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If surface , S is the part of the sphere, x² + y²+ z² =1 ,lies above the cone, x² + y² = z

a)| ⃗rᵩ × ⃗rθ| = ⟨sin²φ cosθ,sin²φsinθ, sinφcosφ⟩

b)∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφdθ,

S

where x = 2π and y = π/4

A surface in space given in Cartesian coordinates as f(x,y,z) = 0, can be parametrized as a vector function with two parameters,

r(u,v)= ⟨r₁(u,v) , r₂(u,v) , r₃(u,v)⟩ , (u,v)∈R²

We have, S is a part of Sphere , x² + y²+ z² =1 , lies above the cone, x² + y² = z and surface S parametrized by following vector equation ,

r(θ, φ) = ⟨sinφ cosθ , sinφsinθ,cosφ⟩ --(1)

⃗rᵩ = ∂r/∂φ =⟨cos φ cosθ,cos φ sinθ,-sin φ⟩

⃗rθ= ∂r/∂θ=⟨- sinφ sinθ , sinφ cosθ ,0⟩

| ⃗rᵩ × ⃗rθ| =| i j k |

|cosφcosθ cosφ sinθ -sinφ |

|- sinφsinθ sinφ cosθ 0 |

= i( 0 + sinφ cosθsinφ) -j(0- (- sinφsinθ)(-sinφ)) + k(sinφ cosθ cosφcosθ - (-sinφ sinθ ) cosφ sinθ)

= (sin²φcosθ )i + (sin²φsinθ)j + (sinφcosφ(sin²θ+cos²θ))k

= sin²φcosθ )i + (sin²φsinθ)j +(sinφcosφ)k

| ⃗rᵩ× ⃗rθ|=⟨sin²φcosθ ,sin²φsinθ,sinφcosφ⟩

b) from part (a) we get,

x = sinφ cosθ

y = sinφsinθ

z = cosφ

are spherical coordinates of S where, 0≤φ≤π/4 and 0≤ θ≤ 2π.

so, ∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφ dθ ,

S

where x = 2π and y = π/4

So,Surface integeral is equals to ₀∫ˣ₀∫ʸcos²φ dφ dθ.

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Complete question:

If Sis the part of the sphere x2 + y2 + x2 = 1 that lies above the cone z= x2 + y2, find the following: a)S can be parametrized by a vector equation

r(θ, φ) = ⟨sinφ cosθ , sinφsinθ , cosφ ⟩ then a)| ⃗rᵩ × ⃗rθ| = ?

b) ∫∫ z² ds = ?

S

Answer:

-9

Step-by-step explanation:

Given function:

[tex]f(x)=-9x+4[/tex]

Given x-values:

Calculate the value of the function for the two given values of x:

[tex]\begin{aligned}\implies f(x_1)&=-9(-5)+4\\&=45+4\\&=49\end{aligned}[/tex]

[tex]\begin{aligned}\implies f(x_2)&=-9(0)+4\\&=0+4\\&=4\end{aligned}[/tex]

[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Average rate of change of function $f(x)$}\\\\$\dfrac{f(b)-f(a)}{b-a}$\\\\over the interval $a \leq x \leq b$\\\end{minipage}}[/tex]

As -5 < 0:

Therefore:

[tex]\begin{aligned} \implies \textsf{Average rate of change}&=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\\\\&=\dfrac{4-49}{0-(-5)}\\\\&=\dfrac{-45}{5}\\\\&=-9\end{aligned}[/tex]

Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]

option D is correct

Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape. Surface area refers to an open surface or the perimeter of a three-dimensional object, whereas plane region or plane area refers to a shape or planar lamina.

given

the width(w) of the rectangular garden= [tex]x[/tex][tex]-2[/tex]

area of the rectangular garden = [tex]x^{3} -2x-4[/tex]

The area of a rectangle is,

area = length × width(w) ,

If l is the rectangle's length,

w is the width of the rectangle,

By substituting values,

[tex]x^{3}-2x-4[/tex] = ([tex]x-2[/tex]) *  length ( l )

[tex]l=\frac{x^{3}-2x-4 }{x-2}[/tex] = [tex]x^{2} +2x+2[/tex]   (by long division method)

Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]

option D is correct

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Given the graph of f(x), the range of f⁻¹(x) is; (−∞, 2) ∪ (2, ∞)

The range of a function is defined as the set that is composed by all the output values on a function. Thus, on the graph, the range of a function is composed by the values of y of the function.

