How many different selections of 4 books can be made from a bookcase displaying 12 books?

If they have a remainder of 4 when divided by six the sequence of terms are: 6n+4

42≤6n+4≤92

subtract 4 from both ends

 38≤6n≤88

divide both ends by 6

 6.3≤n≤14.6

since n must be an integer...

7≤n≤14 so the number of integers is:

14-7+1=8 

they are 6n+4 for n=[7,14]...46,52,58,64,70,76,82,88

There are 8 integers between 42 and 92, inclusive, that have a remainder of 4 when divided by 6. These integers can be expressed in the form of 6n + 4 and can be found by incrementally adding 6, starting from the smallest integer greater than 42 that meets this condition.

To find how many integers from 42 to 92, inclusive, have a remainder of 4 when divided by 6, we need to identify the numbers that fit the condition. These numbers are of the form 6n + 4, where n is an integer. Starting with the smallest number greater than 42 that fits the condition, which is 46 (as 6*7 + 4 = 46), we can find the next by adding 6 to get 52, then 58, and so on, up until we reach the largest number less than or equal to 92 that fits the condition, which is 88 (as 6*14 + 4 = 88).

To find the total count, we calculate (88 - 46) / 6 + 1 = 42 / 6 + 1 = 7 + 1 = 8. Therefore, there are 8 numbers between 42 and 92 that have a remainder of 4 when divided by 6.

83 x 6 = 498

 all six of her scores need to equal at least 498 for an 83 average

83 + 84 + 93 + 77 + 85 = 422

498-422 = 76

 she needs a 76

Final answer:

To find the area of a triangle with sides in a given ratio, first find the value of x by setting up and solving an equation using the perimeter. Then, multiply each side by x to find the base and height of the triangle. Finally, substitute the base and height into the formula for the area of a triangle.

Explanation:

To find the area of a triangle, we can use the formula: Area = 1/2 × base × height. In this case, we don't know the base and height directly, but we do know that the sides of the triangle are in the ratio of 11:16:24. Let's assume the sides are 11x, 16x, and 24x. The perimeter of the triangle is 510 ft, which means 11x + 16x + 24x = 510. Simplifying this equation, we get 51x = 510. Therefore, x = 10. Now we can find the base and height of the triangle by multiplying the corresponding sides by 10. The base is 11x * 10 = 110 ft and the height is 16x * 10 = 160 ft.

Now we can substitute these values into the formula to calculate the area of the triangle. Area = 1/2 × base × height = 1/2 × 110 ft × 160 ft = 8800 square feet.

Answer:0

Step-by-step explanation:

A right triangle can have at most one obtuse angle since it already contains one right angle, and the sum of its angles must be 180 degrees according to geometrical axioms.

The question asks about the maximum number of obtuse angles that can be contained in a right triangle. In a right triangle, one angle is 90 degrees by definition, which is a right angle. According to Theorem 20, the sum of the angles of a triangle is two right angles, which equals 180 degrees. Having an obtuse angle, which is greater than 90 degrees, combined with a 90 degree angle, would exceed 180 degrees when the third angle is added, and this is not possible in a plane geometric figure. Therefore, a right triangle cannot have more than one right angle. As such, the option with the maximum number of obtuse angles a right triangle can contain is:

Answer: B) 1

The answer is

20.7 ft/s

Without the specific function that calculates the object's downward speed on Earth, it is not possible to estimate the impact speed on Mars even with knowing the vertical compression factor.

Estimating Impact Speed on Mars

The question asks us to estimate how fast a rover would hit the surface of Mars after bouncing 15 ft in height, with the Mars function being compressed vertically by a factor of about 2/3 compared to the function used on Earth. Although the specific Earth function isn't provided, we can still proceed with a general understanding. To find this, we would normally use the Earth function for speed and multiply the result by 2/3 to adjust for Mars' weaker gravity. However, since we are only given the option to select an answer, no calculation can be performed without the Earth function. Therefore, it's not possible to provide a confident answer to this question.

Answer:

the input (x) values of the relation

Step-by-step explanation:

(a) No, The output (y) values of the relation are called Range. So it is the wrong option.

(b) Yes, the input (x) values of the relation are called Domain. Thus, it is the correct option.

(c) No, it is not a definition of Domain. Thus, this is an incorrect option.

(d) No, it is not a definition of Domain. It is called the cartesian point. Thus it is also an incorrect option.

Further,

The Domain is the all possible input values of a function that gives defined values.  

The Range is the all defined output values that we get from a function (or y).

To find the time it takes for the object to hit the ground, we can solve the quadratic equation -16t^2 + 1600 = 0 using the quadratic formula. The positive solution to this equation gives us the time in seconds. Therefore, the object will take √102400/32 seconds to hit the ground.

