polar point : (6,3π).
Find three additional polar representations of the point, using −2 < < 2. (Enter your answers in order from smallest to largest first by r-value, then by -value.)

When not considering the order of choice, there are also 3 ways to choose 2 letters from a, b, and c.

When choosing 2 letters without replacement from the 3 letters a, b, and c, the number of ways can be calculated considering the order of choice and without considering the order of choice.

1. Considering the order of choice:

In this case, the order in which the letters are chosen matters. We can think of this as a permutation problem.

To calculate the number of ways when order matters, we use the formula for permutations:

[tex]nPr = n! / (n - r)![/tex]

where n is the total number of items and r is the number of items chosen.

In this case, we have 3 letters and we want to choose 2, so n = 3 and r = 2.

Using the formula, we get:

[tex]3P2 = 3! / (3 - 2)![/tex]

 [tex]= 3! / 1![/tex]

    = 3

Therefore, when considering the order of choice, there are 3 ways to choose 2 letters from a, b, and c.

2. Without considering the order of choice:

In this case, the order in which the letters are chosen does not matter. We can think of this as a combination problem.

To calculate the number of ways when order does not matter, we use the formula for combinations:

[tex]nCr = n! / (r!(n - r)!)[/tex]

Using the same values of n = 3 and r = 2, we get:

[tex]3C2 = 3! / (2!(3 - 2)!)[/tex]

    = [tex]3! / (2! * 1!)[/tex]

    = 3

Therefore, when not considering the order of choice, there are also 3 ways to choose 2 letters from a, b, and c.

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The volume of the shapes are;

6. 11259.47 in³

7. 14476.46 mm³

8.  3744 ft³

9. 473.6mm³

10.  18816.57cm³

To determine the values, we need to know the following;

In a right-angled triangle, a ² + b ² = c ²

Now. for each of the shapes, substitute the values, we have;

6) π (16) ² X 14

Multiply the values

= 11259.47 in³

7) Diameter² = 40² - 32²

Diameter = √(40² - 32²) = 24, Radius = 12.

Substitute the values

Volume = π (12)² X 32 = 14476.46 mm³

8) volume = 8 X 12 X 39

Multiply the values

= 3744 ft³

9) cross-section (triangle) = 1/2 X 5.4 X 11 = 29.7

Volume = 29.7 X 16 = 473.6mm³

10) radius = 33/2 = 16.5

volume = (4/3) π (16.5)³

Multiply the values

= 18816.57cm³

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4.5 mm is the measurement of the given line AB.

In the given graph both lines AE and BD are parallel.

So, the ratio between the two lines will be the same,

Thus, from the above property

AC/AB= EC/ED

in the given case,

AC =36 mm

EC = 72 mm

ED = 9 mm

Substitute the value in the above equation,

36/AB = 72/9

AB = 9/2

AB = 4.5 mm

Therefore, the measurement of line AB is 4.5 mm.

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The mean and standard deviation are less than 88 peanuts: z = (88 - 94) / 3 = -2

between 88 and 91 peanuts: z = (91 - 94) / 3 = -1

between 91 and 94 peanuts: z = (94 - 94) / 3 = 0

between 94 and 97 peanuts: z = (97 - 94) / 3 = 1

greater than 97 peanuts: z = (97 - 94) / 3 = 1

The probability that a 16-ounce can of Nut Munchies will contain between 88 and 97 nuts is 0.8186, or approximately 82%.

The probability of the former event is 0.4772, which is greater than the probability of the latter event, which is 0.1587.

(a) The intervals below the normal distribution can be labeled using z-scores, which are calculated using the formula:

z = (x - μ) / σ

where x is the value of interest, μ is the mean, and σ is the standard deviation.