For the inverse function, the input and the output are exchanged, and as such given the graph of a function, we can say that the range of the inverse function is given by the domain of the initial function, which is, the values of x of the graph of the original function.

From the description of the rational function, we are told that the asymptote is given as: x = 2

Thus, the domain of the graphed function, would be the same as the range of the inverse function,  will be given by the interval:

(−∞, 2) ∪ (2, ∞).

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a) The option that is a negation of the given statement is; The water temperature is not 70.

b) The conclusion of the given conditional statement is; Keisha likes the dress.

In mathematics, a conditional statement is defined as a statement that can be written in the form “If P then Q,” where P and Q are sentences.

a) We are given the statement as;

The water temperature is 70.

Now, negation is a false/opposite form of the given statement and as such among the options, the only one that is a negation is that

"The water temperature is not 70."

b) We are given the conditional statement as;

If the dress is red, then Keisha likes the dress.

Thus, the conclusion of this is that likes the dress.

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Step-by-step explanation:

To solve the equation -16(d + 1) = -21 for d, we can first use the distributive property to expand the expression -16(d + 1):

-16(d + 1) = -16d - 16

Then, we can set the two sides of the equation equal to each other and solve for d:

-16d - 16 = -21

-16d = -5

d = 5/16

Therefore, the value of d that satisfies the equation -16(d + 1) = -21 is d = 5/16.

Properties of log:

Evaluate given using the properties above:

Q1

Q2

Q3

Answer:

[tex]\textsf{1.} \quad 5 \ln x + 5 \ln y[/tex]

[tex]\textsf{2.} \quad \dfrac{1}{7} \ln t[/tex]

[tex]\textsf{3.} \quad \ln x - \dfrac{5}{2}\ln y \;\; \;\;\textsf{or} \;\;\;\; \ln x - 5\ln y[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{6 cm}\underline{Natural log laws}\\\\Product law: \;$\ln xy=\ln x + \ln y$\\\\Quotient law: $\ln \left(\dfrac{x}{y}\right) = \ln x - \ln y$\\\\Power law: \;\;\;\;$\ln x^n=n \ln x$\\\end{minipage}}[/tex]

Apply the power law followed by the product law:

[tex]\begin{aligned}\ln (xy)^5 & = 5 \ln (xy)\\ & = 5\left( \ln x + \ln y \right) \\ & = 5 \ln x + 5 \ln y\end{aligned}[/tex]

[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}[/tex]

then apply the power law:

[tex]\begin{aligned}\ln \sqrt[7]{t} & = \ln t^{\frac{1}{7}} \\ & = \dfrac{1}{7} \ln t\end{aligned}[/tex]

It is not completely clear where the square root sign begins and ends, so I have provided answers for both permutations:

[tex]\begin{aligned}\ln \left(\sqrt{\dfrac{x^2}{y^5}}\right) & = \ln \left(\dfrac{\sqrt{x^2}}{\sqrt{y^5}}\right) \\\\& = \ln \left(\dfrac{x}{y^{\frac{5}{2}}}\right) \\\\&=\ln x - \ln y^{\frac{5}{2}}\\\\&=\ln x - \dfrac{5}{2}\ln y\end{aligned}[/tex]

[tex]\begin{aligned}\ln \left(\dfrac{\sqrt{x^2}}{y^5}\right)& = \ln \left(\dfrac{x}{y^5}\right) \\\\&=\ln x - \ln y^5\\\\&=\ln x - 5\ln y\end{aligned}[/tex]

Using percentages, we know that out of 20 people, 5 people that is 25% of the random sample consider themselves as republicans.

A percentage is a figure or ratio stated as a fraction of 100 in mathematics.

We must divide the value by the entire value to find the percentage, and then multiply the resulting number by 100.

