5x−4<10give your answer as an improper fraction in its simplest form.

Answer: 36

Step-by-step explanation:

[tex]\frac{1}{3} \pi 15^{2} h=2700\pi \\75h=2700\\h=36[/tex]

According to the given condition the two-digit number is 66.

To find the two-digit number that is 11 times its units digit and has a sum of digits equal to 12, we can use the following steps:

1. Let's represent the two-digit number as XY, where X is the tens digit and Y is the units digit.
2. The number is 11 times its units digit, so we can write the equation: 10X + Y = 11Y.
3. The sum of the digits is 12, which means X + Y = 12.
4. Now, we have two equations with two variables:
   - 10X + Y = 11Y
   - X + Y = 12
5. We can solve for X from the second equation: X = 12 - Y.
6. Substitute the value of X in the first equation: 10(12 - Y) + Y = 11Y.
7. Simplify and solve for Y: 120 - 10Y + Y = 11Y.
8. Combine the Y terms: 120 - 9Y = 11Y.
9. Move all the Y terms to one side: 120 = 20Y.
10. Divide by 20 to get Y: Y = 6.
11. Now, substitute the value of Y back into the X equation: X = 12 - 6.
12. Solve for X: X = 6.

So, the two-digit number is 66.

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To calculate the amount of the sales tax on Sally's purchase, we first need to add the prices of the shoes and the shirt together. So, $12.50 + $23.50 = $36. Then, we need to calculate 8.25% of $36, which is done by multiplying 36 by 0.0825. That gives us a sales tax of $2.97. So, the amount of the sales tax on Sally's purchase is $2.97.

In summary, Sally wants to buy shoes for $12.50 and a shirt for $23.50, and the sales tax is 8.25% on a purchase of 50 points. The amount of the sales tax on Sally's purchase is $2.97. In order to calculate the sales tax, we added the prices of the items together and then calculated 8.25% of that total.

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Answer:

a b e

Step-by-step explanation:

The expected value of x is given as follows:

E(X) = 1.6.

The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.

The distribution for this problem is given as follows:

Hence the expected value is given as follows:

E(X) = -7 x 0.2 - 3 x 0.1 + 3 x 0.4 + 7 x 0.3

E(X) = 1.6.

The table is given by the image presented at the end of the answer.

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The combinations of side lengths that would NOT form a triangle with vertices X, Y, and Z is 7 mm , YZ = 14 mm , XZ = 25 mm.

option A.

The lengths of triangle are determined base a given set of rules;

let a, b, and c be the side lengths of a triangle;

Based on the rules of side lengths of triangles, the sum of length a and b must be greater than c, or the sum of a and c must be greater than b or the sum of b and c must be greater than a.

For option A;

7 mm + 14 mm < 25 mm (this cannot be)

For option B;

11 mm + 18 mm > 21 mm (this will work)

For option C;

11 mm + 14 mm > 21 mm (this will work)

For option D;

7 mm + 14 mm > 17 mm (this will work)

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Answer:

a_48000cm^3

b_48000000mm^3

Step-by-step explanation:

first of all, let's find the base area:

Ab=((b+B)h)/2=((10cm+30cm)20cm)/2=400cm^2

then, to find the volume, we need to multiplicate the base area to the height of the prism:

V=Ab*H=400cm^2*120cm=48000cm^3=48000000mm^3

The surface area of octagonal pyramid with height of 12 and a side length of 4.24 is 281.477 unit²

Height of octagonal pyramid = 12

Side length of octagonal pyramid = 4.14

The surface area of octagonal pyramid is  

SA = 2s²( 1 + √2) + 4sh

Here, s is side length of the octagonal pyramid = 4.14

h is height of the octagonal pyramid = 12

putting the values in the equation we get

SA = 2 × 4.14 ( 1 + √2 ) + 4 × 4.14 × 12

SA = 82.757 + 198.72

SA = 281.477

The surface area of octagonal pyramid is 281.477

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The maximum height the golf ball reached before landing back on the ground is 3.75 yards.

To find the maximum height the golf ball reached before landing back on the ground, we need to find the vertex of the quadratic function[tex]h(t) = -0.6t^2 + 3t.[/tex] The vertex of a quadratic function in the form of[tex]f(x) = ax^2 + bx + c[/tex] is given by the formula x = -b/(2a).