The intervals and their corresponding z-scores are:

less than 88 peanuts: z = (88 - 94) / 3 = -2

between 88 and 91 peanuts: z = (91 - 94) / 3 = -1

between 91 and 94 peanuts: z = (94 - 94) / 3 = 0

between 94 and 97 peanuts: z = (97 - 94) / 3 = 1

greater than 97 peanuts: z = (97 - 94) / 3 = 1

(b) To find the probability that a 16-ounce can of Nut Munchies will contain between 88 and 97 nuts, we need to find the area under the normal distribution curve between the corresponding z-scores. Using a standard normal distribution table or a calculator, we can find that:

P(88 ≤ X ≤ 97) = P(-2 ≤ Z ≤ 1) = 0.8186

Therefore, the probability that a 16-ounce can of Nut Munchies will contain between 88 and 97 nuts is 0.8186, or approximately 82%.

(c) To determine which event is more likely, we can compare the probabilities of each event. Using the same method as in part (b), we can find that:

P(88 ≤ X ≤ 94) = P(-2 ≤ Z ≤ 0) = 0.4772

P(X > 97) = P(Z > 1) = 0.1587

Therefore, it is more likely for a 16-ounce can of Nut Munchies to have between 88 and 94 nuts than to have over 97 nuts, since the probability of the former event is 0.4772, which is greater than the probability of the latter event, which is 0.1587.

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Answer:

-3,-2,-1 is less than -4 and not greater than

Step-by-step explanation:

To find the percentage of M2 that M1 represents, we can use the following formula:(M1 / M2) x 100%Substituting the given values, we get:(1050 / 3500) x 100% = 30%Therefore, M1 represents 30% of M2.

The graph of the translations are attached

The red is the preimage while the blue is the image

The absolute value graph has equation f(x) = |2x| and the translation we want to obtain is y = f(x) + 4 and this is a translation 4 units up

This results to a graph of equation  y = f(x) + 4 = |2x| + 4

For the parabolic graph, the equation is written as

y = a(x - 2)² - 3

solving for a using (0, -5)

-5 = a(0 - 2)² - 3

-2 = 4a

a = -1/2, then the equation is g(x) = -1/2 (x - 2)² - 3

Applying the translation, the equation will be

y = g(x + 2) =  -1/2 (x - 2 + 2)² - 3 = -1/2 (x)² - 3

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The probability that a randomly selected resident is in favor of the dog park is option

B. 7/18

To find the probability that a randomly selected resident is in favor of the dog park  we  have  to determine the number of residents who are in favor and divide it by the total number of residents surveyed

the number of residents in favor of the dog park

= in favor both male and female

=  19 + 16

= 35.

total number of residents surveyed is  90

the probability

P(In Favor) = Number of residents in favor / total number of residents surveyed

= 35 / 90

= 7/18

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The correct answer is Add all of the numbers, then divide.

Given that we need to determine which choice demonstrates the process for calculating the mean,

We know that the mean is calculated by,

You must add up all the values in a dataset and divide the total by the overall number of values in order to determine the mean, also known as the average.

The following steps should be followed to determine the mean:

Add up all the figures.

Subtract the amount from the overall value set.

In order to determine the mean, the option "Add all of the numbers, then divide" is the proper course of action.

Hence the correct answer is Add all of the numbers, then divide.

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The total length of the chord PR is 30.

Given a circle Q.

Given a chord named as PR.

We have to find the length of the chord.

Here Q is the center of the circle.

Given that ∠QSP is a right angle.

Then clearly, ∠QSR is also a right angle.

Consider the triangle QSR and QSP.

QP = QR, since they are radius of the circle.

So, QP = QR = 17

QS ia a common side.

So QS = 8

Then it is clear that,

PS = SR

Using Pythagoras theorem,

(PS)² = 17² - 8²

        = 225

PS = √225 = 15

So the length of the chord, PR = 15 + 15 = 30.

