Find the distance between the points (-3, -6) and (1, -2).

One positive number is 5 more than another. The sum of their squares is 53. Then the larger number is 8.


Lets take x be the smaller number, then x + 5 = the larger number. Thus, we have the equation:
x2 + (x + 5)2 = 53
By Simplifying the equation, we get: 2x2 + 10x + 25 = 53
By Solving this quadratic equation, we get x = 3 and the larger number is 8.

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).

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The volume of Sphere A is V(A) = (4/3)πr³ and the volume of Sphere B is V(B) = 125(4/3)πr³. So the volume of Sphere B is 125/1 = 125 times larger than that of Sphere A.

what is sphere?

A sphere is a three-dimensional geometric shape that is perfectly round, like a ball. It is defined as the set of all points in space that are equidistant from a given point, called the center of the sphere.

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius.

Let the radius of Sphere A be denoted by "r", then the radius of Sphere B is 5r.

The volume of Sphere A is V(A) = (4/3)πr³.

The volume of Sphere B is V(B) = (4/3)π(5r)³ = (4/3)π125r³ = 5³(4/3)πr³ = 125(4/3)πr³.

Therefore, the volume of Sphere B is 125/1 = 125 times larger than that of Sphere A.

So, the volume of Sphere B is 125 times larger than that of Sphere A.

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The exact value of tangent of 11 times pi over 12 is  [tex]-2+\sqrt{3}[/tex].

We have to find the exact value of [tex]tan\frac{11\pi}{12}[/tex]

[tex]tan\frac{11\pi}{12}=tan\frac{6\pi+5\pi}{12}[/tex]

[tex]tan\frac{11\pi}{12}=tan(\frac{6\pi}{12}+\frac{5\pi}{12})[/tex]

[tex]=tan(\frac{\pi}{2}+\frac{5\pi}{12})[/tex]

We have [tex]tan(\frac{\pi}{2}+x)=-cotx[/tex]

[tex]tan\frac{11\pi}{12}=-cot(\frac{5\pi}{12})[/tex]

[tex]tan\frac{11\pi}{12}=-cot(\frac{2\pi+3\pi}{12})[/tex]

[tex]tan\frac{11\pi}{12}=-cot(\frac{\pi}{6}+\frac{\pi}{4})[/tex]

We have identity [tex]cot(x+y)=\frac{cotxcoty-1}{coty+cotx}[/tex]

[tex]tan(\frac{\pi}{2}+x)= -\frac{cot\frac{\pi}{6}cot\frac{\pi}{4}-1}{cot\frac{\pi}{4}+ cot\frac{\pi}{6}}[/tex]

[tex]=\frac{-\sqrt{3}(1) - 1}{1+\sqrt{3}}[/tex]

Rationalize the denominator and simplify, we get:

[tex]=-2+\sqrt{3}[/tex]

Therefore, the exact value of  [tex]tan\frac{11\pi}{12}[/tex] is [tex]-2+\sqrt{3}[/tex].

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Complete question is:

Find the exact value of  [tex]tan\frac{11\pi}{12}[/tex] ?

Answer:

20

Step-by-step explanation:

you add them and divide by how ever many you added

The circumference and area of the circle with the arc of length 21 meters and a central angle of 287° are 4.35 meters and 1.16 square meters.

An arc of length 21 meters subtends a central angle of 287° in a circle. The circumference of the circle and area are required to be determined.

Here is the solution:

Let us suppose that the radius of the circle is r. Thus, we can write:

r = (L/θ) r = (21/287)

Let us substitute the values of r in the formulae of the circumference of the circle and area of the circle.

Circumference of the circle:

C = 2πr

C = 2π (21/287)

C ≈ 4.35 meters

Area of the circle:

A = πr²

A = π (21/287)²

A ≈ 1.16 square meters

Therefore, the circumference of the circle is 4.35 meters and the area of the circle is 1.16 square meters.

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Answer:

M<E = 63

M<F = 63

M<G = 117

Step-by-step explanation:

M<F:

= 180-117

=63

M<E:

=180-117

=63

M<G:

=180-63

=117

The x-intercepts are approximately (-1.08, 0) and (-0.42, 0), and the y-intercept is (0, -5).

