In this triangle, the product of sin B and tan C is _____ , and the product of sin C and tan B is _______.

To approximate functions near zero using the linear approximation (1+x)^k ≈ 1 + kx, we find an approximation for each given function.

To approximate the function f(x) for values of x near zero using the linear approximation (1+x)^k ≈ 1 + kx, we need to find an approximation for each given function:

a.) f(x) = (1-x)^8 ≈ 1 - 8x

b.) f(x) = -8/(1-x) ≈ -8x

c.) f(x) = 1/(sqrt(1+x)) ≈ 1 - 0.5x

d.) f(x) = sqrt(4+x^2) ≈ 2 + x^2/4

e.) f(x) = (6+3x)^(1/3) ≈ 2 + x/3

The probability that you missed your friend’s call is 1/6

The range is given as:

Range = 3:00 P.M. and 5:00 P.M.

The number of minutes in the range is:

Total = 120 minutes

At 3:20 that you notice the phone is off the hook, the number of minutes missed is:

Time missed = 20 minutes

The probability is then calculated as:

P = 20/120

Evaluate

P = 1/6

Hence, the probability that you missed your friend’s call is 1/6

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The value of x is 20 when right triangle is with the hypotenuse labeled x , one angle has a measure of thirty degrees, and the opposite leg has a length of ten.

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

Right ∆ with Hypotenuse is x, Opposite is 10

Reference angle  = 30°.

We need to find the value of x

Since we are given a reference angle with opposite length, and we are to find the hypotenuse length, x, apply the following trigonometric ratios formula:

sin(theta)=opposite side/hypotenuse

Plug in the values into the equation

sin 30 = 10/x

1/2=10/x

Apply cross multiplication

x=20

Hence, the value of x is 20 when right triangle is with the hypotenuse labeled x , one angle has a measure of thirty degrees, and the opposite leg has a length of ten.

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Answer:

Line segment: E

Plane: D

Perpendicular lines: B

Line: A

Parallel lines: C

Step-by-step explanation:

This middle school mathematics question asks for matches between geometric terms and their definitions. Line segments have two endpoints, planes extend infinitely without thickness, perpendicular lines intersect at 90 degrees, lines extend indefinitely in two directions, and parallel lines never intersect.

Explanation:

In the field of mathematics, particularly in geometry, we often deal with various foundational elements such as lines, line segments, planes, and the relationships between them. To understand these concepts, it is essential to know their definitions and how they relate to each other.

A line segment is best described as a part of a line that has two distinct endpoints. This means a line segment is contained and does not extend infinitely in any direction. An example of this is the edge of a ruler, which begins and ends at specific points. The correct match for a line segment in the provided list is E) part of a line that has two endpoints.

A plane in geometry is a flat, two-dimensional surface that extends infinitely in all directions. It's similar to a vast sheet of paper with no edges. It is defined by at least three noncollinear points and is often represented in drawings as a four-sided figure, but it continues without end. The correct definition for a plane is D) a flat surface that extends infinitely and has no thickness.

Perpendicular lines are two lines that intersect at an angle of 90 degrees. The point where they cross forms right angles, and the two lines are always at the same distance from each other at this point. The definition of perpendicular lines is B) two lines that intersect at 90° angles.

Regarding a line, it is an undefined term in geometry that represents a straight path that extends infinitely in both directions with no thickness. It is comprised of an infinite number of points. A simple description of a line is A) a series of points that extend in two directions without end.

Lastly, parallel lines are lines in the same plane that do not intersect, no matter how far they are extended on either side. This is because they are always the same distance apart, maintaining a consistent separation across their entire length. The correct match for parallel lines is C) lines that lie in the same plane and do not intersect.

Answer: The simplified form will be

[tex]\frac{1}{(x+3)(x+4)}[/tex]

Step-by-step explanation:

Since we have given that

[tex]\frac{x+1}{x^2+x-6}\div \frac{x^2+5x+4}{x-2}[/tex]

We need to simplify the above expression, we get that

[tex]\frac{x+1}{x^2+x-6}\div \frac{x^2+5x+4}{x-2}\\\\=\frac{x+1}{x^2+x-6}\times \frac{x-2}{x^2+5x+4}\\[/tex]

Now, we first factorize the quadratic equation:

[tex]x^2+x-6\\\\=x^2+3x-2x-6\\\\=x(x+3)-2(x-3)\\\\=(x+3)(x-2)[/tex]

Similarly,

[tex]x^2+5x+4\\\\=x^2+4x+x+4\\\\=x(x+4)+1(x+4)\\\\=(x+4)(x+1)[/tex]

So, it will become

[tex]\frac{x+1}{x^2+x-6}\times \frac{x-2}{x^2+5x+4}\\\\=\frac{x+1}{(x+3)(x-2)}\times \frac{x-2}{(x+4)(x+1)}\\\\=\frac{1}{(x+3)(x+4)}[/tex]

Hence, the simplified form will be

[tex]\frac{1}{(x+3)(x+4)}[/tex]

The value of integral is  2 * arc sin((x - 2) / 2) + (x - 2) * √(4 - (x - 2)²) / 2 + C.

