A student is saving money to go on a school field trip. The graph shows the relationship between the number of weeks the student has been saving, x, and the total dollar amount they have saved, y.

Using the Pythagorean triple formula the numbers 8, 15 and 17 are Pythagorean triple and as such will form a right-angle triangle.

Pythagorean triples are x² = y² + z² where x, y and z are the three positive integers. These triples are represented as (x, y, z). Here, y is the perpendicular, z is the base and x is the hypotenuse of the right-angled triangle.

In this problem, to determine if the given numbers will form a right-angle triangle, we have to check if the numbers are Pythagorean triple.

In doing so, we would use the previous formula given as;

x² = y² + z²

In a Pythagorean triple, the largest side is the hypothenuse.

17² = 8² + 15²

289 = 64 + 225

289 = 289

The numbers are equal and they're Pythagorean triple.

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Answer:

220 in.²

Step-by-step explanation:

top face - rectangle: A = LW = 4 in. × 2 in. = 8 in.²

bottom face - rectangle: A = LW = 9 in. × 2 in. = 18 in.²

left face - rectangle: A = LW = 12 in. × 2 in. = 24 in.²

right upper face - rectangle: A = LW = 8 in. × 2 in. = 16 in.²

right lower face - rectangle: A = LW = 4 in. × 2 in. = 8 in.²

middle horizontal face - rectangle: A = LW = (9 - 4) in. × 2 in. = 10 in.²

front L-shaped face - 2 rectangles: A = (12 × 4 + 5 × 4) in.² = 68 in.²

back L-shaped face - same as front L-shaped face: A = 68 in.²

total surface area = (8 + 18 + 24 + 16 + 8 + 10 + 68 + 68) in.²

total surface area = 220 in.²

The exact values are:

sin(α + β) = (-6√5 - 8)/25

cos(α + β) = (4√5 - 6)/25

sin(α - β) = (-6√5 + 8)/25

tan(α - β) = (-9√5 + 45)/4

We have,

Recall the trigonometric identities:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)tan(b))

Using these identities, we can find the exact values of the expressions given.

(a) Sin (α + β)

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

= (√5/5)(-3/5) + (2/5)(-4/5) (using the values given for sin and cos)

= -6√5/25 - 8/25

= (-6√5 - 8)/25

(b) Cos (α + β)

cos(α + β) = cos(α)cos(β) - sin(α)sin(β)

= (2/5)(-3/5) - (√5/5)(-4/5) (using the values given for sin and cos)

= -6/25 + 4√5/25

= (4√5 - 6)/25

(c) Sin (α - β)

sin(α - β) = sin(α)cos(β) - cos(α)sin(β)

= (√5/5)(-3/5) - (2/5)(-4/5) (using the values given for sin and cos)

= -6√5/25 + 8/25

= (-6√5 + 8)/25

(d) tan (α - β)

tan(α - β) = (tan(α) - tan(β))/(1 + tan(α)tan(β))

= ((√5/5) - (-4/5))/(1 + (√5/5)(-4/5)) (using the values given for sin and cos)

= (9√5/5)/(1 - 4/5√5)

= (-9√5 + 45)/4

Therefore,

The exact values are:

sin(α + β) = (-6√5 - 8)/25

cos(α + β) = (4√5 - 6)/25

sin(α - β) = (-6√5 + 8)/25

tan(α - β) = (-9√5 + 45)/4

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The value of solution of the expression is,

⇒ - 5 (- 6w + 2u - 5)

Since, Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We have to given that;

Expression is,

⇒ - 5 (- 6w + 2u - 5)

Now, By using distributive property we get;

⇒ - 5 (- 6w + 2u - 5)

⇒ - 5 × - 6w - 5 × 2u - 5 × - 5

⇒ 30w - 10u + 25

Thus, The value of solution of the expression is,

⇒ - 5 (- 6w + 2u - 5)

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The area of the figure is 24.28 cm²

Any figure that we can break and form more than one basic shape from is called a composite figure.

The composite figure can be broken into a right triangle, a rectangle and a semi circle.

area of triangle = 1/2 bh

= 1/2 × 4 × 3

= 2 × 3

= 6 cm²

area of the rectangle = l × w

= 3 × 4

= 12 cm²

area of the semi circle = 1/2 πr²

= 1/2 × 3.14 × 2²

= 3.14 × 2

= 6.28cm²

Therefore the area of the composite figure =

6 + 12 +6.28

= 24.28 cm

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The equation that can be used to calculate the base areas of the rectangular prisms is B = V/6

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

Volume of rectangular prisms

The volume of rectangular prisms is calculated as

V = Bh

Where

B = Base area

h = height

Given that

h = 6 inches,

We have

V = B * 6

Divide both sides by 6

B = V/6

This means that the the equation to calculate the base areas is B = V/6

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Complete question

Mr. James is teaching his studens about the volume of rectangular prisms.He has various rectangular prisms with a height of 6 inches

Which equation can be used to find B, the area of the base with volume V

Answer:

Step-by-step explanation:

7247.1977343

The square root of (35 x 35 x 35) is approximately 207.57.