To find the x-intercepts of a parabola, we set y = 0 and solve for x. Similarly, to find the y-intercept, we set x = 0 and solve for y.

Setting y = 0 in the given equation, we get,

0 = −10x^2 − 16x − 5

We can solve for x by using the quadratic formula:

x = [-(-16) ± sqrt((-16)^2 - 4(-10)(-5))] / (2(-10))

Simplifying, we get,

x = [-(-16) ± sqrt(256 - 200)] / (-20)

x = [-(-16) ± sqrt(56)] / (-20)

x = [8 ± 2sqrt(14)] / (-10)

So the x-intercepts are:

x = [8 + 2sqrt(14)] / (-10) ≈ -1.08

x = [8 - 2sqrt(14)] / (-10) ≈ -0.42

To find the y-intercept, we set x = 0 in the given equation:

y = −10(0)^2 − 16(0) − 5

y = -5

So the y-intercept is -5.

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The slope of the line that passes through the pair of points (7/20,5/19) and (5/9,11/16) is 3/5.

To find the slope of a line passing through two points, we need to use the slope formula. The slope formula is m = (y2 - y1) / (x2 - x1). To calculate the slope, we need to first calculate the x and y values of the two points, (7/20,5/19) and (5/9,11/16). The x values are 7/20 and 5/9 and the y values are 5/19 and 11/16. We can then plug these values into the slope formula to get the slope. The slope is (11/16 - 5/19) / (5/9 - 7/20) = 3/5. Therefore, the slope of the line that passes through the two points (7/20,5/19) and (5/9,11/16) is 3/5.

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According to the given information, the function g(x) that represents a horizontal stretch of 3 on f(x) = 2x is [tex]\rm g(x) = \frac{2}{3}x$[/tex]. Thus, option A is correct.

A functiοn is a relatiοn between a set οf inputs and a set οf pοssible οutputs, with the prοperty that each input is related tο exactly οne οutput.

Tο create a hοrizοntal stretch οf 3 οn the functiοn f(x) = 2x, we can replace x in the functiοn with x/3, which will cause the οutput tο take three times the input value. This gives us the functiοn:

[tex]$$ g(x) = f\left(\frac{x}{3}\right) = 2\left(\frac{x}{3}\right) = \frac{2}{3}x $$[/tex]

Therefore, the function g(x) that represents a horizontal stretch of 3 on f(x) = 2x is [tex]g(x) = \frac{2}{3}x$.[/tex]

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Complete question:

Create a stretch of 3 on function f(x) = 2x to make g(x)

[Group of answer choices]

a. [tex]\rm g(x) = \tfrac{2}{3}x$[/tex]

b.  [tex]\rm g(x) = \tfrac{3}{2}x$[/tex]

c.  [tex]\rm g(x) = {2}x^3$[/tex]

d. [tex]$\rm g(x) = \sqrt[3]{{2}x} $[/tex]

The answer of the given question based on finding the new coordinate of c is  (1, 5). and to determine if the image of Triangle ABC is similar or congruent to the original triangle is AB / A'B' = BC / B'C' = CA / C'A' = √(2).

Congruent triangles are triangles that have the same shape and size. More specifically, if two triangles have exactly the same size and shape, then they are congruent triangles. This means that all the corresponding sides and angles of two triangles are equal. Congruent triangles can be thought of as being identical to each other, but they may be positioned and oriented differently in space.

To rotate a point 180° degrees counterclockwise about the origin, we can simply multiply its coordinates by the matrix:

[ -1  0 ]

[  0 -1 ]

So, the coordinates of the rotated point C would be:

[ -1  0 ]   [ 3 ]

[  0 -1 ]*[ 8 ] =[ -3 ]

           [  1 ]

we translate this point 4 units to the right and 4 units up by adding 4 to the x-coordinate and 4 to the y-coordinate:

[ -3 + 4 ]

[  1 + 4 ] = [ 1, 5 ]

Therefore, the new coordinates of point C are (1, 5).

