1/4 of 36 bottles of water =

Given expression is:

Take 3 common from it:

So the factor are:

The equation of the tangent plane to the parameterized surface at the point (1, 9, 3) is y + 3z = 27.

First, find the normal vector to the surface at that point.

The normal vector to the surface is given by the cross product of the partial derivatives of the surface with respect to u and v, evaluated at the point (1, 9, 3):

\(n = (∂x/∂u, ∂y/∂u, ∂z/∂u) × (∂x/∂v, ∂y/∂v, ∂z/∂v)\\= (1, 3u², 2u) × (-1, 0, 0)\\= (0, -2u, -3u²)\)

To find the normal vector at the point (1, 9, 3),

Evaluate the expression above with u = 2,

since x = u - v = 2 - v,

y = u³ + 1 = 9, and

z = u² - 1 = 3 at the point (1, 9, 3).

Therefore, the normal vector at the point (1, 9, 3) is:

n = (0, -4, -12)

Now, find the equation of the tangent plane by using the point-normal form of a plane equation:

n · (r - r0) = 0

where r = (x, y, z) is a point on the plane, r0 = (1, 9, 3) is the given point,

· denotes the dot product.

Substituting the values of n and r0, we have

(0, -4, -12) · ([x, y, z] - [1, 9, 3]) = 0

-4(y - 9) - 12(z - 3) = 0

or equivalently:

y + 3z = 27

Hence, the equation of the tangent plane to the parameterized surface at the point (1, 9, 3) is y + 3z = 27.

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Answer:

US = 128.15 ft

Step-by-step explanation:

tan 31° = 77/US

0.6009 = 77/US

0.6009US = 77

US = 128.15 ft

Answer:

128.7

Step-by-step explanation:

A triangle whose sides are all equal is called an equilateral triangle and its area is given by A=\(\frac{\sqrt{3} }{4}\) \(x^{2}\), where x is the side of the triangle.

The area of a rectangle is found by using the formula

A=lw, where l is the length and w is the width.

The given figure can be split into two shapes: a rectangle and an equilateral triangle as shown in the diagram. Let l be the length of the rectangle.

Add all of the sides of the window and equate their result to 16. Then solve for l.

x + x + l + x + l = 16

3x + 2l = 16

2l = 16 - 3x

l = \(\frac{16-3x}{2}\)

The total area of the window will be the sum of the area of the equilateral triangle ABE and the area of rectangle BCDE.

A = \(\frac{\sqrt{3} }{4}\)\(x^{2}\) + lx

Subsitute  \(\frac{16-3x}{2}\) for l into the obtained equation and simplify.

A =\(\frac{\sqrt{3} }{4}\)\(x^{2}\) + ( \(\frac{16-3x}{2}\))x

A = \(\frac{\sqrt{3} }{4} x^{2}\) + 8x - \(\frac{3}{2} x^{2}\)

Differentiate the obtained equation with respect to x and equate the first derivative to 0 to calculate the critical point x .

\(\frac{dA}{dx}\) = \(\frac{\sqrt{3} }{4}\) (2x) + 8 - \(\frac{3}{2}\) (2x)

0 = \(\frac{\sqrt{3} }{2}\) x - 3x + 8

x (3 - \(\frac{\sqrt{3} }{2}\)) = 8

x = \(\frac{16}{6-\sqrt{3} }\)≈3.74887

Again, differentiate the equation

\(\frac{dA}{dx}\) = \(\frac{\sqrt{3} }{2}\)x - 3x + 8  with respect to x.

\(\frac{d^{2}A }{dx^{2} }\) = \(\frac{\sqrt{3} }{2}\) - 3

The value of the second derivative is \(\frac{\sqrt{3} }{2}\)−3, which is negative. This implies that the area will be maximum at the critical point.

Subsitute x = 3.74887 into the equation

l = \(\frac{16-3x}{2}\) and simplify

l = \(\frac{16-(3)(.74887)}{2}\) ≈ 2.3767

The maximum light will enter through the window when the triangular portion will have the side length of 3.74887 feet and the dimensions of the rectangular portion will be 3.74887-ft-by-2.3767-ft.

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Answer:

12 sales

Step-by-step explanation:

a and b

65 per sale maximum commision 1300 so no base

40 per sale base 300

Set up equation:

\(65x=40x+300\)

Solve:

\(25x=300\)

\(x=12\)

So they must have had 12 sales.

Hope this helps :D

m∠f to the nearest degree is 83°.

The Law of Cosines is used to find an angle when all triangle sides are known.

f² = d² +e² -2de cos(F)

To find angle ∠F, we need to find the value of cos(∠F), which we can do by rearranging the Law of Cosines as follows:

cos(F) = (d² +e² -f²) / (2de)

cos(F) = (20² + 25² - 30²) / (2 × 20 × 25)

cos(F) = (400 + 625 - 900) / (1000)

cos(F) = 125/1000

∠F = arccos(1/8)

∠F = 82.8°

Rounding to the nearest degree

∠F = 83°

Hence, m∠f to the nearest degree is 83°.

