Use integration to find the area of the figure having the given vertices.
(0, 0), (1, 6), (3, −2), (1, -7)

Answer: B) adult ticket: $13, student ticket: $6

Step-by-step explanation:

Let adults tickets be written as a

Let student tickets be written as s

We then use the information given in the question to form equation. This will be:

3a + 7s = 81 .......... equation i

4a + 7s = 94 .......... equation ii

Subtract equation i from ii

a = 13

Adult tickets cost $13

From equation i

3a + 7s = 81

We put the value of a = 13

3(13) + 7s = 81

39 + 7s = 81

7s = 81 - 39

7s = 42

s = 42/7

s = 6

Students ticket cost $6

Answer:

A scale factor is a number which scales, or multiplies, some quantity.

Answer:

The scale factor is the ratio of the length of a side of one figure to the length of the corresponding side of the other figure.

Step-by-step explanation:

Answer:

1.4

Step-by-step explanation:

enter it and lmk if it's right. it is right

Answer:

A

Step-by-step explanation:

\( - 3(2x + 8) \\ = - 3(2x) - 3(8) \\ = - 6x - 24\)

The expression is negative when 0.13 < k < 9.87. These are the values of k for which the line and the parabola do not intersect at any point.

To find the values of k, we need to solve the system of equations y = kx - 2 and y = 1 + 5x - 2x² by substituting y from one equation into the other. This will give us a quadratic equation in x that depends on k.

y = kx - 2

y = 1 + 5x - 2x²

Substituting y from the first equation into the second equation, we get:

kx - 2 = 1 + 5x - 2x²

Rearranging and simplifying, we get:

2x² - (k - 5)x + 3 = 0

This quadratic equation has two solutions for x, given by the formula:

x = [ (k - 5) ± √ ( (k - 5)² - 24 ) ] / 4

The solutions for x will determine the type of intersection between the line and the parabola. There are three possible cases:

i) The line and the parabola touch at one point. This happens when the quadratic equation has one repeated solution for x, which means that the discriminant (the part under the square root) is equal to zero. Therefore, we need to solve:

(k - 5)² - 24 = 0

Expanding and simplifying, we get:

k² - 10k + 1 = 0

Using the quadratic formula, we get:

k = [10 ± √ (100 - 4) ] / 2

k = [10 ± √ (96) ] / 2

k ≈ 9.87 or k ≈ 0.13

These are the values of k for which the line and the parabola touch at one point.

ii) The line and the parabola intersect at two points. This happens when the quadratic equation has two distinct solutions for x, which means that the discriminant is positive. Therefore, we need to solve:

(k - 5)² - 24 > 0

Expanding and simplifying, we get:

k² - 10k + 1 > 0

To find the values of k that satisfy this inequality, we can use a sign diagram. We first find the zeros of the quadratic expression by setting it equal to zero and using the quadratic formula, as we did in part i). We get k ≈ 9.87 or k ≈ 0.13. These are the values that make the expression change sign. We then test a value in each interval to see if it makes the expression positive or negative.

From the sign diagram, we can see that the expression is positive when k < 0.13 or k > 9.87. These are the values of k for which the line and the parabola intersect at two points.

iii) The line and the parabola do not intersect at any point. This happens when the quadratic equation has no real solutions for x, which means that the discriminant is negative. Therefore, we need to solve:

(k - 5)² - 24 < 0

Expanding and simplifying, we get:

k² - 10k + 1 < 0

To find the values of k that satisfy this inequality, we can use a sign diagram again. We already found the zeros of the quadratic expression in part i), so we just need to test a value in each interval to see if it makes the expression positive or negative.

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Answer: y - 3= -1/2 * (x - 8)

Answer:

y=-1/2x+3

Step-by-step explanation:

Slope intercept form is written: y=mx+b...

y=doesn't change, but you need it by itself

m=slope

x=the variable right next to the slope

b=y-intercept

Also, don't worry! This is super simple ^^

The question gives us -1/2 as the slope, so it needs to be plugged into the equation as m (next to x). Now our equation looks like this:

y=-1/2x+b

Next, we can figure out b. B is the y-intercept and we can figure this out by looking at the ordered pair (8, 3). The y is the second number, which is 3, so that is the y-intercept! We can now plug in 3 into b. Our final equation looks like this:

y=-1/2x+3

Hope this helps!! Have an awesome day ❤

It is most likely that John and Dawn live in a "married couple with children" household. They have been married for over 20 years and have three children residing with them in their home in Northville, MI.