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The volume of each figure is given below:

Figure 1: [tex]819 \ feet[/tex]³

Figure 2: [tex]1,728 \ m[/tex]³

Figure 3: [tex]10,696 \pi[/tex] yards³

Figure 4: [tex]1,079.5[/tex] m³

Figure 5: [tex]4,680[/tex] yards³

Figure 1: In the first figure we can see a Cuboid. So below is the method to find the cuboid's volume:

Given dimensions:

Length (l) = [tex]9[/tex] feet, Breadth (b) = [tex]13[/tex] feet, Width (w) = [tex]7[/tex] feet.

Formula:

[tex]\[ V = l \times b \times w \][/tex]

[tex]\[ V = 9 \, \text{feet} \times 13 \, \text{feet} \times 7 \, \text{feet} \][/tex]

Figure 2: In the given figure we see a Prism. Below is the method to find its volume:

Given dimensions:

Hypotenuse = [tex]15[/tex] m, Base = [tex]9[/tex] m, Height = [tex]12[/tex] m, Width = [tex]16[/tex] m.

Formula:

[tex]\[ V = \text{Base} \times \text{Height} \times \text{Width} \][/tex]

[tex]\[ V = 9 \, \text{m} \times 12 \, \text{m} \times 16 \, \text{m} \][/tex]

Figure 3: Here we have the method to find the volume of a Cylinder:

Given dimensions:

Length (l) = [tex]26[/tex] yards, Radius (r) = [tex]11[/tex] yards.

Formula:

[tex]\[ V = \pi \times r^2 \times l \][/tex]

[tex]\[ V = \pi \times (11 \, \text{yards})^2 \times 26 \, \text{yards} \][/tex]

Figure 4: Here, we have the method to find the volume of a Cuboid:

Given dimensions are:

Length (l) = [tex]9[/tex] m, Breadth (b) = [tex]9[/tex] m, Height (h) = [tex]13.5[/tex] m.

Formula:

[tex]\[ V = l \times b \times h \][/tex]

[tex]\[ V = 9 \, \text{m} \times 9 \, \text{m} \times 13.5 \, \text{m} \][/tex]

Figure 5: Next figure is a Prism.

The dimensions that have been given to us are:

Hypotenuse = [tex]15[/tex] yards, Base = [tex]15[/tex] yards, Height = [tex]13[/tex] yards, Width = [tex]24[/tex] yards.

Formula:

[tex]\[ V = \text{Base} \times \text{Height} \times \text{Width} \][/tex]

Now we will calculate it according to the values given in the question:

[tex]\[ V = 15 \, \text{yards} \times 13 \, \text{yards} \times 24 \, \text{yards} \][/tex]

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I'm also getting 22.9875, but the instructions say to round to 2 decimal places.

22.9875 rounds to 22.99 because 0.9875 is closer to 0.99 than it is to 0.98

In other words, 9875 is closer to 9900 than it is to 9800.

Answer:

The answer is 2,000,

Step-by-step explanation:

Probability :- Probability is a way to gauge how likely something is to happen. It is represented by a number between [tex]0[/tex] and [tex]1[/tex], with [tex]0[/tex] denoting an impossibility and [tex]1[/tex] denoting a certainty. By dividing the number of favorable outcomes by the total number of possible outcomes, the probability of an event is determined.

A total of  [tex]32 + 40 + 300 = 372[/tex]  children either take bus number seven, bus number ten, or walk to school.

The proportion of students that ride bus [tex]7[/tex] or bus [tex]10[/tex] to the total number of students determines the likelihood that a student will board either bus [tex]7[/tex]or bus [tex]10[/tex].

[tex]P(taking bus number seven or ten) = (32 + 40) / 1000[/tex]

[tex]P(taking bus number seven or ten) = 72/1000[/tex]

[tex]P(using bus number 7 or ten) = 0.072[/tex]

Therefore, the probability that the student either rides on bus 7 or rides on bus 10 is 0.072 or 7.2%.

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The value of the missing side length x in the right triangle is 15 inches.

Pythagorean theorem states that the "square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides.