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No, the sum of the magnitudes of two vectors can never be equal to the magnitude of the sum of the same two vectors.

The magnitude of the sum of two vectors is determined by the vector addition process, which takes into account both the magnitudes and the directions of the vectors. The magnitude of the sum of two vectors is generally greater than or equal to the sum of their individual magnitudes.

Mathematically, for two vectors A and B, the magnitude of the sum (|A + B|) is given by the triangle inequality:

|A + B| ≤ |A| + |B|

Equality in the triangle inequality occurs only when the vectors are collinear, meaning they have the same direction or are in opposite directions. In such cases, the vectors can be scaled such that their magnitudes add up to the magnitude of their sum. However, in general, when vectors have different directions, the sum of their magnitudes will always be greater than the magnitude of their sum.

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Answer:

6.6 cm and 14.6 cm

Step-by-step explanation:

(a)

the length of arc AB is calculated as

AB = circumference of circle × fraction of circle

     = 2πr × \(\frac{95}{360}\)

     = 2π × 4 × \(\frac{95}{360}\)

     = 8π × \(\frac{95}{360}\)

     = \(\frac{8\pi (95)}{360}\)

     ≈ 6.6 cm ( to the nearest tenth )

(b)

the perimeter (P) of sector AOB is

P = r + r + AB = 4 + 4 + 6.6 = 14.6 cm

Step-by-step explanation:

3.

the circumference of a circle is

2×pi×r

r = 4 cm

so, we know, the full circle circumference is

2×pi×4 = 8×pi = 25.13274123... cm

a.

the arc length of AB is the part of the whole circumference that corresponds to 95° out of the full 360° of a whole circle.

arc AB = 8×pi × 95/360 = pi × 95/45 = pi × 19/9 =

= 6.632251158... cm

b.

the perimeter of OAB (the "pie slice") is then the arc AB plus 2 radius lengths (from the end points on the arc to the center of the circle) :

pi × 19/9 + 2×4 = pi×19/9 + 8 = 14.632251158... cm

Answer:

A) \(f(x)=5^x+2\)

Step-by-step explanation:

As exponential functions grow faster than polynomial functions and linear function, A is the only answer that makes sense.

The mean of the numbers means that it is average of the given numbers.

According to the question,

We have the following information:

Mean of the numbers

We know that the mean of a given data set means that we add all the numbers in data set and divide the obtained result by the total numbers.

For example, we have a following data set:

2,5 and 8

Total numbers in the set = 3

Now, the mean of these numbers will be:

Mean = (2+5+8)/3

Mean = 15/3

Mean of the numbers = 5

So, in other words, it can be expressed as the average of the numbers.

Hence, the mean of the numbers means that it is average of the given numbers.

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Answer:

11) 5x + 8 = 7x - 16

24 = 2x

x = 12

7(12) - 16 = 68

90 - 68

m<GJK = 22

12) 4x + 15 + 13x + 7 = 90

17x + 22 = 90

17x = 68

x = 4

4(4) + 15 = 31

90 + 31 + ADE = 180

m<ADE = 59

13) 5x - 12 + 2x - 3 = 90

7x - 15 = 90

7x = 105

x = 15

5(15) - 12 = 63

VWZ + 63 = 90

m<VWZ = 27

14) 9x + 3 = 14x - 27

30 = 5x

x = 6

9(6) + 3 = 57

14(6) - 27 =57

90 - 57 = 33

33 + 33 + DHG = 180

m<DHG = 114

Measure of angles of polygons and quadrilaterals:

1) ∠GJK = 22°

2) ∠ADE = 59°

3) ∠VWZ = 27°

4) ∠DHG = 114°

11)

5x + 8 = 7x - 16

24 = 2x

x = 12

Now substitute the value of x,

7(12) - 16 = 68

90 - 68

∠GJK = 22°

12)

4x + 15 + 13x + 7 = 90

17x + 22 = 90

17x = 68

x = 4

Now substitute the value of x,

4(4) + 15 = 31

90 + 31 + ADE = 180

∠ADE = 59°

13)

5x - 12 + 2x - 3 = 90

7x - 15 = 90

7x = 105

x = 15

Now substitute the value of x,

5(15) - 12 = 63

∠VWZ + 63 = 90

∠VWZ = 27°

14)

9x + 3 = 14x - 27

30 = 5x

x = 6

Now substitute the value of x,

9(6) + 3 = 57

14(6) - 27 =57

90 - 57 = 33

33 + 33 + DHG = 180

∠DHG = 114°

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The value of integer k is -5.