John and Dawn have been married for over 20 years and have lived in their home in Northville, MI with their three children for most of those years. To determine the type of household they live in, we can consider the various household classifications provided by the Census Bureau.

Married couple with children: This classification applies to households where a married couple resides with one or more children who are under the age of 18. Since John and Dawn have three children, it is possible that they fall under this category.

Married couple without children: This classification includes households where a married couple resides without any children under the age of 18. However, since John and Dawn have three children, this classification is not applicable to their situation.

Single-parent household: This classification applies when a single parent resides with one or more children under the age of 18. Since John and Dawn are married, this classification is not relevant in their case.

Other household types: There are several other household types defined by the Census Bureau, such as multi-generational households, unmarried partner households, and non-family households. However, based on the given information, none of these classifications seem to apply to John and Dawn's situation.

Considering the information provided, it is most likely that John and Dawn live in a "married couple with children" household. They have been married for over 20 years and have three children residing with them in their home in Northville, MI.

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the answer: 2/10 or 1/5

The area and perimeter of a polygon with vertices (1,-2) , (1,5) , (7,-2)

A=23.27644 units^2

Let

A= (1,-2) ,

B=(1,5) ,

C=(7,-2)

Therefore, the equation for distance is  mathematically given as

d=\sqrt {(x^2-x^1)^2+(y^2-y^1)^2}

Therefore For AB

\(d=\sqrt {(-2-1)^2+(5-1)^2}\)

d=5

For BC

\(d=\sqrt {(5-1)^2+(-2-7)^2}\)

d=9.84

for CA

\(\sqrt {(5-1)^2+(-2-7)^2}\)

9.486

Hence, Perimeter

P=9.486+9.84 +5

P=24.326

In conclusion, the Area of a triangle

A = √s(s−a)(s−b)

Where

s=24.326/2

Therefore

\(A=\sqrt{24.326/2((24.326/2)-5)((24.326/2)-9.486 )((24.326/2)-9.84 )}\)

A=23.27644 units^2

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The equation of the plane through the origin and the points (2, –4, 6) and (5, 1, 3) is - 9x + 12y + 11 x = 0.

The general equation of a plane through (0, 0, 0) is a(x - 0) + b(y - 0) + c(z - 0) = 0

Since the plane passes though the origin, the equation of the plane is given by ( x, y, z ) = 0. Simplify it so that you write the equation of the plane in the form a x + b y + c z = 0

ax + by + cx = 0...(1)

It will pass through B(2, -4, 6) and C(5, 1, 3) if

a(2) + b(-4) + c(6) = 0

2a - 4b + 6c = 0

a - 2b + 3c = 0 ...(2)

a(5) + b(1) + c(3) = 0

5a + b + 3c = 0 ... (3)

Solving (2) and (3) by cross-multiplication, we have

\(& \frac{a}{-6-3}=\frac{b}{15-3}=\frac{c}{1+10} \\\)

\(& \Rightarrow \frac{a}{-9}=\frac{b}{12}=\frac{c}{11}=\lambda\) (say) }

\(& \Rightarrow\) a = - 9 \(\lambda\), b = 12 \(\lambda\)  and c = 11 \(\lambda\)

Substituting the values of a, b and c in (1), we get

- 9 \(\lambda\) x + 12 \(\lambda\) y + 11 \(\lambda\) x = 0

- 9x + 12y + 11 x = 0

which is the required equation of the plane.

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Since, We catch the other car if our speed of our car is 60mph.

When you catch up, the distance traveled is the same

Let us consider x as the velocity.

Therefore, our velocity is x + 10

We know that:

Distance = rate x time

According to the question:

r = uncle's rate

30 min = 30/60 hr = uncle's time

25 min = 25/60 hr =  your time

and

Uncle's distance = Your distance

Now,

(r)(30/60) = (r + 10)(25/60)

⇒ (r)(1/2) = (r + 10)(5/12)

⇒ r/2 = 5r/12 + 50/12  

⇒ 6r = 5r + 50

⇒ r = 50  

Now,

Uncle's rate = 50 mph

and, putting the values:

Your rate = 50 + 10 = 60 mph

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Answer:

412.3 feet

Step-by-step explanation:

A canoe rental shop is being built near a river. An equation of the line representing the river is y=1/4x+8. Each unit in the coordinate plane corresponds to 100 feet. Approximately how far is the canoe rental shop from the river? Round your answer to the nearest whole foot. The rental shop is at (-3,3).

