In six years, Clem Kadiddlehopper will be five times as old as he was 26 years ago. How old is he now?

Unit circle : a circle with radius of one. Unit circle is centered at the origin.

Use the formula cot x = base / perpendicular, where x is the angle. Substituting x,y and t to the formula, we get cot (t) = x / y.

To find the trigonometric function values for a point on the unit circle corresponding to a real number t involves using the x and y coordinates, which are derived from the cosine and sine functions at that angle t.

The student's question involves finding the values of trigonometric functions for a point p(x, y) on the unit circle that corresponds to a real number t. This typically involves understanding the relationship between the coordinates of a point on the unit circle and the trigonometric functions sin, cos, and tan. In this context, x(t) and y(t) are understood as the x and y coordinates of a point on the unit circle at a certain angle t. The point p(x, y) would be given by the cosine and sine functions: x(t) = cos(t) and y(t) = sin(t). In more advanced contexts, these functions can take different forms like x(t) = A cos(wt + p) where A, w, and p are constants. The solution to trigonometric equations may involve differential equations or complex numbers in such cases.

The rate of change of the quadratic function over the interval −4≤x≤−2 is 2.

To find the rate of change of the quadratic function [tex]$f(x)=x^2+4 x+5$[/tex] , over the interval −4≤x≤−2, we need to find the average rate of change over that interval.

The average rate of change of a function f(x) over an interval [a,b] is given by:

[tex]Average Rate of Change =\frac{f(b)-f(a)}{b-a}$[/tex]

So, in this case:

a=−4

b=−2

We calculate f(−4) and f(−2):

[tex]$\begin{aligned} & f(-4)=(-4)^2+4(-4)+5=16-16+5=5 \\ & f(-2)=(-2)^2+4(-2)+5=4-8+5=1\end{aligned}$[/tex]

Now we can find the rate of change:

Rate of Change = [tex]$\frac{1-5}{-2-(-4)}=\frac{-4}{-2}=2$[/tex]

The average rate of change of the function f(x)=x² + 4x + 5 over the interval from x = -4 to x = -2 is -2.

The rate of change for the quadratic function f(x)=x² + 4x + 5 over the interval from x = -4 to x = -2 can be calculated using the average rate of change formula. This formula is given by:

Rate of Change = [f(x2) - f(x1)] / (x2 - x1)

Where x1 = -4 and x2 = -2. We first calculate f(-4) and f(-2) by plugging these values into the function:

f(-4) = (-4)² + 4(-4) + 5 = 16 - 16 + 5 = 5

f(-2) = (-2)² + 4(-2) + 5 = 4 - 8 + 5 = 1

Now we use these results in our rate of change formula:

Rate of Change = (1 - 5) / (-2 + 4) = -4 / 2 = -2

The average rate of change of the function f(x) over the interval from x = -4 to x = -2 is -2.

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Hope this helps...

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Because we are finding DOLLAR amounts, and not a total, we are simply going to do this:

Salary * (percent /100) = $Value

20,000 * 3%  [or 0.03] = $600

20,000 * 4%  [or 0.04] = $800

20,000 * 6%  [or 0.06] = $1,200

Now to find his new salary if he were to get a raise, we will add the individual amounts to 20,000 is he were to only get a SINGLE percent raise increase...

If his salary (currently) is $20,000 and gets a 3% raise, his new salary is:

$20,600

If his salary (currently) is $20,000 and gets a 4% raise, his new salary is:

$20,800

If his salary (currently) is $20,000 and gets a 6% raise, his new salary is:

$21,200

If he were to get all THREE raises at the same time (per say) [or one after the other going off of a $20,000 salary], his new salary is:

$22,600

The value of the x is greater than negative of 32 over 9.

Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.

The inequality is given as,

–5/2(3x + 4) < 6 – 3x

Simplifying the equation,

–5(3x + 4) < 2(6 – 3x)

  5(3x + 4) > 2(3x – 6)

  15x + 20 > 6x – 12

            9x > –12 – 20

              x > –32 / 9

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Final answer:

The equation for Omar's total pay after working h hours at $8 per hour is p = 8h. This linear equation indicates a direct proportion between hours worked and total pay, reflecting a total of $64 for an 8-hour day.