Polynomial equations formed with variables, exponents and coefficients .

Since the polynomial has three integer roots, we can express it as:

(x - r1)(x - r2)(x - r3) = 0

where r1, r2, and r3 are the three integer roots.

Expanding the left-hand side, we get:

x^3 - (r1 + r2 + r3)x^2 + (r1r2 + r1r3 + r2r3)x - r1r2r3 = 0

Comparing this with the given polynomial, we see that:

r1 + r2 + r3 = 1

r1r2 + r1r3 + r2r3 = k

r1r2r3 = 3

First we need to find the value of k. From the first equation, we see that one of the roots must be 1, since the sum of three integers that are not 1 cannot be 1. Without loss of generality, assume that r1 = 1. Then we have:

r2 + r3 = 0

r2r3 = 3

Since the roots are integers, the only possibility is r2 = -3 and r3 = 1.

Therefore, we have:

k = r1r2 + r1r3 + r2r3 = 1(-3) + 1(1) + (-3)(1) = -5

Therefore, the value of integer k is -5.

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Answer: Cube root of 27(top right)

Step-by-step explanation:

take the cube root of each side, to get x by itself and the other side ends of the cube root of 27

x^3 = 27

x = 3√27

= 3

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We can be 95% confident that the true proportion of people who would say yes is between 79.2% and 84.8%.

The 95% confidence interval for the population proportion who would say yes is 0.792 to 0.848. This means that we can be 95% confident that the true proportion of people who would say yes is between 79.2% and 84.8%.

To calculate the confidence interval, we use the following formula:

Confidence interval = (Sample proportion) ± (Zα/2 * Standard error)

where:

The sample proportion is the proportion of respondents who said yes in the sample.

Zα/2 is the critical value for a 95% confidence interval.

The standard error is the standard deviation of the sample proportion.

In this case, the sample proportion is 0.792, the critical value is 1.96, and the standard error is 0.027.

Substituting these values into the formula, we get:

Confidence interval = 0.792 ± 1.96 * 0.027

= 0.792 ± 0.052

= 0.792 to 0.848

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Answer:

(1,0)

Step-by-step explanation:

\(\boxed{midpoint = ( \frac{x1 + x2}{2}, \frac{y1 + y2}{2}) }\)

Thus, midpoint of the line segment is

\( = ( \frac{3 - 1}{2} , \frac{1 - 1}{2} ) \\ = ( \frac{2}{2} , \frac{0}{2} ) \\ = (1,0)\)

The quality which a taylor polynomial have that the tangent line does not necessarily have is that Taylor polynomial has a far better approximation of f(x) near x = a than is the tangent line

Taylor polynomial function simply refers to an infinite sum of terms that are expressed in terms of the function's derivatives at a single point.

So therefore, the quality which a taylor polynomial have that the tangent line does not necessarily have is that Taylor polynomial has a far better approximation of f(x) near x = a than is the tangent line

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Answer:

A) x = -4

Step-by-step explanation:

\(3x-2\left(6-x\right)=7x+2\left(5+x\right)-6\\\\\mathrm{Expand\:}3x-2\left(6-x\right):\quad 5x-12\\\\\mathrm{Expand\:}7x+2\left(5+x\right)-6:\quad 9x+4\\5x-12=9x+4\\\\\mathrm{Add\:}12\mathrm{\:to\:both\:sides}\\5x-12+12=9x+4+12\\\\Simplify\\5x=9x+16\\\\\mathrm{Subtract\:}9x\mathrm{\:from\:both\:sides}\\5x-9x=9x+16-9x\\\\Simplify\\-4x=16\\\\\mathrm{Divide\:both\:sides\:by\:}-4\\\frac{-4x}{-4}=\frac{16}{-4}\\\\Simplify\\x=-4\)

The question asks us to find the a graph which represents the width of the rectangle increasing by 1 unit while the area remains constant. The best graph to model this, would be answer choice C

The graph in option 3 models the length of the new rectangle.

According to the problem,

Now if width becomes (x+1) units

∴ Length will be represented as 10/(x+1)

Now from the given options we need to find the exact graph of              f(x) = 10/(x +1)

Here if x = 4 , y =2

So the point (4 , 2) is satisfied which is only happening in the graph of option 3

Option 3 represents the correct graph

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Answer:

C=2πr

r=C/2π

 198/2π

≈31.51268cm

The solution is, 28 miles can he go.

Given that,

The Yellow Cab Taxi charges a flat rate of $3.50 for every cab ride, plus $0.95 per mile.

Tofi needs a ride from the airport.