As an illustration, 1% of 1,000 chickens is equal to 1/100 of 1,000, or 10 birds, and 20% of the quantity is equal to 20% of 1,000, or 200.

So, we have a random sample of 20 people.

n = 20

And we also know that 25% of Americans identify themselves as republicans.

Then the number of republicans in 20 people would be:

20/100 * 25

.2 * 25

5

Therefore, using percentages, we know that out of 20 people, 5 people that is 25% of the random sample consider themselves as republicans.

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Answer:

124 and 228

Step-by-step explanation:

Two numbers will be presented as x and y

x + y = 352

x - y = 104

x + y + x - y = 352 + 104

2x = 456

Divide both side by 2

x = 228

x + y = 352

y = 352 - x

y = 352 - 228

y = 124

A)  A linear function that will yield the compensation y of the sales executive for a given amount of monthly sales x is;

B) The account executive's compensation for $90,000 in monthly sales is; $300

The general equation of a line in slope intercept form is;

y = mx + b

where;

m is slope

b is y-intercept

x and y are provided in the problem: x = monthly sales and y = compensation

We will have to find m (Slope), which represents the commission percentage, and b (y-intercept), which represents the the base salary.

    x₁        y₁         x₂        y₂      

(50000, 6500) (60000, 6800)

Slope;

m = (y₂ - y₁) / (x₂ - x₁)

m = (6800 - 6500) / (60000 - 50000)  

m = 300/10000

m = 3/100 = 0.03 = 3%

To find b, we have to take one of our ordered pairs [ (50000, 6500) or (60000, 6800) ] and plug it into the equation, y=mx+b.

Let's use (60000, 6800)

6800 = 0.03(60000) + b

6800 = 1200 + b  

b = 6800 - 1200

b = $5600

The base salary = $5600

The equation is y = 0.03x + 5600

b) For $90000 in monthly sales, the executive's compensation is;

y = 0.03(90000) + 5600

y = 2700 + 5600

y = $300

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The absolute value equation represented in the graph is

|x + 3|  

Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.

A number can never have a negative absolute value.

The equation is written in the form below

|x|

The a transformation which involves a translation of 3 units to the left took place.  the transformation rule for 3 units to the left is addition of 3, hence we have the equation

|x + 3|

We can therefore conclude that the absolute value equation of the function is |x + 3|

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Answer:

.

Step-by-step explanation:

sorry i hit my daily limit and i need more answers

Therefore ,the the  the work done in propelling a 10-ton satellite to a height of 200 miles above Earth is 1904.7619 million ton.

An expression is composed of a number, a variable, both, or neither, and specific operation symbols. An equation is made up of two expressions, which are separated by an equal sign.

Here,

Use F(x) = C/[tex]x^{2}[/tex]

Where c is a constant and x is the radius of the earth

F(x) is satellite weight

We have to find c.

thus,

=> 10 = c /[tex]4000^{2}[/tex]

=> c =160000000

Therefore,

=> F(x) =  160000000/  [tex]x^{2}[/tex]

Workdone ,

=> W = [tex]\int\limits^{4200}_{4000} {F(x)} \, dx[/tex]

=>W = [tex]\int\limits^{4200}_{4000} {160000000 /x^{2} } \, dx[/tex]

=>W =[tex]\left \{ {{4200} \atop {4000}} \right. \frac{-160000000 }{x}[/tex]

=> W =1904.7619 million ton

Therefore ,the the  the work done in propelling a 10-ton satellite to a height of 200 miles above Earth is 1904.7619 million ton.

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Answer: Measure less 90 degrees

Step-by-step explanation: Less than 90 degrees measure acute angle. 90 degrees in right corner. Obtuse corners are larger than 90 degrees

The estimated distance traveled by the car is 73.5 mi.

To use the Midpoint Rule, we need to divide the time interval into subintervals and evaluate the average velocity over each subinterval.

The time interval is from 0 seconds to 100 seconds, so we will divide this interval into 5 subintervals of length 20 seconds each. The midpoint of each subinterval is the average time, which we can use to estimate the average velocity over that subinterval.