It is expressed as;

( hypotenuse )² = ( leg 1 )² + ( leg 2 )²

The image in the diagram is a right triangle:

Hypotenuse = 17 inches

Leg 1 = 8 inches

Leg 2 = x

To solve for x, we use the pythagorean theorem.

( hypotenuse )² = ( leg 1 )² + ( leg 2 )²

( leg 2 )² = ( hypotenuse )² - ( leg 1 )²

( leg 2 )² = ( 17 )² - ( 8 )²

( leg 2 )² = 289 - 64

( leg 2 )² = 225

Take the square roots

Leg 2 = √225

Leg 2 = 15 inches.

Therefore, the value of x is 15 inches.

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The required perimeter of the equilateral triangle is 12.72 cm.

An equilateral triangle has all three sides equal in length. Therefore, if the edge of an equilateral triangle is 4.24 cm, then all three sides are 4.24 cm.

To find the perimeter of the equilateral triangle, we need to add the length of all three sides together:

Perimeter = 4.24 cm + 4.24 cm + 4.24 cm

Perimeter = 12.72 cm

Therefore, the perimeter of the equilateral triangle is 12.72 cm.

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At the end of 2 years, Jackson's account balance is $700.40.

We have,

The formula for calculating simple interest is:

I = Prt

Where:

I is the interest earned

P is the principal amount

r is the interest rate

t is the time period

In this case,

Jackson invested $680.00 at an annual interest rate of 1.5% for 2 years. Therefore:

P = $680.00

r = 1.5% = 0.015

t = 2 years

Plugging these values.

I = Prt = $680.00 x 0.015 x 2 = $20.40

So,

Jackson earned $20.40 in interest over 2 years.

To find his total account balance at the end of 2 years, we need to add the interest earned to the initial principal:

Total balance

= $680.00 + $20.40

= $700.40

Therefore,

At the end of 2 years, Jackson's account balance is $700.40.

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Answer:

The approximate value of √329 to the nearest integer is 18.

Approximation is the process of using rounded values when presenting numerical data and making rough calculations.

To approximate √329 to the nearest integer, we can find the perfect squares that are less than or equal to 329.  

The perfect squares less than or equal to 329 are 16, 25, 36, and 49. But the nearest perfect square to 329 is 324, which is the square of 18 (i.e. 18² = 324).

Therefore, the nearest integer to √329 is 18.

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The equation of the polynomial is f ( x ) = -13.5 ( x + 2 )²( x - 3 )

Given data ,

The equation of degree 3 polynomial function with real coefficients

And ,  zeros x = - 2 with multiplicity 2 and x = 3 with multiplicity 1

where function passes through the point (1, 54)

If a polynomial function has a zero x = a with multiplicity k, then the factor (x - a)^k appears in its factored form.

Therefore, a degree 3 polynomial function with zeros x = -2 with multiplicity 2 and x = 3 with multiplicity 1 can be written in factored form as:

f(x) = a(x + 2)²(x - 3)

where a is a constant factor. To find the value of a, we use the fact that the function passes through the point (1, 54):

f(1) = a(1 + 2)²(1 - 3) = 54

a(-1)²(-2) = 54

-4a = 54

Divide by -4 on both sides , we get

a = -13.5

Hence , the equation of the degree 3 polynomial function with the given zeros and passing through the point (1, 54) is f ( x ) = -13.5 ( x + 2 )²( x - 3 )

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Answer:

972 square feet

Step-by-step explanation:

length, a, of rectangle with  width, b, will have area a X b (simplified to just ab).

so area = 36 (feet) X 27 (feet) = 972 (square feet)  

Answer:

i think this is the correct answer

sorry if its wrong

It is a simple method for transforming 2D shapes. By separating between planar transformations and spaces, the transformation can be grouped

Step-by-step explanation:

Answer:

See below for proof.