Explanation:

The equation to represent Omar's total pay, p, after working h hours, at the rate of $8 per hour, is given by p = 8h. This linear equation shows that the total pay is directly proportional to the number of hours worked. Therefore, if Omar works for a standard 8-hour day, he would earn 8 hours times $8 per hour, which equals $64 for that day.

Answer: Option C i.e. 22.1 is correct

Step-by-step explanation:

The given expression follows pemdas rule

PEMDAS:

P for Paranthesis

E for Exponents

M for Multiplication

D for division

A for Addition

S for Subtraction

Given :24 -12 /2 *3.2

=  24-12/2*3.2     [Multiplication]

= 24- 12/6.4      [Division]

=24-1.875         [Subtraction]

=22.125

So the answer is C. 22.1

f(x) = 3x^2-x

F(-2)

 replace x with -2

3(-2)^2 - -2 =

3*4 +2 = 14

 Answer is 14

The number line for [tex]\fbox{\begin\\\ \bf \text{option 1}\\\end{minispace}}[/tex] represents the solution for the equation [tex]|x+4|=2[/tex].

Further explanation:

The equation is given as follows:

[tex]|x+4|=2[/tex]

In the above equation [tex]||[/tex] represents the modulus function.

Modulus function is defined as a function which gives positive value of the function for any real value of [tex]x[/tex].

For example: The function [tex]y=|x|[/tex] is a modulus function in which [tex]y>0[/tex] and [tex]x<0[/tex] or [tex]x>0[/tex].

In the given equation [tex]|x+4|[/tex]  is a modulus expression.  

There are two cases formed for [tex]|x+4|[/tex].

First case:  [tex]x>-4[/tex]

If [tex]x>-4[/tex] then [tex]|x+4|\rightarrow (x+4)[/tex].

Substitute [tex]|x+4|=(x+4)[/tex] in [tex]|x+4|=2[/tex].

[tex]\begin{aligned}x+4&=2\\x&=2-4\\&=-2\end{aligned}[/tex]

Therefore, for [tex]x>-4[/tex] the value of [tex]x[/tex] which satisfies the equation [tex]|x+4|=2[/tex] is [tex]x=-2[/tex].

Second case:  [tex]x<-4[/tex]

If [tex]x<-4[/tex] then [tex]|x+4|\rightarrow -(x+4)[/tex].

Substitute [tex]|x+4|=-(x+4)[/tex] in [tex]|x+4|=2[/tex].

[tex]\begin{aligned}-(x+4)&=2\\-x-4&=2\\-x&=2+4\\-x&=6\\x&=-6\end{aligned}[/tex]

Therefore, for [tex]x<-4[/tex] the value of [tex]x[/tex] which satisfies the equation [tex]|x+4|=2[/tex] is [tex]x=-6[/tex].

This implies that the solution for the equation [tex]|x+4|=2[/tex] or the value of [tex]x[/tex] which satisfies the given equation are [tex]\fbox{\begin\\\ \math x=-2\ \text{and}\ x=-6\\\end{minispace}}[/tex].

Option 1:

The number line in option 1 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].

As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].

This implies that option 1 is correct.

Option 2:

The number line in option 2 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=2[/tex] and [tex]x=4[/tex].

As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].

This implies that option 2 is incorrect.

Option 3:

The number line in option 3 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=2[/tex] and [tex]x=6[/tex].

As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].

This implies that option 3 is incorrect.

Option 4:

The number line in option 4 shows that the solutions of the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-4[/tex].

As per our calculation made above the solutions for the equation [tex]|x+4|=2[/tex] are [tex]x=-2[/tex] and [tex]x=-6[/tex].

This implies that option 4 is incorrect.

Therefore, the number line for [tex]\fbox{\begin\\\ \bf \text{option 1}\\\end{minispace}}[/tex] represents the solution for the equation [tex]|x+4|=2[/tex].

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Functions

Keywords: Functions, modulus, modulus function, number line, real line, |x+4|=2, equation, root, zeroes, solutions,  absolute function, x=-6 and x=-2.

Option A is correct, the values of [tex]x[/tex] staisfying the given equation [tex]|x+4|=2[/tex] are [tex]-2[/tex] and [tex]-6[/tex].

Modulus function always returns a positive value to the equation.

Here, [tex]|x+4|[/tex] will give positive result if [tex]x>-4[/tex] and negative value if, [tex]x<-4[/tex].

