A?

b) What are the coordinates of B?

if A (-3, 2) and B (5, 0)

The area of the lawn that will be watered by the sprinkler is 191.1 square feet, rounded to the nearest tenth.

Area is a measurement of the size of a two-dimensional surface, such as a square, rectangle, or circle. It is calculated by multiplying the length of the surface by its width. Area can also be measured in terms of square units, such as square feet or square meters. Area is an important concept in mathematics and is used to measure the size of shapes and objects. It is also used to find the total area of a group of shapes.

To calculate the area, use the formula A = r2θ, where A is the area, r is the radius, and θ is the central angle in radians. In this scenario, the radius is 10 ft and the central angle is 30°, or π/6 radians. Plugging these numbers into the formula, we get A = (10 ft)2(π/6 radians), or 191.1 square feet.

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Answer:

Tan u = sin u/cos u, csc u = 1/sin u, sec u = 1/cos u, and cot u = cos u/sin u. The denominators of the fractions can be rationalized by multiplying the numerator and denominator by the conjugate of the denominator.

Step-by-step explanation:

Answer:

3.2 miles

Step-by-step explanation:

To approximate the height of the mountain, we need to use the tangent function in trigonometry. We have two angles of elevation and the distance between the observer and the mountain, which we can use to create a right triangle.

Let h be the height of the mountain and d be the distance from the observer to the foot of the mountain.

At the first observation, we have:

tan(3.5°) = h/d

And at the second observation, when the observer is 11 miles closer to the mountain:

tan(9°) = h/(d-11)

Dividing the two equations, we have:

tan(9°) / tan(3.5°) = h/(d-11) / h/d

Solving for h, we have:

h = (d * tan(9°)) / (tan(3.5°) + 1)

We don't have an exact value for d, so we can use an estimate based on the difference in distance between the two observations.

Since d-11 = 11 miles, we have:

d = 11 * 2 = 22 miles

Using this estimate and the given angles, we can approximate the height of the mountain:

h = (22 * tan(9°)) / (tan(3.5°) + 1) = (22 * 0.156434465) / (0.066012183 + 1)

= 22 * 0.156434465 / 1.066012183 = 22 * 0.146838446 = 3.23 miles

The height of the mountain is approximately 3.2 miles to one decimal place.

R and S are points on the number line. If RS= 11 and S lies at 5, where could R be located 16 or -6.

A number line is a visual representation of numbers arranged in a continuous sequence along a straight line. It is used to represent the real numbers and to help visualize relative sizes and order of numbers. The numbers on a number line are placed at equal intervals, and the distance between two numbers represents the magnitude of the difference between them.

A typical number line starts from a specific number on the left, such as zero, and extends to the right, with positive numbers increasing as you move to the right and negative numbers decreasing as you move to the left. Points on the number line correspond to specific numbers, and the position of a point on the line represents the value of that number.

The number line is a useful tool in mathematics for solving problems, such as determining the solution to an equation, estimating the value of a decimal or fraction, or visualizing the magnitude of a negative or positive number. It is also used in geometry to represent the positions of points and to understand concepts such as distance and midpoint.

If S lies at 5 on the number line and RS = 11, then R could be located at either 5 + 11 = 16 or 5 - 11 = -6. So, R could be at either 16 or -6 on the number line.

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Answer:

see explanation

Step-by-step explanation:

using the rules of exponents/ radicals

• [tex]a^{m}[/tex] × [tex]a^{n}[/tex] = [tex]a^{(m+n)}[/tex]

• [tex]a^{\frac{m}{n} }[/tex] = [tex](\sqrt[n]{a} )^m[/tex]

then

(fg)(x)

= f(x) × g(x)

= [tex]x^{\frac{1}{2} }[/tex] × 2x²

= 2 × [tex]x^{(\frac{1}{2}+2) }[/tex]

= 2[tex]x^{\frac{5}{2} }[/tex]

when x = 9 , then

= 2 × [tex]9^{\frac{5}{2} }[/tex]

= 2 × [tex](\sqrt{9})^5[/tex]

= 2 × [tex]3^{5}[/tex]

= 2 × 243

= 486

The value of b set to 2 is the third option, which is the correct one.

The provided equation is h(x) = -x + 5.

We obtain b=5 when comparing with the slope-intercept form y = mx+b.

Consequently, the y-intercept is at 5.

To lower the graph by three units, we must make changes.

As per the transformational laws, f(x)+c shifts c units upward, whereas f(x)-c shifts c units downward.

Therefore, we need to subtract the supplied function by 3 in order to move the graph down 3 units, and we get

y = -x+5 -3

y = -x +2.

Here, the y-intercept is 2.