The midpoints of the subintervals are:

Using the data from the table, we can calculate the average velocity over each subinterval:

We can now use the Midpoint Rule to estimate the distance traveled by the car:

distance = (20/6) * (17 + 53 + 53 + 49.5 + 47)

              = (20/6) * 220.5

              = <<20/6*220.5=73.5>>73.5 mi

So, the estimated distance traveled by the car is 73.5 mi.

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First, multiply [tex]18*5 = 90[/tex]

Then, divide [tex]90 / 6 = 15[/tex]

Therefore, each group has 15 students.

Let A be the amount of money invested at a 12% annual interest rate and B be the amount of money invested at a 8% annual interest rate.

We know that A = B + 1500, and the total interest earned by the two investments is $1880.

The interest earned by the investment at a 12% annual interest rate is 0.12 * A = 0.12A

The interest earned by the investment at a 8% annual interest rate is 0.08 * B = 0.08B

Therefore, we can write the following equation to represent the situation:

0.12A + 0.08B = 1880

Since A = B + 1500, we can substitute this into the equation to get:

0.12(B + 1500) + 0.08B = 1880

Solving for B, we get:

0.12B + 180 + 0.08B = 1880

Combining like terms, we get:

0.2B + 180 = 1880

Subtracting 180 from both sides, we get:

0.2B = 1700

Dividing both sides by 0.2, we get:

B = 8500

Since A = B + 1500, we can substitute this value into the equation to find the value of A:

A = 8500 + 1500 = 10000

Therefore, the investor invested $8,500 at an 8% annual interest rate and $10,000 at a 12% annual interest rate.

Answer:

Account A (12%)= $10,000

Account B (8% stock fund) = $8,500

Step-by-step explanation:

Annual Interest Formula

[tex]\large \text{$ \sf I=P\left(1+r\right)^{t} -P$}[/tex]

where:

Account A:

[tex]\implies \sf Interest=(x+1500) (1+0.12)^1-(x+1500)[/tex]

[tex]\implies \sf Interest=(x+1500) (1+0.12)-(x+1500)[/tex]

[tex]\implies \sf Interest=(x+1500)+0.12(x+1500)-(x+1500)[/tex]

[tex]\implies \sf Interest=0.12(x+1500)[/tex]

[tex]\implies \sf Interest=0.12x+180[/tex]

Account B (stock fund):

[tex]\implies \sf Interest=x (1+0.08)^1-x[/tex]

[tex]\implies \sf Interest=x (1+0.08)-x[/tex]

[tex]\implies \sf Interest=x+0.08x-x[/tex]

[tex]\implies \sf Interest=0.08x[/tex]

If the total interest from the two investments was $1880:

[tex]\implies \sf 0.12x+180+0.08x=1880[/tex]

[tex]\implies \sf 0.2x+180=1880[/tex]

[tex]\implies \sf 0.2x=1700[/tex]

[tex]\implies \sf x=8500[/tex]

Therefore, the money invested in each of the two accounts is:

The values of v and w that will make ΔPQR congruent to ΔKJI are:

v = 12; w = 20.

Based on the CPCTC, every corresponding sides and corresponding angles of two congruent triangles are equal to each other.

Therefore, given that ΔPQR ≅ ΔKJI

PR = KI, and QR = JI

Substitute the values:

5v = 6v - 12

5v - 6v = -12

-v = -12

v = 12

w - 1 = 3w - 41

w - 3w = 1 - 41

-2w = -40

w = -40/-2

w = 20

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The required inequality expressions are 7x + 7y ≤ 49 and x + 2y ≤ 13 an the graph has been plotted with clarity.

The expression for inequality can be written by taking the suitable sign for the given problem and then relating the variable part to the constant terms.

(a) Suppose the number of short film and full movie are x and y respectively.

Then, the inequality expression for the given case can be written as follows,

The inequality for money raised is,

7x + 7y ≤ 49

And, the inequality for script writing is,

x + 2y ≤ 13

(b) These inequalities can be graphed as follows,

Hence, for the given case the required system of inequalities is 7x + 7y ≤ 49 and x + 2y ≤ 13 and the graph for the solution is drawn properly.