Case 1: When [tex]x>-4[/tex]

According to the given equation,

[tex]|x+4|=2\\x=2-4\\x=-2[/tex]

So, the value of [tex]x[/tex] which satisfies the given equation [tex]|x+4|=2[/tex] for [tex]x>-4[/tex] is [tex]x=-2[/tex].

Case 2: When [tex]x<-4[/tex]

According to the given equation,

[tex]|x+4|=2\\-x-4=2\\x=-6[/tex]

So, the value of [tex]x[/tex] which satisfies the given equation [tex]|x+4|=2[/tex] for [tex]x<-4[/tex] is [tex]x=-6[/tex].

Hence, the values of [tex]x[/tex] staisfying the given equation [tex]|x+4|=2[/tex] are [tex]-2[/tex] and [tex]-6[/tex].

Now, according to the options, Option A is correct.

Learn more about number line here:

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Final answer:

To find the equation of a line perpendicular to another, determine the slope of the original line and use the negative reciprocal of that slope for the perpendicular line. Choose a point on the perpendicular line, and apply the point-slope form to get the equation.

Explanation:

The process of finding a perpendicular line involves first understanding the slope of the given line. To find the equation of a line that is perpendicular to another, you need to identify the slope (m) of the original line and use the fact that the slopes of perpendicular lines are negative reciprocals of each other (meaning if the slope of the first line is m, the slope of the line perpendicular to it will be -1/m). In the context of vectors and analytical methods in physics, components can help describe forces or directions. If we consider the example of a skier on a slope, breaking down the weight force into components parallel and perpendicular to the slope helps analyze the motion. However, for strictly finding a perpendicular line in mathematics, we focus on the slopes of the lines. Suppose you have the equation of the original line. If it is in the format y = mx + b, where m is the slope and b is the y-intercept, you can find the slope of the perpendicular line by taking the negative reciprocal of m. If the slope is not readily apparent, you might need to rearrange the equation into this format. Once you have the slope of the perpendicular line, choose a point through which the line passes (this could be the original point or any particular point you're given). Then, use the point-slope form (y - y1) = m(x - x1) to write the equation of the line.

There were 6 trucks.

if you multiply 6 * 5 = 30.

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Answer:

You need to buy 6.5 cubic yards.

Step-by-step explanation:

You want to put a 5 inch thick layer of topsoil for a new 23 ft by 18 ft garden.

We know 1 yard = 3 feet or 1 yard = 36 inches

Then 1 inch = [tex]1/36[/tex] yard.

1 feet = 1/3 yard

The volume of the layer of the topsoil is given by:

= [tex]5(1/36)(23)(1/3)(18)(1/3)[/tex] cubic yards

[tex]2070/324=6.39[/tex] cubic yards

Now, the store only sells increments of 1/4 = 0.25 cubic yards.

So, we need to buy [tex]6.39/0.25=25.56[/tex] increments

Rounded to 26 increments.

Therefore, we need to buy 26 increments of 1/4 cubic yards.

That becomes 6.5 cubic yards.

To apply the Pythagorean Theorem, you need the lengths of the two legs of a right triangle. Trigonometric ratios involve the angles and ratios of sides in a right triangle. Examples are provided for both methods.

In order to apply the Pythagorean Theorem, you need the lengths of the two legs of a right triangle. The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For example, if we have a triangle with legs of lengths 3 and 4, we can use the theorem to find the length of the hypotenuse. The square of the hypotenuse is 3^2 + 4^2 = 9 + 16 = 25, so the hypotenuse has a length of 5.

Trigonometric ratios, on the other hand, involve the angles of the right triangle and ratios of its sides. The three main trigonometric ratios are sine, cosine, and tangent. For example, if we have a right triangle with an angle of 30 degrees and one leg of length 5, we can use trigonometry to find the length of the other leg. The sine of the angle is given by the ratio of the opposite side (the leg we want to find) to the hypotenuse. So, sin(30 degrees) = opposite / hypotenuse = x / 5. Solving for x, we get x = 5 * sin(30 degrees) = 5 * 0.5 = 2.5.

The equation q=r+rst can be rearranged to isolate t, with the final solution being t=(q-r)/rs.

To solve for t in the equation q=r+rst, first, you need to isolate t on one side of the equation. You can do this by subtracting r from both sides so that you have q-r=rst. Next, divide both sides by rs, which gives you the final solution: t=(q-r)/rs.

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