To find the time it takes for the object to hit the ground, we need to solve the equation -16t^2 + 1600 = 0. This is a quadratic equation, so we can use the quadratic formula. Plugging in the values, we get t = (-b ± √(b^2 - 4ac))/(2a). In this case, a = -16, b = 0, and c = 1600. Plugging in these values, we get t = (± √(0 - 4(-16)(1600)))/(2(-16)). Simplifying the equation further, we get t = (± √(0 + 102400))/(32). This gives us two possible values for t: t = √102400/32 and t = -√102400/32. Since time cannot be negative in this context, we can discard the negative solution. Therefore, the object will take √102400/32 seconds to hit the ground.

20+0.15x = 35+0.10x

0.15x=15+0.10x

0.05x=15

x= 15/0.05

x= 300minutes

check:

300*.015 = 45+20=65

300*0.10 = 30 + 35 = 65

they equal

 so number of minutes would be 300

The most appropriate statistical test for examining the relationship between temperature and the number of ice cream cones sold would be a correlation analysis, specifically a Pearson correlation coefficient.

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

Correlation coefficient test measures the strength and direction of the linear relationship between two continuous variables, which is suitable for examining the relationship between temperature and the number of ice cream cones sold.

Additionally, a scatterplot could be used to visually assess the relationship between the two variables before conducting the statistical test.

Hence, statistical test for examining the relationship between temperature and the number of ice cream cones sold would be a correlation analysis

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Answer:

Angles are 34.59°, 85.08° and 60.33°

Step-by-step explanation:

Let ABC is a triangle, ( that show the dog park)

In which,

AB = 150 feet

BC = 98 feet

CA = 172 feet,

By the cosine law,

[tex]BC^2=AB^2+AC^2-2(AB)(AC)cos A[/tex]

[tex]2(AB)(AC)cos A=AB^2+AC^2-BC^2[/tex]

[tex]\implies cos A=\frac{AB^2+AC^2-BC^2}{2(AB)(AC)}-----(1)[/tex]

Similarly,

[tex]\implies cos B=\frac{AB^2+BC^2-AC^2}{2(AB)(BC)}-----(2)[/tex]

[tex]\implies cos C=\frac{BC^2+AC^2-AB^2}{2(BC)(AC)}-----(3)[/tex]

By substituting the values in equation (1),

[tex]cos A=\frac{150^2+172^2-98^2}{2\times 150\times 172}[/tex]

[tex]=\frac{22500+29584-9604}{51600}[/tex]

[tex]=\frac{42480}{51600}[/tex]

[tex]\approx 0.8233[/tex]

[tex]\implies m\angle A\approx 34.59^{\circ}[/tex]

Similarly,

From equation (2) and (3),

m∠B ≈ 85.0°, m∠C ≈ 60.33°

To find h(-12), substitute -12 into the function h(x)=2x²-2. The value of h(-12) is 286.

To find h(-12), we substitute -12 into the function h(x)=2x²-2:

h(-12) = 2(-12)² - 2

h(-12) = 2(144) - 2

h(-12) = 288 - 2

h(-12) = 286

Therefore, h(-12) is equal to 286.

The sine of negative eleven pi divided by twelve is -0.5.

To find the sine of negative eleven pi divided by twelve, we need to determine the exact value of this trigonometric function. The sine function represents the ratio between the length of the side opposite the angle and the hypotenuse in a right triangle. In this case, the angle is negative, which means it lies on the negative y-axis. Remember that the unit circle can help us determine the values of trigonometric functions for any angle. By mapping the angle on the unit circle, we can see that -11π/12 is equivalent to -330 degrees. Since the sine function has a period of 360 degrees, we can find the reference angle by subtracting 360 degrees. So, the reference angle is 30 degrees (360 - 330).The sine of 30 degrees is 0.5. However, the angle is negative, so the sine function is negative. Therefore, the exact value of sine of -11π/12 is -0.5.

Final Answer:

The exact value of [tex]\(\sin\left(-\frac{11\pi}{12}\right)\)[/tex] is:
[tex]\[\frac{\sqrt{2} + \sqrt{6}}{4}\][/tex]

Explanation:

To find the sine of an angle that is not one of the commonly known angles, such as [tex]\(-\frac{11\pi}{12}\)[/tex], we can use sine sum and difference identities. In this case, we know the sine of [tex]\(-\frac{\pi}{12}\)[/tex] is not a standard angle, but we can express it as a sum or difference of angles like [tex]\(-\frac{\pi}{4}\)[/tex] and [tex]\(-\frac{\pi}{3}\)[/tex] whose sines we do know.