Step-by-step explanation:

[tex]\boxed{\textsf{Prove that}\;\;\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A) = \sin^3A}[/tex]

Rewrite 240° as (180° + 60°):

[tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(180^{\circ}+60^{\circ}+ A)[/tex]

As sin(180° + x) = -sin(x), we can rewrite sin³(180° + 60° + A) as:

[tex]\sin^3(180^{\circ}+60^{\circ}+ A)=-\sin^3(60^{\circ}+ A)[/tex]

Substitute this into the expression:

[tex]\sin^3A + \sin^3(60^{\circ} + A) -\sin^3(60^{\circ}+ A)[/tex]

As the last two terms cancel each other, we have:

[tex]\sin^3A[/tex]

Hence proving that:

[tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A) = \sin^3A[/tex]

As one calculation:

    [tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A)[/tex]

[tex]=\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(180^{\circ}+60^{\circ}+ A)[/tex]

[tex]=\sin^3A + \sin^3(60^{\circ} + A) -\sin^3(60^{\circ}+ A)[/tex]

[tex]=\sin^3A[/tex]

[tex]\hrulefill[/tex]

[tex]\boxed{\textsf{Prove that}\;\;\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A) = -\dfrac{3}{4}\sin 3A}[/tex]

Use the sine and cos double angle identities to rewrite sin(3x) in terms of sin(x):

[tex]\begin{aligned}\sin(3x)&=\sin(2x+x)\\&=\sin2 (x)\cos (x)+\sin (x)\cos2 (x)\\&=(2\sin (x)\cos (x))\cos (x)+\sin (x)(1-2\sin^2 (x))\\&=2\sin (x)\cos^2 (x)+\sin (x)-2\sin^3 (x)\\&=2\sin (x)(1-\sin^2 (x))+\sin (x)-2\sin^3 (x)\\&=2\sin (x)-2\sin^3 (x)+\sin (x)-2\sin^3 (x)\\&=3\sin (x)-4\sin^3 (x)\end{aligned}[/tex]

Rearrange to isolate sin³x:

[tex]\begin{aligned}\sin(3x)&=3\sin (x)-4\sin^3 (x)\\\\4\sin^3 (x)&=3\sin (x)-\sin (3x)\\\\\sin^3 (x)&=\dfrac{3\sin (x)-\sin (3x)}{4}\end{aligned}[/tex]

Use this expression to rewrite the terms in sin³A on the left side of the equation:

   [tex]\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A)[/tex]

[tex]=\dfrac{3\sin A-\sin3A}{4}+ \dfrac{3\sin (120^{\circ} + A)-\sin (3(120^{\circ} + A))}{4}+\dfrac{3\sin (240^{\circ}+ A)-\sin (3(240^{\circ}+ A))}{4}[/tex]

[tex]=\dfrac{3\sin A-\sin3A+3\sin (120^{\circ} + A)-\sin (360^{\circ} + 3A)+3\sin (240^{\circ}+ A)-\sin (720^{\circ}+ 3A)}{4}[/tex]

As sin(x ± 360°n) = sin(x), we can simplify:

[tex]\sin(360^{\circ}+3A) = \sin (3A)[/tex]

[tex]\sin(720^{\circ}+3A) = \sin (3A)[/tex]

Therefore:

[tex]=\dfrac{3\sin A-\sin3A+3\sin (120^{\circ} + A)-\sin (3A)+3\sin (240^{\circ}+ A)-\sin (3A)}{4}[/tex]

[tex]=\dfrac{3\sin A-3\sin3A+3\sin (120^{\circ} + A)+3\sin (240^{\circ}+ A)}{4}[/tex]

Factor out the 3 in the numerator:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (120^{\circ} + A)+\sin (240^{\circ}+ A)\right)}{4}[/tex]

Rewrite 240° = 180° + 60°:

[tex]\sin(240^{\circ} + A) = \sin(180^{\circ} + 60^{\circ} + A)[/tex]