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Answer:

This is a statement. Not a question.

Step-by-step explanation:

The true statements about the graph are 1, 2, and 3.

A line of the best fit is a straight line that reduces the distance between it and some data. The line of best fit is used to represent a relationship in a scatter plot with numerous data points.

Given:

The temperature above 52° F affects the sale of hot chocolate at her shop,

As you can see from the graph, the y-axis shows the orders for hot chocolate on a day when the outside temperature was 52° F.

From the graph,

at x = 0, y = 160

Thus, Sheryl gets 160 orders for hot chocolate on a day when the outside temperature is 52° F

At x = 0, y = 160 and at x = 16, y = 128

So the difference is 160-128 / 16 - 0 = 2

Thus, Sheryl gets 2 fewer orders for each degree increase above 52° F in the outside temperature.

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Answer:

C) x = 9

Step-by-step explanation:

[tex]\frac{12}{27}[/tex] = [tex]\frac{4}{x}[/tex]

[tex]\frac{12}{27}[/tex] ÷ [tex]\frac{3}{3}[/tex] = [tex]\frac{4}{9}[/tex]  This is the fraction simplified

[tex]\frac{4}{9}[/tex] = [tex]\frac{4}{x}[/tex]  Simplified, I think that it is easier to see that x = 9

Answer:

6x+6x2+3

Step-by-step explanation:

6x2+5x+1+x+2

Combine 5x and x to get 6x.

6x2+6x+1+2

Add 1 and 2 to get 3.

6x2+6x+3

The range of possible values for x is 4/11 < x < 8

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

Triangles = 2

On these triangles, we have the following angles

Angle 1 = 11x - 4

Angle 2 = 84

Angle 1 cannot exceed angle 2

So, we have

11x - 4 < 84

Add 4 to both sides

11x < 88

Divide

x < 8

Angle 1 cannot exceed be negative or 0

So, we have

11x - 4 > 0

Add 4 to both sides

11x > 4

Divide

x > 4/11

Hence, the range is 4/11 < x < 8

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The formula to find the slope is m = (y₂ - y₁)/(x₂ - x₁), hence option B is correct and the slope of the line will be -1/2.

It is possible to determine a line's direction and steepness by looking at its slope. Finding the slope between lines inside a coordinate plane can aid in anticipating if the lines are perpendicular, parallel, or none at all without physically using a compass.

As per the data mentioned in the question,

1.

The formula to find the slope of the line between two points is,

m = (y₂ - y₁)/(x₂ - x₁)

2.

The given points are,

(-4, 3) and (12, -5)

So, the slope of the line that connects given points,

m = (-5 - 3)/(12+4)

m = -8/16

m = -1/2

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Yes, I agree that this equivalent expression has been simplified.

Reorder and gather like terms: (8.4x +6.3 x) +12.6

Collect coefficients for the like terms: (8.4 +6.3)  ×  x +12.6

Calculate the sum or difference: 14.7 x +12.6

Answer : 14.7 x  + 12.6

What exactly is equivalent expression, and how do we recognize it?

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).

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Using combination we get  20475 teams can made up of 3 girls and 2 boys and 37128  teams can be selected with at least 3 girls.

Since this is just a team of 5 children,

we have   no of girls = 15

No of boys = 10

teams made up of 3 girls and 2 boys

girls out of 3 girls can be selected in  ¹⁵C₃ ways and

2 boys out of 10 boys can be selected in  ¹⁰C₂  ways.

[tex]\implies[/tex] ¹⁵C₃ X  ¹⁰C₂

= 455*45 =20475 Ways

Teams can be selected with at least 3 girls

It is the probability of at least 3 girls which means P(3 girls,2 boys) or P(4 girls,1 boy) or P(5 girls, 0 boys)

[tex]\implies[/tex] ¹⁵C₃ X  ¹⁰C₂  + ¹⁵C₄ X  ¹⁰C₁ + ¹⁵C₅ X  ¹⁰C₀

= 455 * 45 +1365*10+3003*1

=20475 +13650 +3003

=37128 ways

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