In this case, a = -0.6 and b = 3. Plugging these values into the formula:

t = -3 / (2 * -0.6) = 3 / 1.2 = 2.5

Now that we have the time at which the ball reaches its maximum height, we can plug this value back into the height function to find the maximum height:

[tex]h(2.5) = -0.6(2.5)^2 + 3(2.5) = -0.6(6.25) + 7.5 = -3.75 + 7.5 = 3.75[/tex]

So, the maximum height the golf ball reached before landing back on the ground is 3.75 yards.

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If the area of a triangle is half of its base times height then the reasonable value of x is = 4.

A triangle-shaped table top with an area of 324 square inches has a base of 8x+4 inches and a height of 4x+2 inches. Given that the area of a triangle is half of its base times height, we can use the formula:

Area = (1/2) * base * height

Plug in the given values:

324 = (1/2) * (8x + 4) * (4x + 2)

Now, we need to solve for x. Follow these steps:

1. Multiply both sides of the equation by 2 to get rid of the fraction:

2 * 324 = (8x + 4) * (4x + 2)

2. Simplify the equation:

648 = (8x + 4) * (4x + 2)

3. Expand the equation by multiplying the terms in the parentheses:

648 = 32x^2 + 16x + 16x + 8

4. Combine like terms:

648 = 32x^2 + 32x + 8

5. Move all terms to one side of the equation to set it equal to zero:

32x^2 + 32x - 640 = 0

Now, we need to find a reasonable value of x. You can solve this quadratic equation using factoring, the quadratic formula, or a graphing calculator. Using a graphing calculator, we find that x is approximately 4.

Therefore, a reasonable value of x in this situation is 4.

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Answer:

Step-by-step explanation:

The probability of drawing 2 red cards from a standard 52-card deck can be calculated as follows:

There are 26 red cards in the deck, so the probability of drawing a red card on the first draw is 26/52.

After the first card is drawn, there are 25 red cards remaining in the deck out of 51 total cards, so the probability of drawing a red card on the second draw is 25/51.

To find the probability of both events happening together (drawing 2 red cards), we multiply the probabilities of each event:

(26/52) * (25/51) = 0.245 or approximately 24.5%

Therefore, the probability of drawing 2 red cards in a standard 52 card deck is approximately 24.5%.

The critical point is c = 2, the function is decreasing on the interval 0 < x < 2, and increasing on the interval x > 2.

To find the critical point of the function y = x^2 - 4x, we first need to find its derivative, which represents the slope of the tangent line at any point on the curve.

The derivative of y with respect to x is:

y' = 2x - 4

Now, we need to find the critical points, which occur where the derivative is zero or undefined. In this case, the derivative is a polynomial, so it is never undefined. To find where it equals zero, we set y' equal to zero:

0 = 2x - 4

Solving for x, we get:

x = 4/2 = 2

So, the critical point is c = 2.

Now, we need to determine if the function is increasing or decreasing on the interval x > 0. To do this, we can analyze the sign of the derivative. If y' > 0, the function is increasing; if y' < 0, the function is decreasing.

For x > 2 (to the right of the critical point), the derivative y' = 2x - 4 is positive (since 2x > 4 when x > 2). Therefore, the function is increasing on the interval x > 2.

For x < 2 (to the left of the critical point), the derivative y' = 2x - 4 is negative (since 2x < 4 when x < 2). Therefore, the function is decreasing on the interval 0 < x < 2.

In summary, the critical point is c = 2, the function is decreasing on the interval 0 < x < 2, and increasing on the interval x > 2.

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The distance between the tip of the ladder to the ground or the value of 'x' is 13.27 ft.

We know that the ladder is leaning on the wall and thus it makes a right-angle triangle, where:

the hypotenuse(h) is the length of the ladder,

the base(b) is the distance between the foot of the ladder and the bottom of the wall,

and the height(x) is the distance between the tip of the ladder to the bottom of the wall which we need to find.

As the question is on right angled triangle we can use the Pythagoras theorem to find the value of 'x':

[tex]Height^2 + Base^2 = Hypotenuse^2\\x^2 + b^2 = h^2[/tex]

Now we know that h= 15ft, and b=7ft.

Substituting the values in the above equation we get :

[tex]x^2 + 7^2 = 15^2\\x^2 = 225 - 49\\x = \sqrt{176}\\x= 13.27 ft.[/tex]

Therefore the distance between the tip of the ladder to the ground or the value of 'x' is 13.27 ft.

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The training program with the least rate of change is Rachel's, with an increase of 3 minutes per day.

