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By using CPCTC theorem ∆ ADC congruence ∆ BCD.

According to CPCTC, which stands for "corresponding parts of congruent triangles are congruent," if two or more triangles are congruent, their corresponding angles and sides must also be congruent.

Corresponding parts of Congruent triangles is the abbreviation for this phrase. According to the CPCT theorem, if two or more congruent triangles are selected, their corresponding sides and angles are also selected as congruent.

AD = BC and angle ADC = angle BCD

we have to prove AC = BD

Let's see ∆ ADC and ∆ BCD

Given AD = BC

angle ADC = angle BCD

CD = CD

from S - A - S, we get

∆ ADC congruence ∆ BCD

Therefore, by using CPCTC theorem ∆ADC congruence ∆BCD.

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The binomial are factors of the given polynomial function f(x) = x³ + 3x² -16x- 48.

Factorization in mathematics is the process of dividing a large number into smaller ones that, when multiplied together, give you the original number.

We can check whether the given binomials are factors of a given polynomial or not by factorizing the given polynomial.

[tex]f(x) = x^3 + 3x^2 -16x- 48[/tex]

Factoring out x² and 16, we get

[tex]f(x) = x^2(x + 3) -16(x+3)[/tex]

Now, factoring out (x+3), we get

[tex]f(x) = (x + 3)(x^2-16)[/tex]

Using Algebraic identity [tex]a^2-b^2=(a+b)(a-b)[/tex], we get

[tex]f(x) = (x + 3)(x+4)(x-4)[/tex]

So, it is clear that only the (x-4) binomial is a factor of the given polynomial and (x-3) and (x+2) are not.

Therefore, (x-4) is a factor of the given polynomial and (x-3) and (x+2) are not.

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The coordinates of the point on the directed line segment are (7/2,21/4).

What is a line segment?

A line segment is a path between two points that can be measured. Since line segments have a defined length, they can form the sides of any polygon.

Here, we have

A line segment with coordinates partitioned in some ratio.

Let's say that the ends of the segment are A and B.

We have to find the coordinates of C.

Coordinates of point A are = (2,9)

Coordinates of point B are = (8,-3)

Considering the point C as (x,y) and the partition ratios as m:n i.e 1:3

By applying the section formula, we get

x = mx₂ + nx₁ /(m+n) ,  y = my₂ + ny₁ /(m+n)

x = 1 × 9 + 3 × 2 / (1+3) , y = 1×(-3) + 3 × 8 / (1+3)

x = 14/4,   y = 21/4

x = 7/2 , y = 21/4

Hence, the coordinates of the point on the directed line segment are (7/2,21/4).

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The arc length of the curve [tex]y = ln(cos x)[/tex] from [tex]x = 0[/tex] to [tex]x = \pi /4[/tex] is [tex]\ln\left|\sqrt{2} + 1\right|[/tex]

To find the arc length of the curve[tex]y = ln(cos x)[/tex] from[tex]x = 0[/tex]to [tex]x = \pi /4[/tex], we will use the arc length formula for a curve given by y = f(x):

[tex]\text{Arc length} (L) = \int_a^b \sqrt{1 + \left[ f'(x) \right]^2} \, dx[/tex]

Where:

- a and b are the limits of integration (starting and ending points of the curve).

- f'(x) is the derivative of the function y = f(x).

First, we find the derivative of y = ln(cos x):

[tex]y = \ln(\cos x)\\y' = \dfrac{d}{dx}[\ln(\cos x)][/tex]

To find the derivative of ln(cos x), we use the chain rule:

[tex]y' &= \frac{1}{\cos{x}} \cdot (-\sin{x}) \\&= \frac{-\sin{x}}{\cos{x}}[/tex]

Now, we can find the arc length from x = 0 to x = π/4:

[tex]\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{1 + \left( \frac{-\sin x}{\cos x} \right)^2} \, dx\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{1 + \frac{\sin^2 x}{\cos^2 x}} \, dx\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{\frac{\cos^2 x}{\cos^2 x} + \frac{\sin^2 x}{\cos^2 x}} \, dx\text{Arc length } (L) = \int_0^{\frac{\pi}{4}} \sqrt{\frac{\cos^2 x + \sin^2 x}{\cos^2 x}} \, dx[/tex]