The provided equation is h(x) = -x + 5.

The value of b set to 2 is the third option, which is the correct one.

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The solution to the equation 3 / 5x-2 = 2 / 3x-1 is x = 1.

Given the equation in the question;

3 / 5x-2 = 2 / 3x-1

To solve for x, cross multiply the equation

3 / 5x-2 = 2 / 3x-1

3( 3x - 1 ) = 2( 5x - 2 )

Apply distributive property

3( 3x - 1 ) = 2( 5x - 2 )

3×3x + 3×-1 = 2×5x + 2×-2

9x - 3 = 10x - 4

Subtract 10x from both sides

9x - 10x - 3 = 10x - 10x - 4

9x - 10x - 3 = - 4

Add 3 to both sides

9x - 10x - 3 + 3 = - 4 + 3

9x - 10x  = - 4 + 3

-x = -1

Divide both sides by -1

-x/-1 = -1/-1

x = 1

Therefore, the value of x is 1.

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Answer: 8xw=a

Step-by-step explanation:

w=width

a=area

The expression (3a⁸)⁹ * (3b⁵)² is solved to get 3¹¹a⁷²b¹⁰

power or exponents as used in mathematics are number written on top of a base number. The power says how many times the base number multiplies itself.

The problem is solved as follows

= (3a⁸)⁹ * (3b⁵)²

raising the terms by power of 9 and 2 accordingly

= (3^9 * a^(8 * 9)) * (3^2 * b^(5 * 2) )

= (3^9 * a^(72)) * (3^2 * b^(10) )

collecting base 3

= (3^9 * 3^2 * a^(72) * b^(10))

= (3^(9 + 2) * a^(72) * b^(10))

= (3^(11) * a^(72) * b^(10))

rewriting the expressions

= 3¹¹a⁷²b¹⁰

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The length of LN for the triangle is 18.

A triangle is a three-sided polygon because it has three edges and three vertices. The most important attribute of a triangle is the sum of its internal angles to 180 degrees

Given triangle JKL,

and MN is parallel to JK,

taking ΔJKL and ΔLMN

∠L = ∠L  (common angle)

because MN is parallel to JK, so corresponding angles are equal,

so ∠K = ∠N

and ∠J = ∠M

ΔJKL ≈ ΔLMN(triangles are similar)

since both triangles are similar the ratio of their corresponding sides is equal,

JL/LM = JK/MN = LK/LN

and JK = 55 and LK = 33

and MN = 12 + LN

JK/MN = LK/LN

substitute the values

55/(12 + LN) = 33/LN

55LN = 33(12 + LN)

55LN - 33LN = 396

22LN = 396

LN = 396/22

LN = 18

Hence the length of LN is 18.

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The distance betwen the spots are

From the question, we have the following parameters that can be used in our computation:

The coordinates of the spots

The distance between them is then calculated as

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Using the above as a guide, we have the following:

AD = √[(-4 + 4)² + (3 - 1)²] = 2

DF = √[(-4 + 4)² + (1 + 3)²] = 4

EG = √[(5 - 5)² + (1 + 3)²] = 4

AC = √[(-4 - 1)² + (3 - 3)²] = 5

BD = √[(2 + 4)² + (1 - 1)²] = 6

The above values represent the distance betwen the spots

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32 thousands 6 hundreds 8 ones can be rewritten as 32608

According to the BODMAS rule, the brackets have to be solved first followed by powers or roots (i.e. of), then Division, Multiplication, Addition, and at the end Subtraction. Solving any expression is considered correct only if the BODMAS rule or the PEMDAS rule is followed to solve it.

Given here: 32 thousands 6 hundreds 8 ones

Thus we can rewrite as 32×1000+6×100+8×1=32608

Numbers in words are written using the English alphabet. Numbers can be expressed both in words and figures. For example, 100,000 in words is written as One Lakh or One hundred thousand. Numbers in words can be written for all the natural numbers, based on the place value of digits, such as ones, tens, hundreds, thousands, and so on.

Hence, 32 thousands 6 hundreds 8 ones can be rewritten as 32608

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For x=2, the functions g(x)=[tex]f^{-1}(x)[/tex] are true.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

The given function is f(x)=(x-1)/4 and g(x)=2x+5.

Here, y=(x-1)/4

Interchange x with y and y with x, we get

x=(y-1)/4

4x=y-1

y=4x+1

[tex]f^{-1}(x)[/tex]=4x+1

Now, g(x)=[tex]f^{-1}(x)[/tex]

2x+5=4x+1

2x=4

x=2

Therefore, for x=2, the functions g(x)=[tex]f^{-1}(x)[/tex] are true.

