The sum of two numbers is 84. The square of the first number is 6 more than the second number.
Write a system of equations to find the value of x, the first number, and y, the second number.
y = -x +
y = x² +

Answer:

=======================

Connect the center with one end of the chord, this gives us a right triangle with hypotenuse of x, one leg of 6 and another leg of 24/2 = 12.

Use Pythagorean to find the value of x:

Correct choice is C.

Answer:

x = 13.4

Step-by-step explanation:

Definitions

Radius: a straight line from the center of the circle to its circumference.

Chord: a straight line joining two points on the circle.

Perpendicular:  at a right angle (90°).

Isosceles triangle: a triangle with 2 sides of equal length and 2 congruent base angles.

From inspection of the given diagram:

If lines are drawn from the center of the circle to the endpoints of the chord, an isosceles triangle is created with base of 24 units and height of 6 units.  

The isosceles triangle is made up of two right triangles with base of 12 units and height of 6 units.  The radii are the hypotenuse of these right triangles.  Therefore, to calculate the radius of the circle (x), use Pythagoras Theorem.

Pythagoras Theorem

[tex]\sf a^2+b^2=c^2[/tex]

where:

Therefore:

[tex]\sf \implies 6^2+12^2=x^2[/tex]

[tex]\implies \sf x^2=36+144[/tex]

[tex]\implies \sf x^2=180[/tex]

[tex]\implies \sf \sqrt{x^2}=\sqrt{180}[/tex]

[tex]\implies \sf x=\pm 6\sqrt{5}[/tex]

As length is positive:

[tex]\implies \sf x = 6\sqrt{5}=13.4\:units\:(nearest\:tenth)[/tex]

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35miles Rick have to run to burn one pound of adipose tissue.

According to the given question.

Rick runs 5miles to burn 500 calories.

⇒ 1 mile burns the amount of calories = 500/5 = 100 calories.

As we know that

1 pound = 3,500 calories.

So the number of miles Rick needed to run to burn 1 pound of adipose tissue = 3500/100 = 35miles

Hence, 35miles Rick have to run to burn one pound of adipose tissue.

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Using a calculator, the line of best fit for the function is given by:

y = 51.7x - 5.7.

To find the equation, we need to insert the points (x,y) in the calculator. For this problem, a linear regression is used because the data only increases.

From the given table, the points are:

(1, 68), (2,97), (3, 134), (4, 176), (5, 241), (6,335).

Inserting these points on the calculator, the line of best fit for the function is given by:

y = 51.7x - 5.7.

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Answer:

(-1, 5) and (3, -3)

Step-by-step explanation:

The solutions are all the points where the two functions intersect. Since the lines intersect at two different points, there are two solutions.

The two points at which the functions appear to intersect are (-1, 5) and (3, -3).

Answer:

-41

Step-by-step explanation:

5(14-23)+8 ÷ 2

=5(-9)+4

=-45+4

=-41

The solution of the given expression 5(14-23)+8 ÷ 2 is -41.

Usually, simplification involves proceeding with the pending operations in the expression.

Example; 5 + 2 is an expression whose simplified form can be obtained by doing the pending addition, which results in 7 as its simplified form.

Simplification usually involves making the expression simple and easy to use later.

WE have given an expression as;

5(14-23)+8 ÷ 2

Solve the bracket then division;

=5(-9) + 4

=-45+4

=-41

Therefore, the solution of the given expression is -41.

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The value of dividing x^2-3x+9 by x-2 is x + 3

There are several ways to divide polynomial functions; some of these ways include

The expression for the polynomial division is given as:

x^2-3x+9/x-2

To divide polynomial functions, we make use of the division by factorization method

Start by expanding the numerator of the polynomial division

x^2-3x+9/x-2 = x^2 + 3x - 6x + 9/x-2

Factorize the equation

x^2-3x+9/x-2 = x(x + 3) - 2(x + 3)/x - 2

Factor out x + 3 from the numerator

x^2-3x+9/x-2 = (x- 2)(x + 3)/x - 2

Cancel out the common factors

x^2-3x+9/x-2 = x + 3

Hence, the value of dividing x^2-3x+9 by x-2 is x + 3

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we verified the intermidiate value theorem applies to the function f(x) = x^2 + 7x  + 1 . And the value of c is 2.

According to the given question.

We have a function.

f(x) = x^2 + 7x  + 1

As, we know that "the Intermediate Value Theorem (IVT) states that if f is a continuous function on [a,b] and f(a)<M<f(b), there exists some c∈[a,b] such that f(c)=M".

Now, we will apply the theorem for the given function f(x).

