If f(x)=3x, g(x)=x+4, and h(x)=x2-1, find the value of h[f(5)]

Solutions refers to the answer to the problem, Real Solutions are realistic and practical whereas ideal solutions are the best cases for the solutions.

In mathematics, a solution refers to a value or set of values that satisfies a given equation, system of equations, or problem. For example, if the equation is x + 2 = 4, then the solution is x = 2, because substituting 2 for x in the equation makes it true. In the case of systems of equations or more complex problems, a solution may be a set of values that satisfies all the equations or conditions of the problem. In such case, the solution may be represented graphically or as coordinates of a point.

In mathematics, an ideal solution refers to a solution that meets all the desired criteria or requirements without any constraints. It is the "best case" scenario. A real solution, on the other hand, refers to a solution that takes into account all the limitations and constraints of a problem. It is a more practical and realistic solution.

Ideal Solutions - Best cases, Usually not possible.

Real Solutions - Practical and possible to attain.

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The value of a is [tex]-\frac{12}{5}[/tex] if 2 is a zero of the polynomial [tex]$4 x^2-2 x+5 a$[/tex].

As per the given data, the polynomial is p(x) = [tex]$4 x^2-2 x+5 a$[/tex]

The polynomial is zero polynomial if x is 2

The places where a polynomial becomes zero overall are known as the zeros of the polynomial. The term "zero polynomial" refers to a polynomial with a value of zero (0).

Replace x with 2 we get

Therefore [tex]4(2)^2-2(2)+5 \mathrm{a}=0[/tex]

[tex]\Rightarrow[/tex] 4 (4) - 2 (2)+5 (a) = 0

[tex]& \Rightarrow[/tex] 16 - 4 + 5 (a) = 0

[tex]& \Rightarrow[/tex] 12 + 5 (a) = 0

Add -12 on both sides we get

12 + 5 (a) - 12 = 0 - 12

5 (a) = -12

Divided by 5 into both sides we get

a = [tex]-\frac{12}{5}[/tex]

Therefore the value of a is [tex]-\frac{12}{5}[/tex]

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let y = number of child tickets so the total 0f 34 child tickets is sold at that day.

To calculate an expression's value by substitution, integers are substituted for letters in the expression.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

let x = number of adult tickets

let y = number of child tickets

x = 4y  (four times as many adult tickets as child tickets

8.4x + 5.1y = 1315.8

but we can substitute 4y for x

8.4(4y) + 5.1y = 1315.8

33.6y + 5.1y = 1315.8

38.7y = 1315.8

y = 34 and x = 4Y

x = 136

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A) Direction of the wind = (bearing of AS + bearing of WS) / 2 = (82 + 79) / 2 = 80.5 ° NE

B)The resulting ground speed and direction of the plane would be 535.5 miles per hour at a bearing of 80.7°E.

To find the speed and direction of the wind, we can use the fact that the ground speed vector (GS) is the sum of the air speed vector (AS) and the wind speed vector (WS).

Given that:

AS = 480 miles per hour at a bearing of N82°E

GS = 518 miles per hour at a bearing of 79°E

We can use the law of cosines to find the wind speed (WS) and the angle between AS and WS.

WS = √(GS² - AS² + 2AS * GS * cos(GS - AS))

WS = √(518² - 480² + 2(480)(518)cos(79 - 82)) = √(268816 - 230400 + 466560cos(-3)) = √(28672) = 168 miles per hour

Then we can use the angle between AS and WS to find the direction of the wind:

Direction of the wind = (bearing of AS + bearing of WS) / 2 = (82 + 79) / 2 = 80.5 ° NE

To find the resulting ground speed and direction of the plane if the pilot increased the air speed to 500 miles per hour, we can use the same method as before.

Given that:

AS' = 500 miles per hour at a bearing of N82°E

GS' = √(AS'² + WS² - 2AS'WS * cos(direction of WS - direction of AS'))

GS' = √(500² + 168² - 2(500)(168)cos(80.5 - 82)) = √(250000 + 28672 + 118400cos(1.5)) = √(284872) = 535.5 miles per hour

The resulting ground speed and direction of the plane would be 535.5 miles per hour at a bearing of 80.7°E.

