What theorem is used to find the side of a triangle in a right triangle?
A. midsegment theorem
B. base angle theorem
C. triangle proportionality theorem
D. Pythagorean theorem

Answer:

39 bad checks because due to the math, 7 bad checks are received in a day

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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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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Answer:

Step-by-step explanation:

u=31/3

y=-11/9

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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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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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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Answer:

7 cm

Step-by-step explanation:

assuming you require the length of the 3 equal sides.

let one of the equal sides be x then perimeter is

P = 3x + 5

given P = 26 , then

3x + 5 = 26 ( subtract 5 from both sides )

3x = 21 ( divide both sides by 3 )

x = 7

the 3 equal sides are 7 cm each in length

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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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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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 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 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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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:

Step-by-step explanation:

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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a. For selling 1000 shoes, profit earned will be $140,000

Given,

Danielle is engineering a new brand of shoes. For x shoes sold,

in thousands, a profit of p(x) = -3x4 + 4x³ - 2x² + 5x + 10 dollars, in ten thousands, will be earned.

we are asked to determine the profit for selling 1000 shoes = ?

when sold 1000 shoes

X = 1 (in thousands)

substitute X=1 into p(x) = -3x⁴+4x³-2x²+5x+10

thus p(1) = -3×1⁴ + 4×1³ - 2×1² + 5×1 + 10

= -3+4-2+5+10

= 14 (in ten thousand)

so the profit is 14×10000 = 140,000

b. X intercepts means that when he sold X shoes, the profit p(x) equals zero, but the value doesn't make sense, because as long as he sold shoes, the profit would not be zero.

y intercepts means that when he sold zero shoes, the profit he earns,

calculate p(0)=10, the profit is greater than zero , it doesn't make sense, because if he sold zero shoes, there would not be income and profit.

Hence we get the required answers.

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Answer:

5

Step-by-step explanation:

Length is a positive number.  Count the spaces from -2 to -7

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 side x is 6 cm when the triangle inside the parallelogram.

Given that,

In the picture we have triangle inside the parallelogram.

We have to find x side.

The parallelogram is ABCD.

Taking E as a midpoint of the AD.

The triangle is CDE.

The side CD is 6cm and ∠E is 90°.

The side ED is x cm.

Taking the trigonometric ratios that is

SinE= Opposite side/ hypothesis

Sin90° = 6cm /x cm

1= 6cm /x cm

x=6cm

Therefore, The side x is 6 cm when the triangle inside the parallelogram.

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Answer:

Where is it

Step-by-step explanation:

we are here to help you

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 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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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 solution to the questions are:

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

Length = 350 - 2x

Width = x

The area of a rectangle is the product of its dimensions

So, we have

Area = Length x Width

Substitute the known values in the above equation, so, we have the following representation

A = (350 - 2x) * x

Evaluate

A = 350x - 2x²

Differentiate and set to 0

350 - 4x = 0

So, we have

4x = 350

Divide by 4

x = 87.5

Recall that

Length = 350 - 2x

So, we have

Length = 350 - 2 x 87.5

Evaluate

Length = 175

The area is then calculated as

Area = Length x Width

So, we have

Area = 175 x 87.5

Evaluate

Area = 15312.5 square meters

Hence, the area is 15312.5 square meters

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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 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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The equations after the transformations are given as follows:

a) Shifted 7 units up: [tex]y = 3^x + 7[/tex].

b) Shifted 6 units left: [tex]y = 3^{x + 6}[/tex]

c) Reflected over the x-axis and over the y-axis: [tex]y = -3^{-x}[/tex].

There are four cases for the translation of an image, and their rules are listed as follows:

In this problem, the function is:

y = 3^x.

For a shift of 7 units up, we have that:

y = f(x) + 7 = 3^x + 7.

For a shift of 6 units left, we have that:

y = f(x + 6) = 3^(x + 6).

The rules for reflections over the axes are given as follows:

Hence, for a reflection over both axes, the rule is:

y = -f(-x).

Considering the function in this problem, the rule is:

[tex]y = -3^{-x}[/tex]

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The equation which is quadratic in form is, [tex]6(2x+4)^2 = (2x+4)+2[/tex].

What is quadratic equation?

The polynomial equations of degree two in one variable of type

f(x) = ax^2 + bx + c = 0 and with a, b, c ∈  R and a ≠ 0 are known as quadratic equations. It is a quadratic equation in its general form, where "a" stands for the leading coefficient and "c" for the absolute term of f. (x).

Consider the given equations,

[tex]x^4-6x^2-27=0\\3x^4=2x^2\\2(x+5)^4+2x^2+5=0\\6(2x+4)^2=(2x+4)+2\\6x^4=-x^2+5\\8x^4+2x^2-4x=0[/tex]

The first, second, third, fifth and sixth equation are all having 4th degree.

So, by the definition of quadratic equation, these equations are not a quadratic.

Consider, the fourth equation,

[tex]6(2x+4)^2=(2x+4)+2\\[/tex]

Simplifying this, we get

[tex]6(4x^2 + 16x + 16 = 2x + 6\\24x^2 + 96x + 96 = 2x + 6\\24x^2 + 96x - 2x + 96 - 6 = 0\\24x^2 + 94x + 90 = 0[/tex]

This equation having degree 2.

Therefore the equation [tex]6(2x+4)^2 = (2x+4)+2[/tex] is a quadratic equation.

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