So,

f(0) = 0^2 +7(0) + 1 = 1

And,

f(9)=9² + 7(9) + 1 = 81 + 63 + 1 = 145

Here,  f(0) = 1< 19< 145 = f(9).

So, f is continous since it is a polynomial. Then the IVT applies, and such c exists.

To find, c,

We have to solve the quadratic equation f(c) =19.

This equation is

c² + 7c + 1 = 19.

Rearranging, c²+ 7c - 18=0.

Factor the expression to get

c² + 9c - 2c -18 = 0

⇒ c(c + 9) - 2( c + 9) = 0

⇒ (c - 2)(c + 9) = 0

⇒ c = 2 or -9

c = -9 is not possible beacuse it is not in the interval [0, 9].

So, the value of c is 2.

⇒ f(2) = 2^2 + 7(2) + 1 = 4 + 14 + 1 = 19

Hence, we verified the intermidiate value theorem applies to the function f(x) = x^2 + 7x  + 1 . And the value of c is 2.

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Answer: [tex]\frac{2a+6}{8a}[/tex]

Step-by-step explanation:

[tex]\frac{2(a+3)}{8a}=\frac{2a+6}{8a}[/tex]

Answer:

463833

expand

Medium

Solution

verified

Verified by Toppr

In △ABD and △ACD, we have

DB=DC ∣ Given

∠ADB=∠ADC ∣ since AD⊥BC

AD=AD ∣ Common

∴ by SAS criterion of congruence, we have.

△ABD≅△ACD

⇒AB=AC ∣ Since corresponding parts of congruent triangles are equal

Hence, △ ABC is isosceles.

Answer:

A.

Step-by-step explanation:

With the given information, triangles ABD and ACD can be proved congruent, and by CPCTC, segments AB and AC are congruent.

Answer:

Transitive property

The reason that best supports statement 2 in the given proof is division property of equality. Therefore, the correct answer is option D.

Given that, 3(m∠A)=90° and m∠B=m∠A.

We need to prove m∠B=30°.

According to the division property of equality, the quotients of an equation remain equal when both sides are divided by a common real integer that is not equal to 0.

Here,

    Statement:               Reason:

1. 3(m∠A)=90°              1. Given

2. m∠A=30°                 2. Division property of equality

3. m∠B=m∠A              3. Reflexive property

4. m∠B=30°                 4. Transitive Property

Therefore, the correct answer is option D.

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The slope of the scenario is calculated as: -250.

Slope (m) = change in y / change in x = (y2 - y1)/((x2 - x1).

Given the points:

(4, 7,000) = (x1, y1)

(6, 6,500) = (x2, y2)

Plug in the values

Slope (m) = (6,500 - 7,000) / (6 - 4)

Slope (m) = (-500) / (2)

Slope (m) = -250

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Answer:

cot x

Step-by-step explanation:

using the identities

csc x = [tex]\frac{1}{sinx}[/tex]

sin([tex]\frac{\pi }{2}[/tex] - x) = cos x

then

csc x × sin ([tex]\frac{\pi }{2}[/tex] - x)

= [tex]\frac{1}{sinx}[/tex] × cosx

= [tex]\frac{cosx}{sinx}[/tex]

= cot x

Answer:

x = 18

y = 18

Step-by-step explanation:

We are given a right triangle (notice the 90 degree angle), with one of the angles being 45°.

We also know that the hypotenuse (the side opposite of the 90° angle) is 18√2.

We want to find the value of x and y.

First, let's figure out what the value of the other angle is.

The angles in a triangle all add up to 180 degrees.

Let's call the value of the angle we don't know s.
s + 45 + 90 = 180

Add 45 and 90 together.

s + 135 = 180

Subtract 135 from both sides.

s = 45

So the value of the other angle is 45 degrees.

When a right triangle is a 45-45-90 triangle (the numbers referring to the measures of the degrees), there is actually something special about it; if the legs (the sides that make up the 90 degree angle) have the value a (they would both be congruent because a 45-45-90 triangle is isosceles), then the hypotenuse will be a√2.

Keeping this in mind, let's solve for a.

Set 18√2 equal to a√2.

18√2=a√2

Divide both sides by √2.

18 = a

The value of a, aka the legs is a.

Both x and y would be equal to a in this case.

Therefore, x = 18, and y = 18.

The way to solve this is to try to get the vales that would best fill the empty spaces in the diagram. This has been done below

a. log6⁰, log 2 . 1, log 8/3

b. log 1 .4 , 1 , log 4. 6

c. log 9/2, log 3. 5 , log 5⁷

To do this, you have to decide on that specific  number whose logarithm you are trying to determine. Next you have to find the base of that number.