As sin(180° + x) = -sin(x), we can rewrite sin(180° + 60° + A) as:

[tex]- \sin(60^{\circ} + A)[/tex]

Therefore:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (120^{\circ} + A)-\sin (60^{\circ}+ A)\right)}{4}[/tex]

As sin(120° + x) = sin(60° - x) then:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (60^{\circ} -A)-\sin (60^{\circ}+ A)\right)}{4}[/tex]

As sin(60° - x) - sin(60° + x) = -sin(x), then:

[tex]=\dfrac{3\left(\sin A-\sin3A-\sin A\right)}{4}[/tex]

Simplify:

[tex]=\dfrac{-3\sin3A}{4}[/tex]

[tex]=-\dfrac{3}{4}\sin3A[/tex]

Hence proving that:

[tex]\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A) = -\dfrac{3}{4}\sin 3A[/tex]

The solutions to the original equation are:

x = π - 3π/2 = -π/2

x = 3π/2 - 3π/2 = 0

So the values of x that satisfy the equation are -π/2 and 0.

To solve the given equation, we can use the following             trigonometric identity:

sin(A) + sin(B) = 2*sin((A+B)/2)*cos((A-B)/2)

Applying this identity to sin(3π/2+x) + sin(3π/2+x), we get:

sin(3π/2+x) + sin(3π/2+x) = 2*sin((3π/2+x + 3π/2+x)/2)cos((3π/2+x - 3π/2+x)/2)

= 2sin(3π/2+x)cos(0)

= 2(-cos(x))

Therefore, the equation can be simplified to:

-2*cos(x) = 2

Dividing both sides by -2, we get:

cos(x) = -1

This means that x is either π or 3π/2, since those are the values where cos(x) = -1.

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The angles that we can determine their measure are ∠1, ∠5, ∠6, and ∠7.

option A is the correct answer.

The measure of the angles of the polygon is calculated as follows;

The following angles were given;

angle 3 = 60⁰

angle 8 = 135⁰

The following angles can be determined as follows;

angle 5 = 180 - 60⁰  ( sum of angles on a straight line )

angle 5 = 120⁰

angle 7 = angle 3 = 60⁰ (corresponding angles )

angle 5 = angle 1 = 120⁰  (corresponding angles )

The value of angle 6 is calculated as;

∠5 + ∠6 + ∠7 + ∠8 = 360 (sum of angles in quadrilateral)

120 + ∠6 + 60 + 135 = 360

315 + ∠6 = 360

∠6 = 360 - 315

∠6 = 45⁰

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Answer:

15

if you count the length and breadth with the height and multiply them it'll give you 15

Answer: 5, 9

Step-by-step explanation:

factors of 45:

1, 3, 5, 9, 15, 45

Sums: 46, 24, 14

the factors 5 and 9 multiply to get 45 and add to get 14

voila!

Answer:

Step-by-step explanation:

5 minutes is the outlier. It sticks out from the rest.

Answer:

Hope this helps ^^

Step-by-step explanation:

Let's define the variables:

Let's call the cost of a burger "B" (in dollars).

Let's call the cost of a hotdog "H" (in dollars).

Now, we can set up a system of equations based on the given information:

Equation 1: 2B + H = 4.82 (Two burgers and one hotdog cost $4.82)

Equation 2: B + 2H = 3.70 (One burger and two hotdogs cost $3.70)

To solve this system of equations, we can use the method of substitution or elimination.

Using the method of elimination, we can multiply Equation 2 by 2 to eliminate the "B" term:

2(B + 2H) = 2(3.70)

2B + 4H = 7.40

Now, we can subtract Equation 1 from this new equation to eliminate the "2B" term:

(2B + 4H) - (2B + H) = 7.40 - 4.82

2H - H = 2.58

H = 2.58

Therefore, the cost of a hotdog is $2.58.

Labeling the answer: The cost of a hotdog is $2.58.