Solution:

The distance between a point and a line is the perpendicular distance. The perpendicular distance is the shortest distance between the point and the line.

Given a point (\(x_1,y_1\)) and a line with the equation Ax + By + C = 0, the perpendicular distance (d) is given as:

\(d=\frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2} }\)

Given the equation of the line y = 1/4 x + 8

y - 1/4 x - 8 = 0

Therefore A = -1/4, B = 1 and C = -8

The point is (-3,3), therefore \(x_1=-3,y_1=3\)

\(d=\frac{|-1/4(-3)+(1*3)+(-8)|}{\sqrt{(-1/4)^2+1^2} }\\\\d=4.123\ unit\\\\but\ 1\ unit=100\ feet\\\\ d=4.123\ unit*1\ feet=412.3\ feet\)

Answer:

412.3 feet

Step-by-step explanation:

Answer:

1. 29 2. 35

Step-by-step explanation:

check the attached file

Answer:

a. f(x)

b. (2, -1)

c. f(x); y= -2

   g(x); y= -5.1

   h(x); y= -5

d. h(x)

Step-by-step explanation:

a. greatest y-intercept :

set x = 0

f(x) = 7

g(x) = -2

h(x) = -4

b. (2, -1)

c. minimum value (The minimum value of a function is the place where the graph has a vertex at its lowest point.)

f(x); y= -2

g(x); y= -5.1

h(x); y= -5

d. largest ROC = largest rate of change

h(x)

The value of the iterated integral, evaluated by converting to polar coordinates, is √2/3.

What is integral?
An integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over an interval. It is used to find the total value or the net change of a function and has applications in areas such as calculus, physics, and economics.

To convert the given integral to polar coordinates, we make the following substitutions:

x = rcosθ

y = rsinθ

dx dy = r dr dθ

The limits of integration also need to be adjusted accordingly. For the innermost integral, the limits become:

0 ≤ y ≤ √(x² - y²)

0 ≤ y ≤ √(r²cos²θ - r²sin²θ)

0 ≤ y ≤ r√(cos²θ - sin²θ)

0 ≤ y ≤ r√(cos²θ - (1 - cos²θ)) [Using the identity sin²θ = 1 - cos²θ]

0 ≤ y ≤ r√(2cos²θ - 1)

For the middle integral, the limits become:

0 ≤ x ≤ √x² + y²

0 ≤ x ≤ √(r²cos²θ + r²sin²θ)

0 ≤ x ≤ r

Finally, for the outermost integral, the limits are:

0 ≤ r ≤ √2

0 ≤ θ ≤ π/2

Now we can express the integral in polar coordinates:

∫ √2 0 ∫ x 0 √x²+y² dy dx = ∫ π/2 0 ∫ r 0 r√(2cos²θ - 1) r dr dθ

Simplifying the expression and integrating:

∫ π/2 0 ∫ r 0 r√(2cos²θ - 1) r dr dθ = ∫ π/2 0 ∫ r 0 r²√(2cos²θ - 1) dr dθ

Integrating the inner integral with respect to r:

∫ π/2 0 [1/3 r³√(2cos²θ - 1)] evaluated from 0 to r dθ

= ∫ π/2 0 (1/3 r³√(2cos²θ - 1) - 0) dθ

= ∫ π/2 0 (1/3 r³√(2cos²θ - 1)) dθ

Integrating the outer integral with respect to θ:

[1/3 r³√(2cos²θ - 1)] evaluated from π/2 to 0

= [1/3 r³√(2cos²(0) - 1)] - [1/3 r³√(2cos²(π/2) - 1)]

= [1/3 r³√(2 - 1)] - [1/3 r³√(0 - 1)]

= [1/3 r³] - [1/3 r³(-1)]

= [1/3 r³] + [1/3 r³]

= 2/3 r³

Now, substitute back r = √2:

2/3 (√2)³ = 2/3 (2√2)

= √2/3

Hence, the value of the given iterated integral, evaluated by converting to polar coordinates, is √2/3.

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The first law of thermodynamics is a fundamental principle in physics that states energy cannot be created or destroyed, only converted from one form to another. It can be expressed in terms of heat and work through the equation:

ΔU = q - w

where ΔU represents the change in internal energy of a system, q represents the heat added to the system, and w represents the work done on or by the system.

Now, let's address when the quantities q and w would be negative numbers.