To determine if the image of triangle ABC is similar or congruent to the original triangle, we can check if the ratios of the corresponding sides are the same. Let's calculate the lengths of the sides of the original triangle:

AB = √((8-3)² + (3-3)²) = 5

BC = √((3-8)² + (8-3)²) = 5*√(2)

CA = √((3-3)² + (3-8)²) = 5

Now, let's calculate the lengths of the sides of the image triangle, using the new coordinates:

A'B' = √((8-1)² + (3-5)²) = 5*√(2)

B'C' = √((1-1)² + (5-8)²) = 3

C'A' = √((1-8)² + (5-3)²) = 5*√(2)

We can see that the ratios of the corresponding sides are the same:

AB / A'B' = BC / B'C' = CA / C'A' = √(2)

Therefore, the image of triangle ABC is similar to the original triangle.

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Answer:

In the equation X + y = 9, X and y are variables. Variables are used to represent unknown quantities or quantities that can vary. In this case, X and y represent two unknown numbers whose sum is equal to 9. We can use algebraic techniques to solve for either X or y, or both, depending on the situation. The values of X and y could represent anything, such as the number of apples and oranges in a basket, the length and width of a rectangle, or the ages of two people.

Option 1 is correct, The point will be on 10 for the cost $240.

=> 0—————————-10—————————-20

By multiplying the data οbtained with such a nanοsectiοn variable whilst alsο twο that used the unitary methοd , the jοb can be finished. In essence, this mean that whenever a desired item appears, the specified entity is set οr the cοlοur space οf bοth prοductiοn runs is skipped. A varying fee οf Inr ($1.01) might have been necessary fοr fοrty pens.

Here,

Greg wants tο cοllect enοugh cash tο spend $210 οn a new bicycle. He needs tο make an extra $150 because he already has $60. He needs tο mοw the fοllοwing lawns in οrder tο make his $15 per lawn:

=> 150 ÷ 15 = 10 acres

Greg must mοw at least 10 yards in οrder tο accοmplish his οbjective. The number line shοwing hοw many fields he can mοw is as fοllοws:

=> 0—————————-10—————————-20

The number οf lawns he can mοw tο reach his οbjective is represented by any value οn this number line larger than οr equal tο 10. He will, fοr instance, be cοmpensated:

=> 12 × $15 = $180

He already has $60, sο after adding this, he will have:

=> $60 + $180 = $240

which will cοver the cοst οf the new mοtοrbike.

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Answer:

hey, good afternoon!

Step-by-step explanation:

here is an example: the cost of a pizza is 7.50 and you can add a topping for 0.65 cents. find the cost of the pizza with 6 toppings

y=mx+b

the x stands for the toppings and y is the cost. the rate is 0.65 cents and the orginal amount of money is b. so the equation would look like this:

y= .65x+7.50. x=6

y = 65(6)+7.50

    =3.9+ 7.50= $11.50(answer)

the m is the rate of change in this question is 0.65 cents

b is initial/ fixed/ value once again in this qeustion it 7.50$  

hope this helps. have a great day!

Answer:

the velocity of the man is 5 m/s.

Step-by-step explanation:

The kinetic energy of a moving object is given by the equation:

K.E. = 0.5 * m * v^2

where

K.E. is the kinetic energy

m is the mass of the object

v is the velocity of the object

In this problem, we are given that the man has a mass of 60 kg and a kinetic energy of 750 J.

So,

750 J = 0.5 * 60 kg * v^2

Solving for v:

v^2 = (2 * 750 J) / 60 kg

v^2 = 25 J/kg

v = sqrt(25 J/kg)

v = 5 m/s

Therefore, the velocity of the man is 5 m/s.

Answer:

x + y = 110

Step-by-step explanation:

[tex]x=\sqrt{(65^{2} - 56^{2} } \\x=33\\\\\\y=\sqrt{(85^2-36^2} \\y=77\\\\x+y=33+77\\x+y=110[/tex]

The height of the cylinder is 27 yards.

To find the height of a cylinder, we need to use the formula V = πr^2h, where V is the volume, π is the constant pi (3.14), r is the radius, and h is the height.