Rachel:
Starting minutes: 30
Increase in rate: 3 minutes per day
Equation: f(x) = 3x + 30

Nicole:
Starting minutes: 30 (since f(0) = 5(0) + 30 = 30)
Increase in rate: 5 minutes per day
Equation: f(x) = 5x + 30

To find the training program with the least rate of change, we need to find the derivative of each equation and set it equal to zero:

f'(x) = 3 for Rachel's equation
f'(x) = 5 for Nicole's equation

Since 3 is less than 5, Rachel's training program has the least rate of change. Therefore, the rate of change in minutes per day for Rachel's training program that has the least rate of change is 3 minutes per day.

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Manuel can type 68 words in two minutes at a rate of 34 words per minute.

Given that; Manuel types at a rate of 34 words per minute.

To determine how many words Manuel can type in two minutes, we simply need to multiply his typing rate by the number of minutes he is typing.

Since Manuel is typing for two minutes

Hence;

Number of words = Typing rate × Time

Plugging in the values we have from the problem.

Number of words = 34 words/minute × 2 minutes

Simplifying

Number of words = 34 words × 2

Number of words = 68 words

Therefore, he can type 68 words in two minutes.

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Now we need to apply the given boundary conditions to obtain the specific solution for u(x, y):

Boundary conditions in x-direction:
X(2) = X(27) = 0

Boundary conditions in y-direction:
Y(0) = Y(2m) = 0

1. Identify the Poisson equation and boundary conditions.
2. Use the method of separation of variables to solve the equation.
3. Apply the boundary conditions to obtain the specific solution.

Step 1: Identify the Poisson equation and boundary conditions
The given Poisson equation is:
Δu + 2g = 27 - 1,
where Δu is the Laplacian of the potential function u(x, y).

The provided boundary conditions are:
u(2, 0) = u(2, 2m) = 0,
u(0, y) = u(27, y) = 0.

Step 2: Use the method of separation of variables
We assume that the solution u(x, y) can be written as a product of two functions, one depending on x and the other depending on y, i.e., u(x, y) = X(x)Y(y).

Now, let's substitute this into the Poisson equation:
Δu + 2g = 27 - 1,
which becomes
(X''(x)/X(x) + Y''(y)/Y(y)) + 2g = 26.

Separate the variables:
X''(x)/X(x) = -Y''(y)/Y(y) - 2g = λ,
where λ is the separation constant.

This gives us two ordinary differential equations:
X''(x) = λX(x),
Y''(y) = -(λ + 2g)Y(y).

Step 3: Apply the boundary conditions
Now we need to apply the given boundary conditions to obtain the specific solution for u(x, y):

Boundary conditions in x-direction:
X(2) = X(27) = 0

Boundary conditions in y-direction:
Y(0) = Y(2m) = 0

Solving these equations with their respective boundary conditions will give us a specific solution for the potential function u(x, y). However, it is important to note that solving these equations involves solving eigenvalue problems and possibly infinite series expansions. The full solution process is quite involved and goes beyond the scope of this answer.

Nevertheless, I hope this outline of the solution method helps you understand the process of solving the Poisson equation with given boundary conditions.

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The derivative of y=f(x) = x⁴ - 7x + 5 is dy/dx = 4x³ - 7. For x = 5 and Δx=0.2, dy = 1.986 and Δy = -54.5504.

Given the function y = f(x) = x⁴ - 7x + 5, we can find its derivative with respect to x using the power rule of differentiation:

dy/dx = d/dx(x⁴) - d/dx(7x) + d/dx(5) = 4x³ - 7

Now, we can use the given value of x = 5 and Δx = 0.2 to find the values of dy and Δy:

dy = (4x³ - 7) dx, evaluated at x = 5 and Δx = 0.2

dy = (4(5)³ - 7) (0.2) = 198.6 × 10^(-2)

This means that a small change of 0.2 in x results in a change of about 1.986 in y.

To find Δy, we use the formula:

Δy = f(x + Δx) - f(x)

Substituting x = 5 and Δx = 0.2, we get:

Δy = ((5 + 0.2)⁴ - 7(5 + 0.2) + 5) - (5⁴ - 7(5) + 5)

Simplifying this expression gives:

Δy = (122.4496 - 177) = -54.5504

This means that a small change of 0.2 in x results in a change of about -54.5504 in y.

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B. Whole number enlarges, fraction reduces.