1) When q is negative: This occurs when heat is removed from the system, indicating an energy loss. For example, when a substance is cooled, heat is extracted from it, resulting in a negative value for q.

2) When w is negative: This occurs when work is done on the system, decreasing its energy. For instance, when compressing a gas, work is done on it, leading to a negative value for w.

In both cases, the negative sign indicates a reduction in energy or the transfer of energy from the system to its surroundings.

In summary, the first law of thermodynamics can be expressed as ΔU = q - w, and q and w can be negative numbers when energy is lost from the system through the removal of heat or when work is done on the system.

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The equation, in slope-intercept form, of the line that is perpendicular to the line y – 4 = –two-thirds(x – 6) is y = 3/2x - 1

The equation of a lins is a linear equation and expressed as:

y = mx + b

m is the slope

b is the y-intercept

Determine the equation in point slope form

y - y1 = -1/m(x - x1)

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

Write in slope-intercept form

2(y+2) = 3(x+2)

2y + 4 = 3x + 6
2y = 3x - 2

y = 3/2x - 1

Hence the equation, in slope-intercept form, of the line that is perpendicular to the line y – 4 = –two-thirds(x – 6) is y = 3/2x - 1

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a) The marked price of the article was Rs. 1,872.975.

b) The gain or profit made by the shopkeeper was Rs. 187.92.

The marked price refers to the cost price plus the markup.

The markup is the additional cost put on the cost of an item to cover profits and other expenses.

On the other hand, the gain or profit refers to the difference between the retailer's selling price, excluding VAT and the cost price.

The markup rate = 30%

Discount offered to a customer = 15%

VAT = 15%

The selling price of the item = Rs. 1,872.975

The discounted price of the article (excluding VAT) = Rs. 1,628.67 (Rs. 1,872.975/1.15)

The marked price, including discount = Rs. 1,872.975 (Rs. 1,628.67 x 1.15)

The cost price = Rs. 1,440.75 (Rs. 1,872.975/1.3).

The gain or profit made from the sale = The discounted price less the cost price

= Rs. 187.92 (Rs. 1,628.67 - $1,440.75)

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Susan has more candy in weight compared to Isabel.

To compare the candy weights between Susan and Isabel, we need to ensure that both weights are in the same unit of measurement. Let's convert Isabel's candy weight to ounces for a fair comparison.

Given:

Susan: 4 bags x 6 ounces/bag = 24 ounces

Isabel: 1 bag x 16 ounces/pound = 16 ounces

Now that both weights are in ounces, we can see that Susan has 24 ounces of candy, while Isabel has 16 ounces of candy. As a result, Susan is heavier on the candy scale than Isabel.

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Answer:

x - 3y + 12 = 0

Step-by-step explanation:

To write the equation 1/4x - 3/4y = -3 in standard form, we need to eliminate the fractions and rearrange the terms so that the variables are on one side of the equation and the constant is on the other side.

One way to eliminate the fractions is to multiply both sides of the equation by the least common multiple of the denominators, which is 4. This gives:

4(1/4x) - 4(3/4y) = 4(-3)

Simplifying each term, we get:

x - 3y = -12

Now we can rearrange the terms so that the variables are on the left side and the constant is on the right side:

x - 3y + 12 = 0

1945 men and 2849 women register to audition for a singing competition. The number of participants who are not successful in their auditions what’s five times the number of those who are successful. There are 799  participants were successful.

The successful  participants  can be calculate by solving a linear equation as follows

First, it's crucial to understand linear equations.

Equation connects the two algebraic expressions with an equal to sign to demonstrate the equality between the two algebraic expressions.

Linear equations are those with one degree.

In this case, a linear equation must be resolved.

1945 for the men's total

Women are present in 2849.

Participants in total: 1945 + 2849 = 4794

Let x be the proportion of participants that were successful.

Men who failed were 5 times as numerous.

Participants in total: 5x + x = 6x

Due to the issue,

The linear formula is

6x = 4794

x = 4794/6

x = 799

799 of the participants had success.

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Hey there,

We have,

f(x) = x² + 11x + 30g(x) = x + 6

Now,

(f.g)(x) = f(g(x))

Substituting...

g(x) = x + 6f(g(x)) = f(x + 6)

Now,

f(x + 6)= (x + 6)² + 11(x + 6) + 30= x² + 12x + 36 + 11x + 66 + 30= x² + 23x + 132

So,The value is x² + 23x + 132

To determine the equation for the exponential function modeled by the data, you need to identify the pattern and use regression analysis or logarithms. Please provide the data in the table, and I will help you find the equation.