The logarithm of the number is the power that it would have to be raised for us to obtain a different number.

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The expressions 14, 6 + 8, and (6 + 8) are equivalent because they are all equal to 1

14

6 + 8

= 14

(6 + 8)

= 14

The coefficient of (6 + 8) is 1, so, the expression remains the same.

Equivalent expressions refers to those expressions that work the same even though they look different.

The importance or significance of equivalent expression is that its makes expressions easier to work with and solve.

Therefore, the expressions 14, 6 + 8, and (6 + 8) are all equivalent to 14.

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The answer is 25.

The expression is equivalent to [tex]\frac{125^{2} }{125^{\frac{4}{3}} }[/tex] :

[tex]\frac{125^{2} }{125^{\frac{4}{3}} } =125^{2-(\frac{4}{3}) }[/tex]

       [tex]=125^{\frac{2}{3} }[/tex]

 [tex]125=5x^{3}[/tex]

[tex](5^{3} )^{\frac{2}{3}} =5\frac{6}{3} =5^{2} =25[/tex]

Therefore, The answer is 25.

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Answer:

Step-by-step explanation:

The equivalent expression of 21√(3)(15) - 9√(3)(15) is 36√5

The expression is given as:

21√(3)(15) - 9√(3)(15)

Express 15 as 3 * 5

21√(3)(15) - 9√(3)(15) = 21√(3)(3 * 5) - 9√(3)(3 * 5)

Evaluate the product of 3 and 3

21√(3)(15) - 9√(3)(15) = 21√(9 * 5) - 9√(9 * 5)

Take the square root of 9

21√(3)(15) - 9√(3)(15) = 21*3√(5) - 9*3√5)

Evaluate the product

21√(3)(15) - 9√(3)(15) = 63√(5) - 27√5)

Evaluate the difference

21√(3)(15) - 9√(3)(15) = 36√5

Hence, the equivalent expression of 21√(3)(15) - 9√(3)(15) is 36√5

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Answer:

Step-by-step explanation:

rate of change = change in y/ change in x

The rate of change is 0 if the change in y = 0

                        0 = 0 / change in x

The graph is a horizontal line since the y values do not change.

The factorization of given cubic polynomial 6x³ - 11x² - 12x + 5 is:  

6x³ - 11x² - 12x + 5 = (x + 1)(x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])

For given question,

We have been given the cubic polynomial 6x³ - 11x² - 12x + 5

We need to factorize given cubic polynomial.

By the rational roots theorem, any rational zero of f(x) is expressible in the form ± [tex]\frac{p}{q}[/tex] for integers p, q with p a divisor of the constant term 5 and q a divisor of the coefficient 6 of the leading term.

Factors of p = 5: 1, 5

Factors of q = 6: 1, 2, 3, 6

That means that the only possible rational zeros are:

±{ [tex]\frac{1}{1} ,\frac{1}{2} ,\frac{1}{3} ,\frac{1}{6} ,\frac{5}{1} ,\frac{5}{2} ,\frac{5}{3} ,\frac{5}{6}[/tex] }

= ±{ [tex]1 ,\frac{1}{2} ,\frac{1}{3} ,\frac{1}{6} ,5 ,\frac{5}{2} ,\frac{5}{3} ,\frac{5}{6}[/tex] }

We need to find the exact zeros of given cubic polynomial.

For x = 1,

6(1)³ - 11(1)² - 12(1) + 5 = -12

This means, x = 1 is not a zero of given cubic polynomial.

For x = -1,

6(-1)³ - 11(-1)² - 12(-1) + 5 = 0

This means, x = -1 is a zero of given cubic polynomial and (x + 1) is a factor.

To factorize given cubic polynomial we use synthetic division.

The synthetic division (6x³ - 11x² - 12x + 5) ÷ (x + 1) is as shown in following image.

⇒ 6x³ - 11x² - 12x + 5 = (x + 1)(6x² - 17x + 5)

The factors of above quadratic polynomial 6x² - 17x + 5 are:

⇒ 6x² - 17x + 5 = (x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])

So, the factors of given cubic polynomial are:

⇒ 6x³ - 11x² - 12x + 5 = (x + 1)(x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])

Therefore, the factorization of given cubic polynomial 6x³ - 11x² - 12x + 5 is:  6x³ - 11x² - 12x + 5 = (x + 1)(x - [tex]\frac{5}{2}[/tex])(x - [tex]\frac{1}{3}[/tex])

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3.35 feet have a maximum inaccuracy of 0.005 feet.

The correct option is C.