Remember that [tex]cos^2 x + sin^2 x = 1[/tex], so the expression becomes:

[tex]\text{Arc length (L)} = \int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos^2 x}} \, dx[/tex]

Now, simplify the expression inside the square root:

[tex]\text{Arc length (L)} = \int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{\cos x}} \, dx[/tex]

Integrate with respect to x:

[tex]\text{Arc length (L)} = \int_0^{\frac{\pi}{4}} \sqrt{Sec x} \, dx[/tex]

Now, evaluate the integral:

[tex]\text{Arc length} (L) = \ln\left| \sec x + \tan x \right| ~\text{from}~ 0 ~\text{to}~ \frac{\pi}{4}\text{Arc length} (L) = \ln\left| \sec \left( \frac{\pi}{4} \right) + \tan \left( \frac{\pi}{4} \right) \right| - \ln\left| \sec (0) + \tan (0) \right|[/tex]

[tex]\sec\left(\frac{\pi}{4}\right) = \sqrt{2}, \quad \tan\left(\frac{\pi}{4}\right) = 1\\\sec(0) = 1, \quad \tan(0) = 0\\\text{Arc length} (L) = \ln|\sqrt{2} + 1| - \ln|1 + 0|\\\text{Arc length} (L) = \ln|\sqrt{2} + 1|[/tex]

Using the property of natural logarithms, ln(a) - ln(b) = ln(a/b), we get:

[tex]\text{Arc length (L)} = \ln\left|\sqrt{2} + 1\right| - \ln\left|1\right|\\\text{Arc length (L)} = \ln\left|\sqrt{2} + 1\right|[/tex]

The arc length of the curve [tex]y = ln(cos x)[/tex] from [tex]x = 0[/tex] to [tex]x = \pi /4[/tex] is [tex]\ln\left|\sqrt{2} + 1\right|[/tex]

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The amount of saline and water in the solution is given as 27.9 cm³ and 282.1 cm³ respectively.

Given that,
A patient is administered 310cm^3 of a saline and water solution that is labeled as 9% saline.

In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication and division,

here,
According to the condition,
Amount of saline = 0.09 × 310 = 27.9
Amount of water = 310 - 27.9 = 282.1

Thus, the amount of saline and water in the solution is given as 27.9 cm³ and 282.1 cm³ respectively.

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Answer:

8.  x = 9

9.  x = 9

10. x = 36

11.  x = 8

Step-by-step explanation:

Question 8

As the base of the triangle has been bisected, set the expressions either side of the perpendicular bisector equal to each other and solve for x:

[tex]\implies 5x-15=2x+12[/tex]

[tex]\implies 5x-15-2x=2x+12-2x[/tex]

[tex]\implies 3x-15=12[/tex]

[tex]\implies 3x-15+15=12+15[/tex]

[tex]\implies 3x=27[/tex]

[tex]\implies \dfrac{3x}{3}=\dfrac{27}{3}[/tex]

[tex]\implies x=9[/tex]

Question 9

As the base of the triangle has been bisected, set the expressions either side of the perpendicular bisector equal to each other and solve for x:

[tex]\implies 8x-63=4x-27[/tex]

[tex]\implies 8x-63-4x=4x-27-4x[/tex]

[tex]\implies 4x-63=-27[/tex]

[tex]\implies 4x-63+63=-27+63[/tex]

[tex]\implies 4x=36[/tex]

[tex]\implies \dfrac{4x}{4}=\dfrac{36}{4}[/tex]

[tex]\implies x=9[/tex]

Question 10

Angles on a straight line sum to 180°.  Therefore, as the other angle on the line is a right angle, equate the given expression to 90° and solve for x:

[tex]\implies (3x-18)^{\circ}=90^{\circ}[/tex]

[tex]\implies 3x-18=90[/tex]

[tex]\implies 3x-18+18=90+18[/tex]

[tex]\implies 3x=108[/tex]

[tex]\implies \dfrac{3x}{3}=\dfrac{108}{3}[/tex]

[tex]\implies x=36[/tex]

Question 11

Interior angles of a triangle sum to 180°.  Therefore, equate the sum of the given angles to 180° and solve for x:

[tex]\implies 90^{\circ}+(3x)^{\circ}+(8x+2)^{\circ}=180^{\circ}[/tex]

[tex]\implies 90+3x+8x+2=180[/tex]

[tex]\implies 11x+92=180[/tex]

[tex]\implies 11x+92-92=180-92[/tex]

[tex]\implies 11x=88[/tex]

[tex]\implies \dfrac{11x}{11}=\dfrac{88}{11}[/tex]

[tex]\implies x=8[/tex]

Answer:

Step-by-step explanation:

we cant see it so this is not able to answer

Answer:5/25 0.18

and then .66 for 2/3 them multiply both of those numbers

Step-by-step explanation:.467

The area of the rectangle that has a width of 2√3 cm and a length of 5√27 cm is calculated as: 90 cm².

The area of a rectangle is calculated by multiplying the length and the width of the rectangle. It is given by the formula: Area = length × width.

Therefore, we are given the following dimensions:

length of rectangle = 5√27 cm

width of rectangle = 2√3 cm

Substitute the values into the area formula:

Area of rectangle = 5√27 * 2√3

= 5*3√3 * 2√3

= 15√3 * 2√3

Area of rectangle = 90 cm²

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Change in y when x =2 and  Δx= 0.1 is 2.05 and the differential dy when x= 2 and dx = 0.1 is 2

Δy and dy can be calculates as follows

We already knew that Δy = f( x + Δx ) - f(x) and y = f(x)

so we just need to subtitute the values

Δy = f( x + Δx ) - f(x)

Δy = f( 2 + 0.1 ) - f(2) and back to y = f(x) we can subtitute the f(x) by [tex]5x^{2}[/tex] so,

Δy = [tex]5(2.1)^{2} -5(2)^{2}[/tex]

Δy = 2.05

and for the differential,

dy = f'(x) dx  and we need to know diferential from y=[tex]5x^{2}[/tex] is 10x

so we just subtitutes the values

dy = f'(x) dx

dy = f'(2) (0.1) and back to  diferential from y=[tex]5x^{2}[/tex] is 10x so subtitute the f'(x)

dy = 10(2) (0.1)

dy =  2

hence, change in y, Δy is 2.05 an the diferenyial dy is 2

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Answer:

which statement is true about these eight angles that are formed from parallel lines being cut by a transversal the answer is C

The factors of (12+21) and (16x-36) using GCF are 3(4+7) and 4(4x-9) respectively.

According to the question,

We have the following two expressions:

(12+21) and (16x-36)

Now, in order to find the factors of given expressions using GCF, first we should know what is GCF.

GCF stands for Greatest Common Factor which means that the highest number which can divide all the terms of an expression.

So, we have:

12+21

3 can divide both 12 and 21:

3(4+7)

For second expression:

16x-36

4 can divide both the terms:

4(4x-9)

Hence, the factors of (12+21) and (16x-36) using GCF are 3(4+7) and 4(4x-9) respectively.

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The value of the y is 18 when x = 5.

Given,

In the question:

If y varies inversely as x

and, y=9 when x =10

To find the value of y when x = 5

Now, According to the question:

Based on the given conditions:

Formulate:

y = k/x

Calculate and substitute y = 9 , x = 10 into y = k/x

9 = k/10

k = 90

To find the equation of the function:

y = 90/x

Calculate and substitution x = 5 into y = 90/x

y = 18

Hence, The value of the y is 18 when x = 5.

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The run is 9x miles long, while the bike race is 22y miles long.

The distance between two points in mathematics is referred to as the distance. The distance formula, which is simply a derivation of the Pythagorean theorem, which is used to find the length of any one side of a right triangle when you know the lengths of the other two sides, can be used to calculate this distance.

Therefore,

d = vt

where:

d = distance (miles)

v = speed (mph)

t = time (hours)

Let  x = run time

Let  y = bike time

Total time:      x + y = 4

We can create an equation that represents the formula using these variables.

d total = d bike + d run

49 = 9x + 22y

The equation is then only expressed in terms of x or y. We're going to get it in terms of x.

49 = 9x +22(4 - x)

49 = 9x + 88 - 22x

-39 = -13x

3 = x

To determine the distance between each part, enter this value of x into the variables.

run distance = 9x

bike distance = 22(4 - x)

Let x be run time, y be the bicycle race time.