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[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{-2}~,~\stackrel{y_2}{3}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{3}-\stackrel{y1}{(-6)}}}{\underset{\textit{\large run}} {\underset{x_2}{-2}-\underset{x_1}{4}}} \implies \cfrac{3 +6}{-6} \implies \cfrac{ 9 }{ -6 } \implies - \cfrac{3 }{ 2 }[/tex]

Answer:

9/-6 or -1.5

Step-by-step explanation:

Follow the steps in the image

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An example of a quadratic function that has a maximum value is the parabola defined by the equation y = -(x-3)^2 + 4.

A quadratic function is a type of polynomial function that can be written in the form y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. The graph of a quadratic function is a parabola, which can either open upward or downward depending on the value of the leading coefficient (a).

An equation is a mathematical statement that asserts the equality of two expressions. It is usually written with an "=" sign between the two expressions, such as y = 2x + 3 or x^2 + y^2 = 1. Equations can be used to model real-world situations, and solving an equation means finding the values of the variables that make the equation true.

We can determine that this function has a maximum value by looking at the leading coefficient of the quadratic term (x^2). In this case, the coefficient is -1, which means that the parabola opens downward. Therefore, since the parabola opens downward, the vertex of the parabola is the highest point of the parabola, which means that it is a maximum point.

Also we can find the vertex of the parabola by using the formula x = -b/2a, in this case x= -0 / 2*-1 = 3/2 =1.5, and using this value in the function, we get the y value which is 4, so the vertex is (1.5,4) which confirms that the function has a maximum value at (1.5,4)

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Answer:

  -4/17

Step-by-step explanation:

You want the slope of the line described by the equation 20x +85y = 120.

Putting the equation in slope-intercept form by solving for y gives ...

  20x +85y = 120

  85y = -20x +120

  y = -20/85x +120/85

  y = -4/17x +24/17

The slope is the coefficient of x: -4/17.

The rate of change is -4/17.

Probability that the calendar turned to month of May or June is 1/6.

Probability is simply the possibility of getting an event. Or in other words, we are predicting the chance of getting an event.

The value of probability will be always in the range from 0 to 1.

Total number of months = 12

Suppose that a calendar is turned.

Probability that the calendar turned to month of May = 1/12

Probability that the calendar turned to month of June = 1/12

Probability that the calendar turned to month of May or June = 1/12 + 1/12

                                                                                                      = 2/12

                                                                                                      = 1/6

Hence the probability that the calendar turned to month of May or June is 1/6.

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Answer: 19

Step-by-step explanation:

56-13=43

43-24=19

Answer: 19

Step-by-step explanation: 13 + 24 = 37

so (56 - 37 = 19)

The two lines intersect at (-3,-5) and thus a common solution.

The solution of a linear equation is defined as the points, in which the lines represent the intersection of two linear equations. In other words, the solution set of the system of linear equations is the set of all possible values to the variables that satisfies the given linear equation.

Given here, the system of linear equations as  Y=4x+7 and 2=x-y

On solving these two equations we get that the two lines intersect at (-3,-5) and thus a common solution.

The two equations are plotted in the attachment below.

Hence, The two lines intersect at (-3,-5) and thus a common solution.

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Answer:

The area of x is 4

The area of the right triangle is 6

Step-by-step explanation:

The Pythagorean Theorem can be used when dealing with right triangles. So, if we use the equation a^2 + b^2 = c^2, we can find that 3^2 + x^2 = 5^2 or 9 + x^2 = 25. If we calculate, we can find that x is equal to 4. Now if we multiply by 3 and divide by 2, we can get the answer of 6 as the area of the right triangle.

Therefore , By dividing by the constant shown to the right, you can change any equation into intercept form:

=> x/3 + y/3 = 1 => x/1.2 +y/(-1) = 1

A mathematical equation is a formula that joins two statements with the equal sign, or =, to express equality. All mathematical formulas, including equations, have an equal sign. Algebra is frequently utilized in equations. Algebra is used in mathematics when we don't know the exact quantity to compute. A mathematical statement demonstrating the equality of two mathematical expressions is the definition of an equation in algebra. For instance, the variables 3x + 5 and 14 are separated by an equal sign in the equation 3x + 5 = 14.