In this problem, we know the radius is 5 yards, and the volume is 1,177.5 cubic yards. We can rearrange the formula to calculate the height as h = V / (πr^2). Substituting the given values into the equation, we get: h = 1177.5 / (3.14 * 25) = 27 yards. Therefore, the height of the cylinder is 27 yards.

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The image of the point after the reflection is (-3, -6).

To reflect a point across the y-axis, we simply negate its x-coordinate while keeping the y-coordinate the same.

Thus, the image of point (3, -6) reflected across the y-axis would be (-3, -6).

We can visualize this by drawing a line passing through the point (3, -6) and the y-axis.

The reflection of the point will be at the same distance from the y-axis but on the opposite side.

So, the reflection of the point (3, -6) across the y-axis is (-3, -6).

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Answer:

Step-by-step explanation:

its b

If a bag contains 3 red, 4 green, and 5 white marbles, then the probability that a green marble will be selected in the first attempt is equal to 4/12 or 1/3.

Probability is defined as the likeliness of an event to happen. It simply means possibility of any event to occur or the possibility of an event to not occur. The sum of the possibilities in any event is always equal to one. In the given case, there are 3 red, 4 green, and 5 white marbles.

Now total number of marbles is equal to 3+4+5 = 12 marbles

Probability that the green marble is picked first = 4/12 = 1/3

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probability-=0.3419

The probability that the number of defective bulbs in a sample of 550 light bulbs is greater than 15 is 0.3419. To calculate this probability, you need to use the binomial probability formula: P(x) = nCx * p^x * q^(n-x), where n is the sample size (550), p is the probability of a defective bulb (0.035), x is the number of defective bulbs you want to calculate (15) and q is the probability of a non-defective bulb (1-0.035). Plugging in the values, we get P(x) = 550C15 * 0.035^15 * 0.965^(550-15) = 0.3419

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Answer:

To recoup the cost of buying 200 canned drinks, Jack needs to make a profit of zero, meaning that the revenue from selling the drinks needs to equal the cost of buying them.

Let's start by setting the profit equation equal to zero and solving for x:

0 = 1.25x - 100

1.25x = 100

x = 80

Therefore, Jack needs to sell 80 cans of drinks to recoup the cost of buying 200 canned drinks.

Step-by-step explanation:

The values of the variables are x = 2, z = 6.92, x = 10.5 and y = 12.6

Figure 12

In this figure, we make use of the following theorem of intersecting secants

AB * AC = AD * AE

By substitution, we have

(x + 4) * (x + 4 + x - 2) = 4 * 9

Evaluate the sum

(x + 4) * (2x + 2) = 36

So, we have

(x + 4) * (x + 1) = 18

Expand

x^2 + 5x + 4 = 18

Evaluate the like terms

x^2 + 5x - 14 = 0

Factorize

(x + 7)(x - 2) = 0

Solve for x

x = -7 or x = 2

x cannot be negative

So, we have

x = 2

Figure 13

In this figure, we make use of the following theorem of intersecting secant-tangent

XU * WU = UV^2

So, we have

8 * 6 = z^2

Evaluate

z^2 = 48

Take the square root

z = 6.92

Figure 14

Using the same theorem used in (13), we have

2 * (2 + x) = 5^2

So, we have

4 + 2x = 25

Evaluate

2x = 21

Divide

x = 10.5

Figure 15

Using the same theorem used above

y^2 = 8 * 20

So, we have

y^2 = 160

Take the square root

y = 12.6

Hence, the value of y is 12.6

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Answer:

72 degrees

Step-by-step explanation:

If a complete angle is 360 degrees, one-fifth of a complete angle would be:

Therefore one-fifth of a complete angle is 72 degrees.

Answer:

x = 11/6 or 1 5/6 or 1.83333...

Step-by-step explanation:

The solution to the equation 5x^2 = 30x – 45 is x = 3.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

We have the Equation: 5x² = 30x - 45.