Once we have the equation, we can use it to predict the approximate number of stores the retail chain will open in year 13. This is done by substituting the value of 13 into the equation and solving for the unknown variable.

To determine the equation for the exponential function modeled by the data, find the common ratio and use it to write the equation. To predict the number of stores in year 13, substitute the value into the equation.

To determine the equation for the exponential function modeled by the data in the table, we need to find the common ratio. The common ratio is the ratio of a term to its previous term. We can calculate the common ratio by dividing any term by the previous term. Once we find the common ratio, we can use it to write the equation of the exponential function. To predict the approximate number of stores the retail chain will open in year 13, we can substitute the value of 13 into the equation we found in question 1.

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#SPJ11Final answer:

To determine the equation for the exponential function modeled by the data, find the common ratio and use it to write the equation. To predict the number of stores in year 13, substitute the value into the equation.

To determine the equation for the exponential function modeled by the data in the table, we need to find the common ratio. The common ratio is the ratio of a term to its previous term. We can calculate the common ratio by dividing any term by the previous term. Once we find the common ratio, we can use it to write the equation of the exponential function. To predict the approximate number of stores the retail chain will open in year 13, we can substitute the value of 13 into the equation we found in question 1.

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Answer:

Hello,

Step-by-step explanation:

\(A=\begin{bmatrix}2 & 2 \\3 & 4 \end{bmatrix}\\\\\\B=\begin{bmatrix}0 & 4 \\2 & -3 \end{bmatrix}\\\\\\C=\begin{bmatrix}4 & 6 \\5 & 6 \end{bmatrix}\\\\\\(X-4B)*A=C\\\\X-4B=C*A^{-1}\\\\X=C*A^{-1}+4B\\\)

\(\begin{array}{|cc|cc|}A&&A^{-1}&\\---&---&---&---\\2&2&1&0\\3&4&0&1\\---&---&---&---\\1&1&\frac{1}{2}&0\\0&1&\frac{-3}{2}&1\\---&---&---&---\\1&0&2&-1\\0&1&\frac{-3}{2}&1\\---&---&---&---\\\end {array}\\\\\)

\(X=C*A^{-1}+4B=\begin{bmatrix}4 & 6 \\5 & 6 \end{bmatrix}*\begin{bmatrix}2&-1\\\frac{-3}{2}&1\end{bmatrix}+4*\begin{bmatrix}0 & 4 \\2 & -3 \end{bmatrix}\)

\(X=\begin{bmatrix}-1 & 2 \\1 & 1 \end{bmatrix}+\begin{bmatrix}0 & 16 \\8 & -12 \end{bmatrix}\\\\\\X=\begin{bmatrix}-1 & 18 \\9 & -11 \end{bmatrix}\\\)

Answer:

Option (2)

Step-by-step explanation:

ΔDEF is a dilation image of ΔABC.

Rule for the dilation,

Scale factor = \(\frac{\text{Length of one side of the image triangle}}{\text{Length of one side of the original triangle}}\)

                    = \(\frac{DE}{AB}\)

                    = \(\frac{4}{8}\)

                    = \(\frac{1}{2}\)

Therefore, scale factor by which ΔABC is dilated is \(\frac{1}{2}\).

Option (2) will be the correct option.

Answer:

\( \sf \: f(5) = - 12\)

Step-by-step explanation:

Given function,

→ f(x) = -2(x + 1)

Now we have to,

→ Find the required value of f(5).

Then the value of f(5) will be,

→ f(x) = -2(x + 1)

→ f(5) = -2(5 + 1)

→ f(5) = -2(6)

→ [ f(5) = -12 ]

Hence, the value of f(5) is -12.

Answer:

The scale factor of the dilation is;

Explanation:

Given that;

Scale factor is image length divided by pre-image length;

From the diagram;

Applying the above formula;

Therefore, the scale factor of the dilation is;

Answer:

\(6 \times {10}^{3} = 6000\)

Step-by-step explanation:

\((1.5 \times {10}^{ - 6} ) \times (4 \times {10}^{9} )\)

\( = (1.5 \times 4) \times ( {10}^{ - 6} \times {10}^{9} )\)

\( = 6 \times ( {10}^{ - 6} \times {10}^{9}) \)

\( = 6 \times {10}^{ - 6 + 9} \)

\( = 6 \times {10}^{3} \)

\( = 6000\)