Here,

Any equation can be changed to intercept form by dividing by the constant on the right:

=> x/3 + y/3 =1

=> x/1.2 + y/(-1) = 1

These inform you that only graph A is appropriate because the x- and y-intercepts of the first equation are 3 in this case. Given that it has an x-intercept of 1.2 and a y-intercept of -1, the second equation also matches with graph A.

Therefore , By dividing by the constant shown to the right, you can change any equation into intercept form:

=> x/3 + y/3 = 1 => x/1.2 +y/(-1) = 1

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There are 749398 5-disc combinations that can be made from the collection.

Combination refers to the number of ways that a certain number of items can be selected from a larger set of items without regard to the order of the items. It is different from permutation, which takes into account the order of the items.

The number of combinations can be calculated using the formula:

nCr = n! / (r! x (n - r)!)

Where n is the total number of items in the collection, and r is the number of items in each combination

Since a movie collection has 6 mystery, 10 comedy, 4 adventure, 9 drama, and 12 science fiction discs, then the total number of discs in the collection is 41.

Plug in the values to the formula for combination.

nCr = n! / (r! x (n - r)!)

41C5 = 41! / (5! x (41 - 5)!)

41C5 = 41! / (5! x 36!)

41C5 = 749398

Therefore, there are 749398 5-disc combinations that can be made from the collection.

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When two lines intersect the vertically opposite angles so formed are equal in measure.

When two lines intersect each other, then the opposite angles, created due to the intersection are called vertical angles or vertically opposite angles.

We can see in the figure two lines intersect each other at a point. Therefore the two pairs of vertically opposite angles are ∠1 and ∠3, ∠2 and ∠4.

From the figure, ∠1 + ∠2 = 180°  [Linear pair]     eqn(1)

∠2 + ∠3 = 180°  [Linear pair]   eqn(2)

From eqn(1) and eqn(2), we get

∠1 + ∠2 = ∠2 + ∠3

∴ ∠1 = ∠3

Similarly, we can prove ∠2 = ∠4 also.

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The equation 2x + 2y = -6 in the form of y = mx + b is y = -x - 3.

The equation of a line is given by:

y = mx + c

where m is the slope of the line and c is the y-intercept.

Example:

The slope of the line y = 2x + 3 is 2.

The slope of a line that passes through (1, 2) and (2, 3) is 1.

We have,

2x + 2y = -6

Subtract 2x on both sides.

2y = -6 - 2x

Divide both sides by 2.

y = -3 - x

y = -x - 3

This is in the form of y = mx + b

Now,

m = -1 and b = -3.

Thus,

y = -x - 3 is the equation.

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Option c: y = -x + 5 represents scatter plot 1, Option g: The relationship is not linear represents scatter plot 2, Option a: y = 3x represents scatter plot 3, and Option f: y = x + 3 represents scatter plot 4.

What is a scatter plot?

The graphs that show the association between two variables in a data set are called scatter plots. It displays data points either on a Cartesian system or a two-dimensional plane.

The given equations are -

a. y = 3x

b. y = 4x - 2

c. y = -x + 5

d. y = -5x

e. y = -2x - 4

f. y = x + 3

g. The relationship is not linear.

The first scatter plot is going from positive y-axis to positive x-axis in a linear form. The graph for the equation that goes from positive y-axis to positive x-axis is y = -x + 5.

The second scatter plot is a curved parabola going and not a linear plot. So, the scatter plot is said to be a non-linear scatter plot.

The third scatter plot is going from positive x-axis to infinity in a linear form. The graph for the equation that goes from x-axis to infinity is y = 3x.

The fourth scatter plot is going from positive y-axis to infinity in a linear form. The graph for the equation that goes from positive y-axis to infinity is y = x + 3.