When a figure is dilated by a whole number, the image is enlarged by a factor of that whole number. For example, if a figure is dilated by a scale factor of 2, the image will be twice as large as the original.

On the other hand, when a figure is dilated by a fraction, the image is reduced. For example, if a figure is dilated by a scale factor of 1/2, the image will be half as large as the original.

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Answer:

38/93

Step-by-step explanation:

11/31 x 38/33

11 x 38 = 418

31 x 33 = 1023

= 418/1023

Simplifying

The simplified form of 418/1023 is 38/93.

38/93 is your final answer.

36 bags of soil are required to spread a 6 feet layer over a garden bed that is 4 feet by 3 feet.

Given information:

The garden bed has a length of 4 feet.

The garden bed has a width of 3 feet.

The layer of soil to be spread over the garden bed is 6 feet.

One bag of soil contains 2 cubic feet of soil.

To find the number of bags of soil required to spread a 6 feet layer over the garden bed, we need to calculate the volume of soil needed and then divide it by the volume of each bag of soil.

The volume of soil needed can be calculated by multiplying the length, width, and height (depth) of the soil:

Volume = length x width x depth

Volume = 4 feet x 3 feet x 6 feet

Volume = 72 cubic feet

This means we need a total of 72 cubic feet of soil to spread a 6 feet layer over the garden bed.

Next, we need to determine the number of bags of soil required. Since each bag contains 2 cubic feet of soil, we can divide the total volume of soil needed by the volume of each bag to get the number of bags required:

Number of bags = Volume of soil needed / Volume of each bag

Number of bags = 72 cubic feet / 2 cubic feet per bag

Number of bags = 36 bags

Therefore, 36 bags of soil are required to spread a 6 feet layer over a garden bed that is 4 feet by 3 feet.

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The height of a cone with a diameter of 12m whose volume is 226m³ is 6 meters.

The formula for the volume of a cone is

V = (1/3) * π * r^2 * h

where V is the volume, r is the radius, h is the height, and π is approximately equal to 3.14.

We know the diameter of the cone is 12m, which means the radius is 6m.

We also know that the volume of the cone is 226m^3.

Substituting these values into the formula, we get:

226 = (1/3) * π * 6^2 * h

Simplifying:

226 = (1/3) * 3.14 * 36 * h

226 = 37.68h

h = 226/37.68

h ≈ 6

Therefore, the height of the cone is approximately 6 meters.

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The number of minutes you can use the card is 9 minutes

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

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

f(x) = 0.25x + 0.75

If the total cost of the card is $5.00, the number of minutes you is

0.25x + 0.75 = 5

So, we have

0.25x = 4.25

Divide by 0.25

x = 9

Hence, the number of minutes is 9

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The area of garage is 36,504 square inches.

The width of Scott's garage is 156 inches (since it is 1.5 times shorter than the length of 234 inches).

so width = 234/1.5

=> 156

To find the area, we multiply the length by the width:

=> 234 inches x 156 inches

=> 36,504 square inches.

To explain, the formula for finding the area of a rectangle is length x width. In this problem, we are given the length of the garage as 234 inches and are told that the width is 1.5 times shorter than the length.

To find the width, we can multiply the length by 1.5 to get 351 inches (which is longer than the length, so we know it must be incorrect). Instead, we need to divide the length by 1.5 to find the width, which gives us 156 inches. Then, we can multiply the length by the width to find the area of the garage, which is 36,504 square inches.

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The distance from P to the airplane is y = x + 100 ≈ 628.38 m.

A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry.

From a certain point Q on a straight road, which forms an angle of 22° with the horizontal, the angle of elevation of an airplane at point A is 57°. At the same instant from another point Q, 100 meters ahead of the first point, the angle of elevation of the airplane is 63°. The points P, Q, and A are in the same vertical plane. Find the distance from P to the airplane.

To solve the problem, we can use the concept of similar triangles. Let's call H the height of the airplane and x the distance from Q to the airplane. Then, the distance from P to the airplane is given by y = x + 100.

From triangle QA1H, we have:

tan(57°) = H / x

From triangle QA2H, we have:

tan(63°) = H / (x + 100)

Dividing these two equations, we get:

tan(57°) / tan(63°) = x / (x + 100)

Solving for x, we get:

x = 100 * tan(63°) / (tan(63°) - tan(57°)) ≈ 528.38 m

Therefore, the distance from P to the airplane is:

y = x + 100 ≈ 628.38 m.