The square root of (35 x 35 x 35) can be found by simplifying the expression under the radical sign:

√(35 x 35 x 35) = √(35³)

On simplifying 35³ to get:

√(35³) = √(42,875) ≈ 207.57

Therefore, the square root of (35 x 35 x 35) is approximately 207.57.

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The sample mean percentage x is 62.3% and the sample standard deviation S is 8.08%.

To find the percentage of hospitals that provide at least some charity care, we need to use the given sample data.

(a) Using a calculator with mean and sample standard deviation keys, we can find the sample mean percentage x and the sample standard deviation S.

[tex]x = \frac{56.9 + 55.9 + 52.5 + 65.7 + 59.0 + 64.7 + 70.1 + 64.7 + 53.5 + 78.2}{10}\\\\ = 62.3% \\\\\\\\S = \sqrt{\frac{((56.9 - 62.3)^2 + (55.9 - 62.3)^2 + ... + (78.2 - 62.3)^2) }{ 9}}\\\\ = 8.08\%[/tex]

Therefore, the sample mean percentage x is 62.3% and the sample standard deviation S is 8.08%.

(b) To find a 90% confidence interval for the population average u of the percentage of hospitals providing at least some charity care, we can use the formula:

[tex]CI = x \pm z*(S/sqrtn)[/tex]

where CI is the confidence interval, x is the sample mean percentage, z is the z-score for the desired confidence level (90% in this case), S is the sample standard deviation, and n is the sample size (10 in this case).

From the z-table, the z-score for a 90% confidence level is 1.645.

Plugging in the values, we get:

CI = 62.3 ± 1.645*(8.08/sqrt(10)) = (55.5%, 69.1%)

Therefore, we can be 90% confident that the true population average percentage of hospitals providing at least some charity care is between 55.5% and 69.1%.

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-12x stands for the amount Kiran takes out each month.

B. any value of x that is less than or equal to 6.25 will work for Kiran.

c. One can use x ≤ 6.25 to show all the values that would work for Kiran.

d. The answer to Kiran's question can be written using mathematical notation as:

Kiran can be able to take out a maximum of $6.25 every month to make sure that he has at least $25 in the account a year from that time now.

Since the inequality  is -12x + 100 ≥ 25,  the term -12x stands for the amount Kiran takes out each month. It is seen as a negative value because it stands for the monthly withdrawals from the  said account.

b. To find values of x that would work for Kiran,  it can be done by:

-12x + 100 ≥ 25.

-12x + 100 ≥ 25

So, subtract 100 from both sides:

-12x ≥ 25 - 100

-12x ≥ -75

Then divide both sides by -12 ( dividing by a negative number changes the inequality sign) Hence it will be:

x ≤ (-75) / (-12)

x ≤ 6.25

So, any value of x that is less than or equal to 6.25.

c. One can use x ≤ 6.25 to show all the values that would work for Kiran. based on the fact that:

Kiran requires to have at least $25 in the account a year from now, he cannot withdraw more than $6.25 per month. If he withdrew more than that, he would not have more money left in the account after a single year.

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See text below

Kiran has $100 saved in a bank account. (The account doesn’t earn interest.) He asked Clare to help him figure out how much he could take out each month if he needs to have at least $25 in the account a year from now.

a. Clare wrote the inequality -12x + 100 ≥ 25, where x represents the amount Kiran takes out each month. What does -12x represent?

b. Find some values of x that would work for Kiran.

c. We could express all the values that would work using either x ≤ _ or x ≥ _. Which one should we use?

d. Write the answer to Kiran’s question using mathematical notation.

Answer:

100

Step-by-step explanation:

square root(L^2 + W^2)

The unlabeled side length for this problem has the measure given as follows:

13.

The Pythagorean Theorem states that in the case of a right triangle, the square of the length of the hypotenuse, which is the longest side,  is equals to the sum of the squares of the lengths of the other two sides.

Hence the equation for the theorem is given as follows:

c² = a² + b².

In which:

The sides for this problem are given as follows:

5 and 12.

Hence the measure of the hypotenuse is obtained as follows:

h² = 5² + 12²

h² = 169

h = 13.

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The division of the polynomial 3x³ - 2x² + x - 4 by x + 3 is:

Quotient: 3x² + 11x + 34

Remainder: -106/(x + 3)

When dividing a polynomial by a linear binomial of the form (x - c) or (x + c), where "c" is a constant, the process is known as synthetic division. This streamlined method of polynomial long division offers a faster and clearer method for dividing polynomials by linear factors.