Total time

x+y = 4

Run distance R = 9x

Bicycle race distance  B = 22y

Total distance

9x + 22y = 49

Replace y in the second equation with 4 - x to solve this system.

You'll obtain

y = 1

x = 3

so

R = 9x = 27

B = 22y = 22

Hence, The run is 9x miles long, while the bike race is 22y miles long.

The complete question is:

Suppose you have entered a 49?-mile biathlon that consists of a run and a bicycle race. During your? run, your average velocity is 9 miles per? hour, and during

Suppose you have entered a 49?-mile biathlon that consists of a run and a bicycle race. During your? run, your average velocity is 9 miles per? hour, and during your bicycle? race, your average velocity is 22 miles per hour. You finish the race in 4 hours. What is the distance of the? run? What is the distance of the bicycle? race?

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Answer:

Decimal: 0.08

Fraction: 2/25

Step-by-step explanation:

To convert a percentage to a decimal, we divide the percentage by 100, or move the decimal point to the left 2 times:

8/100 = 0.08

To convert a percentage to a fraction, we need to understand what a percentage is first. A percentage is a "number or ratio expressed as a fraction of 100". This means that percentages are numbers out of 100 (x% = x/100). Therefore, to convert 8% to a fraction, we divide 8 by 100:

8% = 8/100

Next, simplify the fraction

8/100 = 2/25

The equation of the parabola as described in the task content is; f (x) = -3 (x - 5)² + 9.

It follows from the task content that the equation of the parabola which has the same shape as f(x)=-3 x² but with vertex (5,9) in the form f (x) = a (x-h)² + k be determined.

Recall that the vertex of a quadratic equations of the given form is defined by the coordinates (h, k).

On this note, the required equation of the parabola by substitution of h and k is;

f (x) = -3 (x - 5)² + 9.

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The trigonometric equation cos² 2x is equivalent by reduction formulas to the expression 1 - 4 · cos² x + 4 · cos⁴ x.

Herein we find the case of a trigonometric equation involving a double angle that must be reduced into a single one by means of reduction formulas. First, write the trigonometric equation:

cos² 2x

Second, use the trigonometric fundamental formula:

1 - sin² 2x

Third, reduce the double angle by the sine formula for the double angle:

1 - 4 · sin² x · cos² x

Fourth, use again the trigonometric fundamental formula and expand the resulting expression:

1 - 4 · (1 - cos² x) · cos² x

1 - 4 · (cos² x - cos⁴ x)

1 - 4 · cos² x + 4 · cos⁴ x

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The factorized form of the given polynomial is F(x) = 2x^4 + 3x³ + 4x² + 5x + 6.

We are given a polynomial. The highest degree of the polynomial is four. The polynomial is a biquadratic equation. We will try to find a solution by hit and trial method. We will use the positive and negative of the factors of the constant term. The factors of 6 are 1, 2, 3, and 6. One by one, we will substitute -1, 1, -2, 2, -3, 3, -6, and 6 in the polynomial equation and check if the equation becomes zero. We see that none of these numbers make the equation zero. So, the equation cannot be factorized.

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m=y₂-y₁/x₂-x₁ we use this formula for line with slope 4 and y intercept is negative

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

Let us consider a equation with slope 4 and a negative y intercept

y=4x-3

In the above equation 4 is the slope and -3 is the y intercept

x    0      1        2      3

y     -3    1        5      9

We have to plot this graph and the graph is given in attachment.

We get to know the slope is 4 by taking any two points (2,5) and (3,9) and applying in formula

m=y₂-y₁/x₂-x₁

Hence m=y₂-y₁/x₂-x₁ we use this formula for line with slope 4 and y intercept is negative

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You can the greatest common factor and least common multiple of two numbers by prime factorization by finding common prime numbers and the prime numbers that account for both numbers respectively

A prime number has two factors, 1 and itself e.g. 2, 3, 5, 7, 11, etc.