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We know that the total mass of 5 polished stones would be 7 grams according to this equation

Sure! The equation m = 1.4p represents the mass, m, in grams, of p polished stones.

This means that if you know the number of polished stones, p, you can use this equation to calculate their total mass, m, in grams. For example, if you have 5 polished stones, you can plug in p = 5 and solve for m: m = 1.4 x 5 = 7 grams.

So the total mass of 5 polished stones would be 7 grams according to this equation.

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Based on the equation, the number of grams which the mass increase for every 7 polished stones is: D. 9.8 g.

In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;

y = mx + c

Where:

Based on the information provided above, a linear equation that models the mass in grams is given by;

y = mx + c

m = 1.4p

By substituting the given parameter (x = 7 polished stones), we have the following:

m = 1.4(7)

m = 9.8 g

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

x = √(6^2 + 18^2) = √(36 + 324) = √360

= 6√10

Answer:

[tex]x = 6\sqrt{10}[/tex]

Step-by-step explanation:

You can find the height using [tex]c^2 = a^2 + b^2[/tex] formula.

[tex](6\sqrt{2})^2 = 6^2 + b^2.[/tex]

[tex]b^2 = 72-36=36.[/tex]

[tex]b=6.[/tex]

You can find x using the same formula.

[tex]x^2 = 6^2 + 18^2 = 360.[/tex]

[tex]x = 6\sqrt{10}[/tex]

The equation of the parabola is:

The two points that define the latus rectum are (±9/64, 4).

The equation of the parabola with vertex at (0,0) and axis of symmetry the y-axis can be written in the form x = ay^2, where a is a constant. Since the parabola contains the point (6,4), we can substitute these values to solve for a:

6 = a(4²)

6 = 16a

a = 6/16 = 3/8

So the equation of the parabola is x = (3/8)y².

To find the two points that define the latus rectum, we need to determine the focal length, which is the distance from the vertex to the focus.

Since the axis of symmetry is the y-axis, the focus is located at (0, f), where f is the focal length. We can use the formula f = a/4 to find f:

f = a/4 = (3/8)/4 = 3/32

So the focus is located at (0, 3/32). The two points that define the latus rectum are the intersections of the directrix, which is a horizontal line located at a distance of f below the vertex, with the parabola. The directrix is located at y = -3/32.

To find the intersections, we can substitute y = ±(16/3)x^(1/2) into the equation of the directrix:

y = -3/32

±(16/3)[tex]x^(^1^/^2^)[/tex]= -3/32

x = 9/64

So the two points that define the latus rectum are (±9/64, 4).

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In right triangle RST, the value of tan (s) is 12/5.

To find tan(s), we first need to determine which side is opposite angle S and which side is adjacent to angle S.

In this case, RT is the side opposite angle S, and ST is the side adjacent to angle S. Since tangent (x) or tan(x) is defined as the ratio of the length of the opposite side to the length of the adjacent side, we can write the formula for tan(s) as follows:

tan(s) = (opposite side) / (adjacent side)

Now we can plug in the given side lengths to calculate the value of tan(s):

tan(s) = RT / ST
tan(s) = 12 / 5

Thus, tan(s) = 12 / 5.

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The given statement "A function f(x) = 3x^4 dominates g(x) = x^4" is True, which means that as x gets larger, the value of f(x) will increase much more rapidly than the value of g(x).

As x increases or decreases, the 3x^4 term in f(x) will grow faster or be larger in magnitude than the x^4 term in g(x). Since f(x) grows faster or has a larger magnitude than g(x), we can conclude that f(x) dominates g(x).

Therefore, the function f(x) = 3x^4 has a higher degree than g(x) = x^4

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The photo takes up 30% of the framed area.

To calculate the percentage of the frame that is taken up by the photo, we first need to calculate the dimensions of the framed photo. If the photo measures 4 inches by 6 inches, the dimensions of the framed photo will be 8 inches by 10 inches (adding 2 inches to each side).

To calculate the area of the frame, we need to subtract the area of the photo from the area of the framed photo. The area of the photo is 4 inches x 6 inches = 24 square inches. The area of the framed photo is 8 inches x 10 inches = 80 square inches. So, the area of the frame is 80 square inches - 24 square inches = 56 square inches.

To calculate the percentage of the frame that is taken up by the photo, we divide the area of the photo by the area of the framed photo and multiply by 100.

24 square inches / 80 square inches x 100 = 30%

Therefore, the photo takes up 30% of the framed area.

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