He only has $30.10 cash.

let, x miles can he go.

so, for x miles, it will charge:

$3.50 + $0.95 x

now, we have,

He only has $30.10 cash.

so, the inequality will be:

$3.50 + $0.95 x ≤ $30.10

or, $0.95 x ≤ 26.60

or, x  ≤ 28

Hence, The solution is, 28 miles can he go.

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The indefinite integral of tan(x¹⁸ + 5) dx is (-1/18) ln|cos(x¹⁸ + 5)| + C,

We want to find the indefinite integral of tan(x¹⁸ + 5) dx.

Since the derivative of x¹⁸ + 5 is 18x¹⁷, we can try using substitution to simplify the integral.

We let u = x¹⁸ + 5, so that du/dx = 18x¹⁷ and dx = du/18x¹⁷.

Substituting these expressions into the original integral, we get:

∫tan(x¹⁸ + 5) dx = ∫tan(u) (du/18x¹⁷)

Now we can use the identity dx/x² = (-1) d(1/x) to simplify the integral.

Specifically, if we let v = 1/x, then dv/dx = -1/x² and dx = -dv/v².

Substituting these expressions into dx/x², we get:

dx/x² = (-1) d(1/x) = (-1) dv/v²

Substituting this identity into the integral, we get:

∫tan(x^18 + 5) dx = (1/18) ∫tan(u) (du/18x¹⁷)

= (1/18) ∫tan(u) (18x¹⁷ dx)/(18x¹⁷)

= (1/18) ∫tan(u) dx/x²

= (-1/18) ∫tan(u) d(1/x)

= (-1/18) ln|cos(u)| + C

where C is the constant of integration.

Finally, we substitute back in u = x¹⁸ + 5 to get:

∫tan(x¹⁸ + 5) dx = (-1/18) ln|cos(x¹⁸ + 5)| + C.

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Answer:

what's the question

Step-by-step explanation:

is there a photo you can post we need more info

Answer:

Step-by-step explanation:

-15-9+(-7)

answer: A, -31.

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Answer:

-31

Step-by-step explanation:

you can subtract -15 and -9 which equals -24. After that, you can just subtract it by 7 . because +(-)= - so -15-9-7= -31

Answer:

cant read it at all

Step-by-step explanation:

The expression that correctly calculates the perimeter of the shape is given as follows:

P = 2(6s + πr).

In which:

The perimeter of a square of side length s is given as follows:

P = 4s.

Hence, for three squares, the perimeter is given as follows:

P = 3 x 4s

P = 12s.

The perimeter, which is the circumference of a semicircle of radius r, is given by the equation presented as follows:

C = πr.

Hence the perimeter of two semicircles is given as follows:

C = 2πr.

The perimeter of the entire shape is given by the sum of the perimeter of each shape, hence:

P = 12s + 2πr.

P = 2(6s + πr).

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Answer:

3. Sarah met Charlie  Harris, the grandson of Ed and June, and he had a blond hair

Step-by-step explanation:

Answer:

68.1407 us the answer

Step-by-step explanation:

Just add

To solve the equation 4sin(4x) = 2, we can begin by isolating the sin(4x) term. Divide both sides of the equation by 4:

sin(4x) = 2/4

Simplifying further:

sin(4x) = 1/2

Now, we need to find the two smallest positive solutions for 4x that satisfy the equation sin(4x) = 1/2.

The two smallest positive solutions occur when the sine function has a positive value of 1/2. These solutions can be found by considering the unit circle or using inverse trigonometric functions.

Using the unit circle, we know that the sine function is positive in the first and second quadrants. In the first quadrant, the reference angle whose sine is 1/2 is π/6 radians. In the second quadrant, the reference angle whose sine is 1/2 is 5π/6 radians.

To find the values of x, we divide the reference angles by 4:

For A, A = π/6 / 4 = π/24

For B, B = 5π/6 / 4 = 5π/24

Therefore, the two smallest positive solutions are:

A = π/24

B = 5π/24

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Answer:

( x + 4 )^2 + ( y - 15 )^2 = 1; Option A

Step-by-step explanation:

~ Let us apply the standard circle equation ( x - a )^2 + ( y - b )^2 = r^2, provided r ⇒ radius when centered at point ( a, b ) ~

1. If the center is provided to be ( -4, 15 ) we could substitute this value into the circle equation with variables x and y remaining such:

( x - ( - 4 ) )^2 + ( y - 15)^2 = r^2

2. Simplifying this equation, we have: ( x + 4)^2 + ( y - 15 )^2 = r^2

3. Now by point on the circle, it would be a point on the circumference, to be clear in more mathematical terms. So far we have eliminated two options, with two remain - and each of them has either a radius of 1 or 2. If the point on the circle is ( -4, 16 ) that would mean 1 more than 15, which means that with a radius of 1 ⇒ this point will be on the circle.

Answer: ( x + 4 )^2 + ( y - 15 )^2 = 1