The sum identity for sine is:
[tex]\[\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)\][/tex]

We can express the angle [tex]\(-\frac{11\pi}{12}\)[/tex] as the sum of two angles for which we do know the sine and cosine values, for example,[tex]\(-\frac{\pi}{4}\)[/tex] and [tex]\(-\frac{2\pi}{3}\)[/tex] because:
[tex]\[-\frac{11\pi}{12} = -\frac{3\pi}{4} - \frac{2\pi}{3}\][/tex]
Using the sum identity for sine, we can find the sine value as follows:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \sin\left(-\frac{3\pi}{4}\right)\cos\left(-\frac{2\pi}{3}\right) + \cos\left(-\frac{3\pi}{4}\right)\sin\left(-\frac{2\pi}{3}\right)\][/tex]

Now, let's find the values of sine and cosine for these angles.
[tex]\(\sin\left(-\frac{3\pi}{4}\right) = -\sin\left(\frac{3\pi}{4}\right) = -\sin\left(\frac{\pi}{2} + \frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex]since sine is an odd function.

[tex]\(\cos\left(-\frac{2\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\)[/tex] since cosine is an even function.

[tex]\(\cos\left(-\frac{3\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex] since cosine is an even function.

[tex]\(\sin\left(-\frac{2\pi}{3}\right) = -\sin\left(\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)[/tex] since sine is an odd function.

Substitute these trigonometric values into the sum identity:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \left(-\frac{\sqrt{2}}{2}\right)\left(-\frac{1}{2}\right) + \left(-\frac{\sqrt{2}}{2}\right)\left(-\frac{\sqrt{3}}{2}\right)\][/tex]

Simplify the expression:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\][/tex]
Combine the terms with a common denominator:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \frac{\sqrt{2} + \sqrt{6}}{4}\][/tex]

Therefore, the exact value of [tex]\(\sin\left(-\frac{11\pi}{12}\right)\)[/tex] is:
[tex]\[\frac{\sqrt{2} + \sqrt{6}}{4}\][/tex]

Answer: 3/5 is the probability that person's skin cleared up given that they used the medication.

Step-by-step explanation:

let S represents skin cleared and M represents medicine used.

According to the given Venn diagram,

P(S) = 40

P(M)= 50

[tex]P(M\cap S) = 30[/tex]

Thus by the definition of conditional probability,

If it is given that the skin is cleared then the probability that persons used the medicine,

P(M/S)=[tex]\frac{P(M\cap S)}{P(S)}[/tex]=30/50= 3/5.

P(M/S)=3/5.

The probability that the person's skin will clear up given that they used the medication is:

This refers to the likelihood of an event to occur based on certain conditions.

If we want to find the probability about whether the skin will clear up is:

We would assign S to represent the skin

M to represent medicine

Hence,

P(MnS) = 30

Using the conditional probability rule,

P9M/S)= P(MnS)/P(S) = 30/50

=3/5

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Answer: Dilation is a transformation that proportionally reduces or enlarges a figure.

Step-by-step explanation:

It stretches or shrinks the actual figure. It produces similar figures.

Since the corresponding sides of similar figures are in proportion.

⇒ It proportionally reduces or enlarges a figure.

Hence, A dilation is a transformation that proportionally reduces or enlarges a figure.

Final answer:

A scale transformation or dilation is a linear transformation that proportionally enlarges or reduces a figure, maintaining the proportional size relationships within the figure.

Explanation:

A transformation that proportionally reduces or enlarges a figure is known as a scale transformation or dilation. In such a transformation, lines are transformed into lines, and parallel lines remain parallel, consistent with the requirement for a transformation to be linear. This is important because it maintains the proportionality of the figure, meaning the size relationship of the parts of a figure to each other and to the whole figure remains constant, even though the overall size changes. Dilation can be characterized by a scale factor, which dictates how much larger or smaller the figure will become. If the scale factor is greater than 1, the figure enlarges; if it is between 0 and 1, the figure reduces in size.

Keywords:

Polynomial, classify, degree, greatest exponent

For this case we have the following polynomial: [tex]Q (x) = 3x ^ 2 + x-6[/tex], we must classify the polynomial according to its degree. For this, we must bear in mind, that by definition, a polynomial is of the form:

[tex]P (x) = ax ^ n + bx ^ {n-1} + ... + cx ^ 3 + dx ^ 2 + ex + f[/tex]

Where:

a, b, c, d, e, f: They are the coefficients of the polynomial

n, n-1,3,2,1,0: They are the exponents. This polynomial is of degree "n", because "n" is the largest exponent.

x: It is the variable

Thus, [tex]Q(x) = 3x ^ 2 + x-6[/tex]is of degree "2" because "2" is the largest exponent.

Answer:  

It is a quadratic polynomial

Option C

The correct classification for this polynomial is C. quadratic.

The polynomial 3x^2 + x - 6 is classified by its degree, a fundamental characteristic of polynomials determined by the highest power of the variable 'x.'

In this case, the highest power is 2, making it a quadratic polynomial. Quadratic polynomials represent a U-shaped graph when plotted, often described as a parabola.

They play a significant role in various areas of mathematics, science, and engineering, serving as fundamental models for various real-world phenomena.

Quadratic equations are commonly encountered in physics, engineering, economics, and other fields, making them essential for solving problems and making predictions. Thus, the correct classification for this polynomial is C. quadratic.

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