When students consistently commit mistakes, an approach known as error analysis is frequently employed to determine the root of the problem. Reviewing student work is the first step in the process, after which misunderstood patterns are sought out. Mathematics mistakes can be conceptual, procedural, or factual, and they can happen for a variety of reasons.

The window is 3.35 feet long.

Find the biggest error you can

Half of the measuring unit is considered to be the maximum amount of measurement error.

3.35 feet are calculated to the nearest centimeter, or 0.01 feet

Consequently, the largest conceivable error is equal to half of the measurement unit.

= .01/2

= .005

Therefore, the maximum inaccuracy for 3.35 feet is 0.005 feet.

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The indicated sum for the geometric series is 121.5

The series is given as:

162 − 54 + 18 − 6 + .....

Start by calculating the common ratio of the geometric series.

This is calculated using

r = -54/162

Evaluate

r = -1/3

The indicated sum for the geometric series is then calculated as:

S = a(1 - r^n)/1 - r

Substitute the known values in the above equation

S = 162(1 - (-1/3)^7)/1 + 1/3

Evaluate the exponent and the sum

S = 162(1 + 1/2187)/4/3

Evaluate the sum

S = 162(2188/2187)/4/3

Express as products

S = 162 * (2188/2187)* 3/4

Evaluate the product

S = 162 * 547/729

Evaluate the product

S = 121.5

Hence, the indicated sum for the geometric series is 121.5

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Answer:

1/4

Step-by-step explanation:

To transform PQR into P'Q'R, dilate the preimage by 1/4, or shrink it by a scale factor of 4 because 3/12 = 1/4

Answer:

  m∠B ≈ 51.5°

Step-by-step explanation:

A triangle solver can find this answer simply by entering the data. If you do this "by hand," you need to first find length BC using the Law of Cosines. Then angle B can be found using the Law of Sines.

The Law of Cosines tells us ...

  a² = b² +c² -2bc·cos(A)

  a² = 21² +13² -2(21)(13)cos(91°) ≈ 619.529

  a ≈ 24.8903

The Law of Sines tells us ...

  sin(B)/b = sin(A)/a

  B = arcsin(sin(A)×b/a) = arcsin(sin(91°)×21/24.8903)

  B ≈ 57.519°

The measure of angle B is about 57.5°.

Answer:

C

Step-by-step explanation:

the sum of the exterior angles of a polygon = 360°

sum the 4 exterior angles and equate to 360

8x + 9x + 5 + 5x + 4 + 9x + 3 = 360 , that is

31x + 12 = 360 ( subtract 12 from both sides )

31x = 148 ( divide both sides by 31 )

x ≈ 11.23 ( to 2 dec. places )

then exterior angle at B is

9x + 3 = 9(11.23) + 3 ≈ 104° → C

The approximate measure of exterior angle B in the polygon is 104° option (C) is correct.

A polygon is a geometric figure with a finite number of sides in two dimensions. On the sides or edges of a polygon, straight-line segments are joined end to end to form a closed shape. The vertices, also known as corners, are the points where two line segments meet and form an angle.

[tex]\rm Area = |\dfrac{(x_1y_2-y_1x_2)+(x_2y_3-y_2x_3)....+(x_ny_1-y_nx_1)}{2}|[/tex]

It is given that:

A polygon is given in the picture with angle measures.

As we know,

The sum of the exterior angles of a polygon = 360°

8x + 9x + 5 + 5x + 4 + 9x + 3 = 360

31x + 12 = 360

31x = 148

x ≈ 11.23

= 9x + 3 = 9(11.23) + 3

≈ 104°

Thus, the approximate measure of exterior angle B in the polygon is 104° option (C) is correct.

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Applying the SAS congruence theorem and the definition of angle bisector, the complete proof is shown in the image attached below.

The SAS theorem (side-angle-side theorem) state that two triangles are congruent to each other if they have two pairs of congruent sides that are corresponding to each other and a pair of congruent angles that are corresponding to each other.

A line segment that divides an angle into two equal parts is referred to as an angle bisector.

From the proof given, we know that;

m∠DEK = 2(m∠DEF)

m∠DEK = 2(44) = 88°

With the given information and using the reflexive property, we know that triangles DEF and KEF have two pairs of congruent sides that are corresponding to each other and a pair of congruent angles that are corresponding to each other.

Therefore, ΔDEF ≅ ΔKEF based on the SAS congruence theorem. Because they are congruent triangles, DF = KF based on the CPCTC theorem.

Then, KF = 1/2(DK) = 1/2(16) = 8.

The complete proof is show in the image attached below.

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Answer:

36

Step-by-step explanation:

12 times 2 which is 24 then u add 12 cause it's the amount they have together so you get 36