We can start by simplifying the equation by subtracting 30x and adding 45 from both sides to get:

5x² - 30x + 45 = 0

We can further simplify by dividing both sides by 5 to get:

x² - 6x + 9 = 0

This can be factored as (x - 3)(x - 3) = 0, which means that the only solution is x = 3.

Therefore, the solution to the equation 5x^2 = 30x – 45 is x = 3.

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To solve the equation, factor [tex]\bold{x^2+2x-35}[/tex] use the formula [tex]\bold{x^2 +(a + b) x + ab = (x + a)(x + b)}[/tex]. To find a and b, set up a system to be solved.

[tex]a + b = 2[/tex]

[tex]ab = -35[/tex]

Since ab is negative, a and b have opposite signs. Since [tex]a+b[/tex] is positive, the positive number has a greater absolute value than the negative. Show all pairs of integers whose product is −35.

Calculate the sum of each pair.

The solution is the pair that gives sum 2.

Rewrite the factored expression [tex](x + a)(x + b)[/tex] with the values obtained.

To find solutions to equations, solve [tex]x-5=0[/tex] and [tex]x+7=0[/tex].

The "solutions" of a Quadratic Equation are the values where the equation equals zero. They are also called "roots", or even "zeros".

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Answer:

We can use the Pythagorean theorem to solve this problem. Let's assume that Cari's house is at point A, the airport is at point B, and her grandparents' house is at point C. We know that the distance from A to B is 45 miles and the distance from A to C is 45 miles. We want to find the distance from B to C, which we can call x.

We can form a right triangle with sides AB, AC, and BC. The distance we want to find, x, is the length of the hypotenuse of this triangle (side BC).

Using the Pythagorean theorem, we know that:

AB^2 + AC^2 = BC^2

Substituting in the values we know:

45^2 + 45^2 = x^2

Simplifying:

2025 + 2025 = x^2

4050 = x^2

Taking the square root of both sides:

x = sqrt(4050)

x is approximately equal to 63.64 miles (rounded to two decimal places).

Therefore, the direct distance from the airport to Cari's grandparents' house is approximately 63.64 miles.

If Miller is going to install baseboard around the perimeter of her room's rectangular floor, s.  it will cost Miller $19.20 to purchase baseboard for her room, assuming an average cost of $0.30 per linear foot requiring baseboard.

To find out how much it will cost Miller to purchase baseboard for her room, we can use the CER (Conversion, Equation, and Re-substitution) strategy.

Conversion:

We can calculate the perimeter of the room by adding the lengths of all four sides:

Perimeter = 2(Length + Width) - 2(Doorway Width)

Perimeter = 2(15 + 20) - 2(3)

Perimeter = 64 feet

So, Miller will need 64 feet of baseboard.

Equation:

To calculate the cost of the baseboard, we can use the formula:

Cost = Length * Cost per linear foot

In this case, the length is 64 feet and the cost per linear foot is $0.30.

Cost = 64 * 0.30

Cost = $19.20

Therefore the final answer is that it will cost Miller $19.20 to purchase baseboard for her room, assuming an average cost of $0.30 per linear foot requiring baseboard.

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Thus, cos x/2 = (sqrt(2))/2 has the solution for x =  4kx ± π/2 in the intervals  [0, 2pi).

The ratio of the hypotenuse to the side adjacent to the acute angle together in right triangle is the cosine function, which is a trigonometric function.

This cosine function for only a specified range of angles is represented graphically by the cos graph. A trigonometric graph's horizontal axis is the angle, which is typically denoted by the symbol, and the y-axis is really the sine function of just that angle. The maximum and minimum values are 1 and 1.

There are amplitudes for both the sine and cosine functions. This is due to the scope limitations of each function. Both functions typically have an amplitude of 1, though this can vary depending on the way the function is constructed.

Given function:

cos x/2 = (sqrt(2))/2

OR,

cos x/2 = √2/2

cos x/2 = cos (2kx ± π/4) , k ∈ R (real number)

x/2 = 2kx ± π/4

x =  4kx ± π/2

Thus, cos x/2 = (sqrt(2))/2 has the solution for x =  4kx ± π/2 in the intervals  [0, 2pi).

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