To solve the integral ∫ √(4x - x²) dx, follow these steps:

Complete the Square: Rewrite the quadratic expression under the square root. Start with: 4x - x²

Rearrange it to the standard quadratic form: -x² + 4x. To complete the square, factor out the negative sign: (x² - 4x)

Next, complete the square inside the parentheses:

x² - 4x = (x - 2)² - 4

So, we get:

(x² - 4x) = - [(x - 2)² - 4] = 4 - (x - 2)²

Therefore:

√(4x - x²) = √(4 - (x - 2)²)

Substitute Variables: Let u = x - 2. Then, du = dx, and the integral becomes:

∫ √(4 - u²) du

Apply the Trigonometric Substitution: Use the trigonometric substitution u = 2 sin θ. Then du = 2 cos θ dθ. Substitute these into the integral:

∫ √(4 - (2 sin θ)²) * 2 cos θ dθ

= ∫ √(4 - 4 sin² θ) * 2 cos θ dθ

= ∫ √(4 (1 - sin² θ)) * 2 cos θ dθ

= ∫ 2 cos θ * 2 cos θ dθ

= ∫ 4 cos² θ dθ

Use Trigonometric Identity: Apply the double-angle identity cos² θ = (1 + cos 2θ) / 2:

∫ 4 cos² θ dθ

= ∫ 4 * (1 + cos 2θ) / 2 dθ

= ∫ (4 + 4 cos 2θ) / 2 dθ

= ∫ (2 + 2 cos 2θ) dθ

Integrate each term:

∫ 2 dθ + ∫ 2 cos 2θ dθ = 2θ + sin 2θ + C

Back-substitute: Replace θ with the original variable x. Recall u = 2 sin θ, so θ = arcsin(u / 2). Since u = x - 2, substitute back:

θ = arcsin((x - 2) / 2)

Therefore:

2 * arcsin((x - 2) / 2) + sin(2 * arcsin((x - 2) / 2)) + C

Using the identity sin(2θ) = 2 sin θ cos θ:

sin(2 * arcsin((x - 2) / 2))

= 2 * (x - 2) / 2 * √(1 - ((x - 2) / 2)²)

= (x - 2) * √(4 - (x - 2)²) / 2

5.27 can be converted to a mixed number 5 27/100, where 5 is the whole number and 27/100 is the decimal part expressed as a fraction, without reducing to lowest terms.

To convert the decimal number 5.27 to a proper fraction or a mixed number, we first recognize that .27 is the decimal part of 5.27. We can write .27 as a fraction by considering it as 27 hundredths, which is 27/100. Therefore, we have the whole number 5 and the fraction 27/100. Combining these, we get the mixed number 5 27/100. Since we are instructed not to reduce to lowest terms, the mixed number remains as is.

The total simple interest earned on a $3000 CD over 30 months with an annual interest rate of 5.5% is $412.50 and the balance at maturity is $3412.50.

To calculate the total amount of simple interest earned on a certificate of deposit (CD) we use the formula I = Prt, where I is interest, P is the principal amount (initial amount of money), r is the annual interest rate (in decimal form), and t is the time in years. For the given 30-month CD purchased for $3000 with an annual simple interest rate of 5.5%, we first convert 30 months to years by dividing by 12 (30 / 12 = 2.5 years).

Using the formula: I = $3000 × 0.055 × 2.5 = $412.50. Therefore, the total interest earned is $412.50.

To calculate the balance at maturity, we add the total interest to the principal: Balance = Principal + Interest = $3000 + $412.50 = $3412.50.

Answer:

One Gallon

Step-by-step explanation:

One Imperial gallon is 4.54 litres so in that case the gallon is greater but one US gallon is only 3.79 litres so in that case the litres is bigger. ... or 1 gallon =1/. 22 =4.545 liters. so one gallon is greater than 4 liters.

Among the options given, B. 45 is not a perfect square because no integer squared equals 45.

The question is asking us to identify which number among the given options is not a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself. Here are the given options with explanations:

Therefore, the number that is not a perfect square in the options provided is B. 45.

The final result is 73 divided by 8 equals 9, with a remainder of 9.

When you divide 73 by 8, use long division to find the quotient and remainder.

1. Write the dividend (73) and the divisor (8) as shown:

                                             8 | 73

2. Start dividing:  Since 8 is larger than 7, we move to the next digit.

3. Bring down the next digit (3) to the right of the remainder:

                                                  8 | 73 | 8

                                                     - 64

                                                     _____

                                                          9

4. The remainder is 9.

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[tex]\huge\textsf{Hey there!}[/tex]

[tex]\mathsf{6 \dfrac{19}{25}}[/tex]

[tex]\mathsf{= \dfrac{6\times25 + 19}{25}}[/tex]

[tex]\mathsf{= \dfrac{150 + 19}{25}}[/tex]

[tex]\mathsf{= \dfrac{169}{25}}[/tex]

[tex]\mathsf{= 169 \div 25}[/tex]

[tex]\mathsf{= 6.76}[/tex]

[tex]\huge\textsf{Therefore, your answer should be: \boxed{\mathsf{6.76}}}\huge\checkmark[/tex]

[tex]\huge\textsf{Good luck on your assignment \& enjoy your day!}[/tex]

Answer:

[tex]m{\angle}D=60^{\circ}[/tex]

Step-by-step explanation:

From the given figure, it can be seen that [tex]m{\angle}A=60^{\circ}[/tex], [tex]m{\angle}B=60^{\circ}[/tex], thus

From ΔABD, using the angle sum property, we have

[tex]m{\angle}A+m{\angle}B+m{\angle}D=180^{\circ}[/tex]

⇒[tex]60^{\circ}+60^{\circ}+m{\angle}D=180^{\circ}[/tex]

⇒[tex]120^{\circ}+m{\angle}D=180^{\circ}[/tex]

⇒[tex]m{\angle}D=180^{\circ}-120^{circ}[/tex]

⇒[tex]m{\angle}D=60^{\circ}[/tex]

therefore, the measure of [tex]{\angle}ADB[/tex] is [tex]60^{\circ}[/tex].

Hence, ΔABD is an equilateral triangle.