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The solution is given below.

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.

here, we have,

Remember that the Angle-Side-Angle Postulate (ASA) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

According to this, the correct answer is: given below.

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Answer:

Each piece of gum is 10cents

Step-by-step explanation:

7 x 0.25 then subtract that from 2.05 then divide by the 3pieces of gum

Answer: (0.25 x 7) + 3x = 2.05 and $0.10 per piece of gum

Step-by-step explanation:

Jeff purchased 7 gumballs for $0.25 each, which is [tex](0.25*7)[/tex]

Jeff then purchased 3 pieces of gum, but we don't know for how much so the cost becomes [tex]x[/tex] and then [tex]3x[/tex]

The final cost then comes down to $2.05, therefore [tex](0.25*7)+3x=2.05[/tex] and solve for [tex]x[/tex] to find the cost of each piece of gum

[tex]1.75+3x=2.05[/tex]
[tex]3x=0.3[/tex]

[tex]x=0.1[/tex]

The cosine of the angle  θ  is √5/5  whose terminal side passes through the given point

In mathematics, trigonometric functions are real functions that relate an angle of a right-angled triangle to ratios of two side lengths. The six trigonometric functions are Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent.

Given here: The terminal side goes through the point  (√5/5,2√5/5)

Thus we have cosθ =√5/5 and sin θ =-2√5/5

now tan θ =sinθ /cosθ

        tanθ    =-2√5/5 / √5/5

              θ = 63.434

Hence, cosθ =√5/5

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The given function

g(x)  =  (x√x)/(x - 6)²

is continuous at  x = 36,

In mathematics, The function is called as continuous at the particular value of x,  if right hand limit, left hand limit and the value of function at x all are equal.

RHL   = LHL  = F(x)

Limit exist and continuous

Given that,

F(x)  =   (x√x)/(x - 6)²

To check whether it is continuous at x = 36

First taking LHL,

[tex]\lim_{h \to \00}[/tex]    [(36-h)√(36-h)]/((36-h) - 6)²

[tex]\lim_{h \to \00}[/tex]  [(36-0)√(36-0)]/((36-0) - 6)²

⇒   (36 x 6)/30²

⇒    6/25

Now, taking RHL

[tex]\lim_{h \to \00}[/tex]    [(36+h)√(36+h)]/((36+h) - 6)²

[tex]\lim_{h \to \00}[/tex]  [(36+0)√(36+0)]/((36+0) - 6)²

⇒   (36 x 6)/30²

⇒    6/25

The value at x = 36

F(-1)  =  [(36)√(36)]/((36) - 6)²

⇒   (36 x 6)/30²

⇒    6/25

RHL = LHL = F(36)

Hence Proved,  the function is continuous at x = 36

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The value of the number that occupies the hundreds place in 25,450 is 400

Calculating a set's size, or the number of elements in a finite collection of objects is the process of counting.

In a counting sequence, there are different units used such as:

Hence, in the value of 25,450,

The value that is found in the hundreds place is 400

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The English translation is:

What is the value of the number that occupies the hundreds place in 25,450

Answer:

1/2 i think this is it

Step-by-step explanation:

The values of X1 and X2 that maximize the function Z = 30X1 + 50X2 are given as follows:

The maximum value is given as follows:

4000.

The function for this problem is defined as follows:

Z = 30X1 + 50X2.

The constraints are given as follows:

X2 has the higher multiplier, hence we should:

The maximum value of X2 is obtained when:

X2 = 100 - X1.

Replacing into the first equation, the value of X1 is given as follows:

X1 + 3(100 - X1) = 200

-2X1 = -100

2X1 = 100

X1 = 50.

Hence the value of X2 is of:

X2 = 100 - X1

X2 = 50.

The maximum value of the function is obtained as follows:

Z = 30 x 50 + 50 x 50

Z = 4000.

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Answer:

Y=4cm

Step-by-step explanation:

We can use the relationship between the side length and area of a square to solve for Y. Let's call the area of the small square "A" and the area of the large square "B". From the problem statement, we know:

A = 1/9 * B

Since the area of a square is equal to the side length squared, we can substitute this relationship into the equation above:

Y^2 = 1/9 * 12^2

Solving for Y:

Y^2 = 1/9 * 144

Y^2 = 16

Y = sqrt(16)

Y = 4

So, the side length of the small square is 4 cm.