When a number is written as a product of its prime factors, we have the prime factorization of the number e.g 8 = 2×2×2

By writing the prime factorization of two numbers we can find the greatest common factor(GCF) and least common multiple (LCM) of the numbers. For example 8 and 24:

The prime factorization of  8 and 24 is:

8 = 2×2×2

24 = 2×2×2×3

The greatest common factor of the two numbers is 2×2×2 = 8 because 2×2×2 appears in the prime factorization of both numbers

The least common multiple of the two numbers is 2×2×2×3 = 24 because all the elements of the prime factorization of both numbers must be accounted for

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Using the subtraction operation, the difference between 7x² + 3x - 5 and 2x² - 2x - 6 is 5x² + x + 1.

Mathematically, the difference is the result of a subtraction operation.

The difference is the product that results when the subtrahend is deducted from the minuend.

A subtraction operation is one of the basic mathematical operations, including addition, division, and multiplication.

  7x² + 3x - 5

-  2x² - 2x - 6

= 5x² + x + 1

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The point estimate of the population mean is 30 and margin of error for the confidence interval with lower bound 25 and upper bound 35 is 5.

Point Estimate = Upper bound + Lower bound /2

Point Estimate = 35 + 25 /2

Point Estimate = 60/2

Point Estimate =30

Margin of error = Upper bound − Lower bound /2

Margin of error =  35 -25/2

Margin of error = 10/2

Margin of error =5

Therefore the  point estimate is 30 and the margin of error is 5.

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Answer:

Step-by-step explanation:

u=31/3

y=-11/9

Answer:

Where is it

Step-by-step explanation:

we are here to help you

Answer:

x = 9

Step-by-step explanation:

7x + 18 = 81

     -18     -18

------------------------

7x = 63

÷7    ÷7

-----------

x = 9

I hope this helps!

The total change in the car's speed during that time is 1500 per hour

The difference is that expressions, which are essentially mathematical "phrases," do not contain an equal sign. Equations have an equal sign and demonstrate that two mathematical expressions are equivalent. An algebraic equation is 2x + b = 14, whereas 2x + b is an algebraic expression.

One of the three main types of algebraic expressions is the monomial expression. Binary Expression Expression of a polynomial.

In the mathematics curriculum and in mathematics in general, algebraic expressions are crucial. Students must be proficient in computations and the manipulation of algebraic expressions in order to advance and perform well in mathematics. They must also be able to read and write expressions.

Given ,

total change in car =300× 2

                              = 1500 per hour

Therefore the total change in the car's speed is 1500 per hour

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The function f(x) = 12×[tex]5^{x-1}[/tex], x = week number

Given data,

week 1 = 12 visitors

week 2 = 60 visitors

week 3 = 300 visitors

week 4 = 1500 visitors

week 5 = 7500 visitors

The function can be given by :

f(x) = 12×[tex]5^{x-1}[/tex], x = week number

f(1) = 12×[tex]5^{0}[/tex] = 12

f(2) = 12×[tex]5^{1}[/tex] = 60

f(3) = 12×[tex]5^{2}[/tex] = 300

f(4) = 12×[tex]5^{3}[/tex] = 1500

f(5) = 12×[tex]5^{4}[/tex] = 7500

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The translation rule is T<5, 2> and the image of U' is (0,3)

Translation.

The transformation of a shape in a plane that preserves length, which means that the object is transformed without getting its dimensions affected. i.e., it may just be shifted to left/right/up/down is referred as translation.

Given,

The vertices of a rectangle are R(-5, -5), S(-1,-5), T(-1, 1), and U(-5, 1). A translation maps R to the point (0, -3).

Here we need to find the translation rule and the image of U

Here we know the given the vertices of rectangle which are  R(–5, –5), S(–1, –5), T(–1, 1), and  U(–5, 1).

And it is also given that the translation maps R(-5,-5) to the point (0, -3). we have to find the translation rule and the image of U.

In order to find the translation rule,

First we have to identify the Horizontal shift for the given point

That is calculated as,

=> 0 - (-5) = 5 unit

Which means R is shifted to 5 units horizontally.

Now, the Vertical shift is calculated as,

=> -3 - (-5) = 2 units

Which means R is shifted to 2 units vertically.

Therefore, the translation rule is < 5, 2 >  

So, the image of U(-5, 1) is calculated as,

=> -5 + 5 = 0

=> 1 + 2 = 3

Therefore, the point of the image U' is (0, 3)

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