Therefore, scatter plot 1 is represented by equation y = -x + 5, scatter plot 2 is not linear, scatter plot 3 is represented by equation y = 3x and scatter plot 4 is represented by equation y = x + 3.

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By setting up the equation and solving we get the positive number is 2/5.

A positive number is a number greater than zero. In mathematical terms, it is a number that is greater than zero on the number line. Positive numbers are often represented with a "+" sign in front of them, but this is not always necessary as the positive sign is the default and is usually omitted.

An equation is a mathematical statement that asserts the equality of two expressions. It is a statement that asserts that the two expressions on either side of the "=" sign are equal and have the same value. The expressions on either side of the "=" sign are called the left-hand side (LHS) and the right-hand side (RHS) of the equation.

For example:

2x + 3 = 7

x(2x+x/2)=x

2x^2+x^2/2=x

5x^2/2=x

5x^2=2x

5x=2

x=2/5

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on solving the provided question, we can say that - the graphs which is shown as the straight line graph.

Graphs are visual representations or charts used in mathematics to methodically express data or values. A relationship between two or more objects is frequently represented by a point on a graph. A non-linear data structure called a graph is made up of nodes, or vertices, and edges. Connect the nodes, also known as vertices. This graph comprises a set of vertices V= 1, 2, 3, 5, and a set of edges E= 1, 2, 1, 3, 2, 4, and (2.5), (3.5), (4.5). Statistics graphs (bar charts, pie charts, line charts, etc.) Exponential diagrams. triangle graph, a logarithmic graph

here,

then line which is drawn is straright,

so the graph is straight .

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Answer:

68

Step-by-step explanation:

25% of a number is 20, where the number is 80 because 25% times 4 is 100% and 20 times 4 is 80.

We can check by doing 60% of 80 to see if it is 48. Instead of 60%, we can do 10% of 80, which is 8. Since we divided by 6, we multiply 8 by 6 which is 48.

Since we know 80 is the number, we have to find 85% of 80. First, let's find 5% of 80, which is 4 since 4 x 20 is 80 and 5% x 20 is 100%. Now, 85 divided by 5 is 17, let's multiply 17 by 4 to get 85% of 80. 17 x 4 is 68 which is 85% of 80.

The maximum area of the rectangle inside the ellipse is 2ab.

Given that,

Ellipse = x2 / a2 + y2 / b2 = 1

we know the general coordinates of any point in an ellipse of the equation

x2 / a2 + y2 / b2 = 1 is (acosθ,bsinθ)

So, we take general vertices of rectangles as

(acosθ,bsinθ), (acosθ,-bsinθ), (-acosθ,bsinθ), and (-acosθ,-bsinθ)

So, we can see that length and breadth of the rectangle is 2acosθ and 2bsinθ

So, its area will be ab4sinθcosθ (Length x Breadth)

Applying trigonometric formula 2sinθcosθ=sin2θ, let area equals to A

Now A=2absin2θ

Now as the area is the maximum given in the question, we have to differentiate it with respect to θ

dA/dθ=2abcos2θ×2 (using dsin2θ / dθ=2cos2θ)

Equating dA / dθ=2abcos2θ×2=0, it means cos2θ=0 , which means 2θ =  π / 2 and θ=π / 4

So now we to put θ=π / 4 in the equation of area which is A=2absin2θ

We get as A=2absin2π / 4=2absinπ / 2=2ab

Hence the max area of the rectangle inside the ellipse is 2ab.

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As per the graph, there is no restrictions for the domain but the range has a minimum value at y = - 4.

The domain is any real number:

The range is any real number not less than - 4:

The y-intercept of the line y = 7x -38 is -38. And the x-intercept is 0

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

Given:

The equation of the line,

y-4=7(x-6)

Simplify,

y -4 = 7x - 42

y = 7x -38

And slope-intercept form:

y = mx + b,

where m is the slope and y is the intercept.

Comparing with the slope intercept form,

we get,

m = 7 and b = -38

And the x-intercept is 0

Therefore3, y-intercept is -38.

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According to the following graph, the for the expression y = 5 is horizontal.

The term graph in math is known as the set of vertices and a binary relation between vertices, adjacency.