The process of synthetic division involves arranging the coefficients of the polynomial in descending order and performing a sequence of simple arithmetic operations.

The result of the synthetic division is:

[tex]Quotient: 3x^2 + 11x + 34[/tex]

Remainder: -106/(x + 3)

Kindly find the step by step analysis below.

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The restricted domain of the function is 0≤s≤25

The equation is a(s) = 37.5s - 1.5s²  models the function represented in the table

a(s) = s(37.5 - 1.5s)

The critical points are s=0 or

37.5 - 1.5s

37.5 = 1.5s

s=25

As area can not be negative, the area should be positive

s< 0 and s>25

Therefore, restricted domain is 0≤s≤25

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We can see here the proof of complex functions to geometry was contributed by the Norwegian mathematician Caspar Wessel.

A complex function is one that accepts a complex number as input and outputs another complex number. It is a function that, in other words, has the complex numbers as a codomain and a subset of the complex numbers as its domain.

Wessel demonstrated in his article that complex functions and complex numbers may both be used to represent curves and points in the plane. This research was a significant advancement in the field of complex analysis and opened the path for its growth into a potent instrument for addressing geometric problems.

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∠b and 76° are alternate angles. They are equal.

So, m∠b = 76

That's it :)

The equation of the circle is (x + 6)² + (y - 1)² = 7² and it's center is (-6, 1)

The standard equation of a circle is given by:

(x - h)² + (y - k)² = r²

Where (h,k) is the coordinates of center of the circle and r is the radius.

In the given problem, the equation is;

x² + y² + 12x - 2y - 12 = 0

This can be rewritten as;

(x + 6)² + (y - 1)² = 7²

The center of the circle is (-6, 1)  from the coordinates (h, k) of the equation of the circle.

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The value of the missing angle x is: 38°

We know that the sum of angles in a triangle is 180 degrees and as such:

∠DAE = 180 - (82 + 68)

∠DAE = 30°

Now, we know that alternate angles are congruent. Thus:

∠DAE ≅ ∠ADC = 30°

Opposite angles of a parallelogram are equal (or congruent) Consecutive angles are supplementary angles to each other (that means they add up to 180 degrees).

Thus:

∠BAE = ∠BDE = 82 + 30

= 112°

Thus:

∠ABC = ∠ACB = 68°

Thus:

∠BAC = 180 - 2(68)

∠BAC = 44°

Thus:

x = 112 - (44 + 30)

x = 112 - 74

x = 38°

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The properties of quadratic functions cause this "death ray" effect is discussed below.

The "death ray" effect refers to the phenomenon where a quadratic function appears to have a sharp point or vertex that shoots off to infinity in a graph. This effect is caused by the properties of quadratic functions, specifically:

The degree of the quadratic function: Quadratic functions have a degree of 2, which means they have a leading term that is a squared variable (x²). This leads to a graph that is a parabola.

The coefficient of the squared term: The coefficient of the squared term in the quadratic function determines the shape of the parabola. If the coefficient is positive, the parabola opens upward, and if the coefficient is negative, the parabola opens downward.

However, when the coefficient of the squared term is negative (a < 0), the graph of the quadratic function opens downward, and the vertex represents the maximum point of the parabola.

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The domain and the range of the table of values are Domain = {-15, 0, 9, 33, 44, 45} and Range = {21, 33, 57, 15, 97, 46}

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

The table of values

The above table of values can be represented as a list of ordered pairs

The rule of a table of values is that

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

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The value of the length x is 1.5

Using the sine trigonometric identity, we have that;

sin 45 = y/6√2

cross multiply the values, we have that;

y = 1/√2 × 6/√2

Multiply the values, we get;

y = 6/2

Divide the values

y = 3

Then, using the cosine identity, we get;

cos 60 = x/3

Find the cosine value and cross multiply, we get;

x = 0.5(3)

Now, multiply the values, we get;

x = 1.5

Hence, the value of the length is 1.5

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Answer:12 faces and 20 vertices

Step-by-step explanation:

Dodecahedron:- The three-dimensional geometric geometry of a dodecahedron comprises 20 vertices, 30 edges, and 12 flat faces.

Each face is a regular pentagon, with five equal-length sides and five equal-sized angles. One of the five Platonic solids, regular, convex polyhedrons with identical regular polygon faces and equal angles between them, the dodecahedron is one of these regular, convex polyhedrons. The dodecahedron is frequently utilised in architecture, design, and the arts because of its great degree of symmetry.

A dodecahedron has 12 faces.