Answer:

Equation is

[tex]\mbox {\large \boxed{g(m) = 4.411 (0.94988)^m}}[/tex]

Step-by-step explanation:

We have to translate the information in the table as an equation of the form

[tex]g(m) = P(1 + r)^m\cdots \cdots(1)[/tex]

Let's take the date from the table

For m = 1, g(m) = 4.19

Substituting these values in equation (1) we get:
[tex]g(1) = P(1 + r)^1[/tex]

[tex]4.19 = P(1 + r)^1[/tex]

[tex]P(1 + r)^1 = 4.19 \; \textrm{ (by switching sides)}[/tex]

[tex]P(1+r) = 4.19\; \cdots \cdots (2)\\\textrm{ (an expression raised to the power 1 is the expression itself)}[/tex]

For the next entry in the table, m = 2, g(m) = 3.98

Plugging into (1)

[tex]3.98 = P(1 + r)^2 \cdots\cdots(3)[/tex]
or

[tex]P(1 + r)^2 = 3.98[/tex]

Divide equation (3) by equation (2)

[tex]\dfrac{P(1+r)^2}{P(1+r} = \dfrac{3.98}{4.19}[/tex]

[tex]\textrm{The P terms cancel and } \dfrac{(1 + r)^2}{(1+r)} = 1 + r[/tex]

==> [tex]1 + r = \dfrac{3.98}{4.19}\\\\1 + r = 0.94988\\\\[/tex]

Plugging this value of 1 + r into equation (2):

[tex]P( 0.94988) = 4.19\\\\P = \dfrac{4.19}{0.94988}\\\\P = 4.411\\\\[/tex]

So the equation is

[tex]g(m) = 4.411 (0.94988)^m[/tex]

We can verify this is correct for m = 2, 3 and 4

[tex]m\quad\quad g(m) = 4.411(0.94988)^m\\\------------------\\2\quad\quad 4.411(0.94988)^2 = 3.97992 \approx 3.98\\\\3\quad\quad 4.411(0.94988)^3 = 3.78044 \approx 3.78\\\\3\quad\quad 4.411(0.94988)^4 = 3.59097 \approx 3.59\\\\[/tex]

Answer:

y=1/6x+3

Step-by-step explanation:

THe equation of the graph is y=mx+c

so,

m= { (-3)-(-4) } / (-1)-(-7) = 1/6

intercept = (0,y)

1/6= (y+3)/1

y=3

so, the equation ⇒y=1/6x+3.

The graph that best represents the situation, given the person cycling from home to the library, is Graph C.

The graph that would best represent the situation would show that the person cycling, stopped moving after 8 kilometers because they were at the library.

The graph should also show that after 30 minutes, the person left the library which is shown in Graph C as:

= 50 - 20 minutes

= 30 minutes

The best graph is therefore Graph C.

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Answer:

162/5 hours.

Step-by-step explanation:

Let's call the normal depth of the river "D". Since 2/3 of the way to go back to normal is still flooded, the current depth of the water is D + 2/3D = 5/3D.

The flood waters recede at a rate of 1/9 the flood level per hour, so the change in water level each hour is -1/9 * 5/3D = -5/27D.

To find the number of hours it takes to go from the current depth (5/3D) to the normal depth (D), we need to divide the change in depth (-5/27D) by the change in depth per hour (-5/27D): (5/3D - D) / (-5/27D) = (5/3D - D) / (-5/27D) = -(5/3D - D) / (5/27D) = (2/3D) / (5/27D).

This can be simplified to 2/3 * 27/5 = 162/5 hours. The answer is 162/5 hours.

Answer: Since the flood waters still have 2/3 of the way to go back down to reach the normal level, it means that 1/3 of the flood level has already receded.

If the flood level recedes at a rate of 1/9 the flood level per hour, then it will take 1/9 * 1 hour = 1 hour to recede 1/9 of the flood level.

Therefore, to recede 2/3 of the flood level, it will take 2/3 * 1/9 hours = 2/27 hours.

Since there are 60 minutes in 1 hour, the time it takes for the flood waters to reach the normal level in minutes is: 2/27 hours * 60 minutes/hour = 20 minutes.

Therefore, it will take 20 more minutes for the water to reach its normal depth.

Step-by-step explanation:

x = 2

y = 2 [tex]\sqrt{2}[/tex]

Special right triangles are triangles with specific angle measurements whose side lengths have patterns.

45-45-90 Triangle

The triangle above is a 45-45-90 triangle since it has two 45-degree angles. For this explanation, let a and b represent the legs of a right triangle and c be the hypotenuse. In this type of triangle, the legs are equal, so a=b. The hypotenuse is [tex]\sqrt{2}[/tex] times the leg length. So, c = a√2.

Finding x and y

In this problem, x is one of the legs. This means that x = 2 since 2 is the length of the leg. The y-value represents the hypotenuse, so we can multiply the leg by √2. Thus, y = 2√2.