Here we have given the expression y = 5.

Now, we have to plot these expression on the graph, then we get the following graph.

Through the given graph, we have identified that the expression makes the horizontal line on the graph.

Here we also know that if the equation of a line is y = 5, then it is a horizontal line.

Here we know that the line is parallel to the x-axis and passes through the point y = 5 at the y-axis.

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Answer:

3/20

Step-by-step explanation:

To do this multiply the numerators (the top numbers) together first.

3 * 1 = 3

This is the numerator (the top number) of the answer fraction.

Next we take the bottom numbers:

5 * 4 = 20

We have 3/20 as our fraction. We know 3 is a prime number and can only be divided by 1 or 3. 20 cannot be divided by 1 or 3 so this fraction cannot be simplified any more.

If Elliot purchased 3 trick cameras and 4 fake snakes for $40 and  Kaci bought 7 trick cameras and 8 fake snakes for $85 , then the system of equations that can be used to find the cost of trick camera and fake snakes are [tex]3x+4y=40[/tex] and [tex]7x+8y=85[/tex] .

Elliot purchased 3 trick cameras and 4 fake snakes for $40  ;

Kaci purchased 7 trick cameras and 8 fake snakes for  $85 ;

let the number of trick cameras purchased be = x ;

let the number of fake snakes purchased be = y ;

So , the equation for Elliot's purchase is ⇒ [tex]3x+4y=40[/tex] ;

and the equation for Kaci's purchase is ⇒ [tex]7x+8y=85[/tex] ;

Therefore , the equations to solve for the number of trick cameras and number of fake cameras are  [tex]3x+4y=40[/tex] and [tex]7x+8y=85[/tex] .

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The interquartile range (IQR) of the data in the dot plot is 3.5.

The interquartile range is a measure of where the “middle fifty” is in a data set. Where a range is a measure of where the beginning and end are in a set, an interquartile range is a measure of where the bulk of the values lie. That’s why it’s preferred over many other measures of spread when reporting things like school performance or SAT scores.

In descriptive statistics, the interquartile range is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the difference between the 75th and 25th percentiles of the data.

The  upper and lower median values are shown below which would be the mean of the 2 middle values shown in the brackets below:

43, (44, 44,) 44, | 45, | 45, (47, 48,) 48

Upper median (Q3) = (47+48) ÷ 2 = 47.5

Lower median (Q1) = (44+44) ÷ 2 = 44

=>Step 4: Subtract the lower median from the upper median to get the IQR.

IQR = Upper median (Q3) - Lower median (Q1)

IQR = 47.5 - 44

IQR = 3.5

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By taking derivative, slope of the logarithmic function can be found.

The slope of a logarithmic function can be found by taking the derivative of the function with respect to x using the rules of logarithmic differentiation. The general form of the derivative of a logarithmic function is:

d/dx(log(f(x))) = (1/f(x))*(df/dx)

The slope means change in Y with respect to change in X. The slope is the ratio of the rise to run and it shows the steepness of the plane.

The slope is denoted by m.

Example:

Find the slope of y = log(x^2+1)

d/dx(log(x^2+1)) = (1/(x^2+1))*(2x) = 2x/(x^2+1)

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Answer:2024

Step-by-step explanation:

Answer:

720,000,000,000,000 or 720 trillion picoseconds.

Step-by-step explanation:

1 picosecond = 1 trillionth of a second. Therefore 1 second has 1 trillion picoseconds:

0.000 000 000 001 seconds = 1 picosecond

0.000 000 000 001 * 1,000,000,000,000 seconds = 1 * 1,000,000,000,000 picosecond

1 second = 1,000,000,000,000 picoseconds

Now, one minute is the same as 60 seconds so we multiply the equation by 60:

1 minute = 60 seconds = 60,000,000,000,000 picoseconds

Finally, we need 12 minutes to be converted to picoseconds.

12 minutes = 720 seconds = 720,000,000,000,000 picoseconds.

So therefore, 12 minutes is equal to 720,000,000,000,000 picoseconds or 720 trillion picoseconds.

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