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The options provided for the domain and range of the given graph are incorrect. The correct descriptions for the domain and range are as follows:

Domain: The domain represents the set of all possible input values, or the x-values, of the graph. Since no specific information is given about the graph or its equation, the domain is not specified. In this case, the correct answer is:

Domain: Not specified

Range: The range represents the set of all possible output values, or the y-values, of the graph. Similarly, without any specific information about the graph or its equation, the range is not specified. Therefore, the correct answer is:

Range: Not specified

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The solution of the equation [tex]\rm 5^{-x/16} = 5[/tex] will be negative 16.

Given that:

Equation, [tex]\rm 5^{-x/16} = 5[/tex]

In other words, the collection of all feasible values for the parameters that satisfy the specified mathematical equation is the convenient storage of the bunch of equations.

Take log on both sides, then we have

(-x / 16) log 5 = log 5

-x / 16 = 1

x = -16

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The linear function that is modeled by the given table is:

y = -4x + 7

A general linear function is written as:

y = ax + b

Where a is the slope and b is the y-intercept.

If we know two points on the line (x₁, y₁) and (x₂, y₂) then the slope is:

a = (y₂ - y₁)/(x₂ - x₁)

Here we can use the first two points on the table:

(-11, 51) and (-5.5, 29)

Then the slope is:

a = (29 - 51)/(-5.5 + 11)

a = -4

Then the linear function is:

y = -4x + b

We know that when x = -5.5, the value of y is 29

29 = -4*-5.5 + b

29 = 22 = b

29 - 22 = b

7 = b

The linear function is:

y = -4x + 7

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The required set of ordered pairs that represent a function is {(−5, −4), (−3, −2), (−1, 0), (0, −1), (0, 3)}. Option A is correct.

Given functions,
A=  {(−5, −4), (−3, −2), (−1, 0), (0, −1), (0, 3)}
B= {(−4, −4), (−4, −3), (−4, −2), (−4, −1), (−4, 0)}
C = {(−3, −4), (−3, 0), (−2, −3), (0, −2), (2, −1)}
D = {(−5, −2), (−3, −2), (−2, −2), (−1, −2), (0, 0)}

The other sets have repeated x-values with different y-values, which violates the definition of a function where each x-value can only have one corresponding y-value.

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Answer: The awnser is D

Step-by-step explanation: Hope this helps

The value of c that makes z = 4 an extraneous solution is c = -1.

We need to find the value of c that makes z = 4 an extraneous solution, which means that 4 is not a valid solution of the original equation but appears to be one when substituted into the equation.

First, we substitute z = 4 into the equation:

√ (3/2 x + 10) = cz + 8

√ (3/2 x + 10) = c(4) + 8

√(16) = 4c + 8

4 = 4c + 8

4c = -4

c = -1

Therefore, the value of c that makes z = 4 an extraneous solution is c = -1.

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The value of s in the circle is 11.

We have,

When two chords in a circle have the same arc length intercepted by the chords, it means that the distances from each endpoint of the chords to the center of the circle are equal.

Now,

The arc length XY and WZ are the same.

This means,

Chord XY and Chord WZ are equal.

Now,

2s = s + 11

Solve for s.

2s - s = 11

s = 11

Thus,

The value of s in the circle is 11.

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The volume of cylinder B include the following: D. 80π cubic units.

In Mathematics and Geometry, the volume of a cylinder can be calculated by using this formula:

Volume of a cylinder, V = πr²h

Where:

Note: Radius of cylinder B = 2r

Height of cylinder B = 2h

By substituting the parameters, we have the following:

Volume of cylinder B = π × (2r)² × 2h

Volume of cylinder B = π × 4r² × 2h

Volume of cylinder B = π × 8r²h

Volume of cylinder B = 8 × πr²h

Volume of cylinder B = 8 × 10π

Volume of cylinder B = 80π cubic units.

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1. The two cylinders have the same height and the same cross-sectional area at every point along that height, so they have the same volume. Sue is correct.

2. The two stacks of quarters have the same height and the same cross-sectional area at every point along that height, so they have the same volume.

3. The two triangular prisms have the same height and the same cross-sectional area at every point along that height, so they have the same volume. This can also be explained using Cavalieri's principle.

The two stacks of quarters have the same height and the same cross-sectional area at every point along that height, so they have the same volume.

This can be explained using Cavalieri's principle, which states that two solids have the same volume if they have the same height and the same cross-sectional area at every point along that height.

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It will take the train 53.33 seconds to pass completely through the tunnel.

We have,

First, let's convert the speed from km/h to m/s:

54 km/h = 54000 m/3600 s = 15 m/s

We need to find the total distance the train travels, which is the sum of its length and the tunnel's length:

So,

distance = 180 m + 620 m = 800 m

Now we can use the formula:

time = distance/speed

time = 800 m / 15 m/s

time = 53.33 seconds

Therefore,

It will take the train 53.33 seconds to pass completely through the tunnel.

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