The domain for f(x) and g(x) is the set of all real numbers. Let f(x) = 2x2 + x − 3 and g(x) = x − 1. Find f(x) • g(x).

A. 2x3 − x2 + 4x − 3

B. x2 − 4x + 3

C. 2x3 − x2 − 4x + 3

D. 2x3 − 4x2 + 3

2.25 * 45 = 101.25 miles to work

101.25 / 25 = 4.05 gallons of gas

4.05 * 2.75 = $11.14 total cost of gas

800 students participated at a charity bike rally

What are mathematics operations?

• A mathematical operation is a function that converts a set of zero or more input values (also called "b" or "arguments") into a defined output value. The number of operands determines the operation's arity. Most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive and multiplicative inverses.

• Zero-arity operations, or nullary operations, are constants, and mixed products are arity three operations, or ternary operations.

It is given that :

there are 1200 students population of Heartsworth Middle School.

and, 2/3 of the student participated in a charity bike rally That is :

(2 x 1200)/ 3 = 2400/3 = 800 students

Therefore, 800 students participated at a charity bike rally.

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Answer:

0.625

Step-by-step explanation:

We are given that

Kesia knows =[tex]\frac{1}{4}=0.25[/tex]

We have to determine the decimal equivalent to [tex]\frac{5}{8}[/tex] using given decimal value.

[tex]\frac{5}{8}[/tex] can be written as

[tex]\frac{5}{8}=\frac{2}{8}+\frac{2}{8}+\frac{\frac{1}{4}}{2}[/tex]

[tex]\frac{5}{8}=\frac{1}{4}+\frac{1}{4}+\frac{\frac{1}{4}}{2}[/tex]

[tex]\frac{5}{8}=0.25+0.25+\frac{0.25}{2}[/tex]

[tex]\frac{5}{8}=0.50+\frac{0.25}{2}[/tex]

[tex]\frac{5}{8}=0.625[/tex]

Hence,[tex]\frac{5}{8}=0.625/tex]

Answer:  t represents the the number of seconds after rocket is released.

Step-by-step explanation:

Given: The height of a rocket a given number of seconds after it is released is modeled by [tex]h (t) = 6t^2 + 32t + 10[/tex].

Here h (height) is the dependent variable , which depends on the number of seconds after rocket is released (independent variable).

Since the independent variable in the function is t, then t must represents the the number of seconds after rocket is released.

The variable t represents the number of seconds that have passed since the rocket was released.

Here we know that the height of a rocket a given number of seconds after it is released is modeled by:

h(t) = 6*t^2 + 32*t + 10

So this is a function that relates height with time in seconds, we know that the function models the height, so we must have that:

[h(t)] is equivalent to height.

This means that the other variable, t, must be related to time in seconds.

Then we can conclude that the variable t represents the number of seconds after the rocket has been released.

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Answer:

1) Average velocity = 10/3 m/s

2) Instantaneous velocity = -2 m/s
   Speed = 2 m/s to the left

3) (0, 3) ∪ (6, 8]

4) Going faster: (3, 4.5) ∪ (6, 8]
   Slowing down: (0, 3) ∪ (4.5, 6)

5) Total distance = 35.67 m (nearest hundredth)

Step-by-step explanation:

The relationships between position (displacement), velocity and acceleration are:

[tex]\boxed{\boxed{\begin{array}{c}\textbf{POSITION (s)}\\\\\text{Differentiate} \downarrow\qquad\uparrow\text{Integrate}\\\\\textbf{VELOCITY (v)}\\\\\text{Differentiate}\downarrow\qquad\uparrow \text{Integrate}\\\\\textbf{ACCELERATION (a)}\end{array}}}[/tex]

Given a particle is moving with velocity v(t) = t² - 9t + 18, to find its position s(t) we can integrate v(t):

[tex]\begin{aligned}\displaystyle s(t)=\int v(t)\;\text{d}t&=\int(t^2-9t+18)\;\text{d}t\\\\&=\dfrac{t^{2+1}}{2+1}-\dfrac{9t^{1+1}}{1+1}+18t+C\\\\&=\dfrac{t^{3}}{3}-\dfrac{9t^{2}}{2}+18t+C\end{aligned}[/tex]

As s(0) = 1, then:

[tex]\begin{aligned}s(0)=\dfrac{(0)^{3}}{3}-\dfrac{9(0)^{2}}{2}+18(0)+C&=1\\0-0+0+C&=1\\C&=1\end{aligned}[/tex]

Therefore, the position function s(t) is:

[tex]\large\boxed{s(t)=\dfrac{t^3}{3}-\dfrac{9t^2}{2}+18t+1}[/tex]

Given a particle is moving with velocity v(t) = t² - 9t + 18, to find its acceleration a(t) we can differentiate v(t):

[tex]\begin{aligned}a(t)=\dfrac{\text{d}}{\text{d}t}[v(t)]&=2\cdot t^{2-1}-1\cdot9t^{1-1}+0\\&=2t-9\end{aligned}[/tex]

Therefore, the acceleration function a(t) is:

[tex]\large\boxed{a(t)=2t-9}[/tex]

[tex]\hrulefill[/tex]

To find the average velocity over the interval [0, 8], use the formula:

[tex]\textsf{Average Velocity}=\dfrac{s(t_2)-s(t_1)}{t_2-t_1}[/tex]

In this case:

Calculate the position at t₁ and t₂ by substituting t = 0 and t = 8 into s(t):

[tex]s(0)=\dfrac{(0)^3}{3}-\dfrac{9(0)^2}{2}+18(0)+1}=1[/tex]

[tex]s(8)=\dfrac{(8)^3}{3}-\dfrac{9(8)^2}{2}+18(8)+1}=\dfrac{83}{3}[/tex]

Therefore:

[tex]\textsf{Average Velocity}=\dfrac{s(8)-s(0)}{8-0}=\dfrac{\frac{83}{3}-1}{8}=\dfrac{10}{3}\; \sf m/s[/tex]

Therefore, the average velocity is 10/3 m/s.

[tex]\hrulefill[/tex]

To find the instantaneous velocity at t = 5 seconds, substitute t = 5 into v(t):

[tex]\begin{aligned}v(5)&=(5)^2-9(5)+18\\&=25-45+18\\&=-2\end{aligned}[/tex]

So, the instantaneous velocity at t = 5 seconds is -2 m/s.

Speed is a scalar quantity that measures how fast an object is moving regardless of its direction. Therefore, speed is the magnitude of velocity:

[tex]\textsf{Speed}=|v(5)|=|-2|=2\;\sf m/s[/tex]

Therefore, the speed at t = 5 is 2 m/s to the left.

[tex]\hrulefill[/tex]

The particle changes direction when v(t) = 0.

[tex]\begin{aligned}v(t)&=0\\\implies t^2-9t+18&=0\\t^2-6t-3t+18&=0\\t(t-6)-3(t-6)&=0\\(t-3)(t-6)&=0\\\\t-3&=0\implies t=3\\t-6&=0\implies t=6\end{aligned}[/tex]

Therefore, the particle changes direction at t = 3 and t = 6.

We know that the position of the particle at t = 0 is 1 meter right of zero. Therefore:

Therefore, the time intervals between 0 ≤ t ≤ 8 when the particle is moving right is:

[tex]\hrulefill[/tex]

When a(t) > 0:

[tex]\begin{aligned}a(t)& > 0\\2t-9& > 0\\2t& > 9\\t& > \dfrac{9}{2}\\t& > 4.5\; \sf s\end{aligned}[/tex]

When a(t) < 0:

[tex]\begin{aligned}a(t)& < 0\\2t-9& < 0\\2t& < 9\\t& < \dfrac{9}{2}\\t& < 4.5\; \sf s\end{aligned}[/tex]

Therefore:

(Refer to the attachment).

If velocity and acceleration have the same sign, it means the object is speeding up.

If velocity and acceleration have opposite signs, it means the object is slowing down.

Therefore, the time intervals when the particle is going faster and slowing down are:

[tex]\hrulefill[/tex]

To find the total distance the particle has traveled between 0 and 8 seconds, we need to consider the distance traveled between the intervals when it changes direction.

To do this, find the position of the particle at t = 0, t = 3, t = 6 and t = 8.

[tex]s(0)=\dfrac{(0)^3}{3}-\dfrac{9(0)^2}{2}+18(0)+1=1[/tex]

[tex]s(3)=\dfrac{(3)^3}{3}-\dfrac{9(3)^2}{2}+18(3)+1=23.5[/tex]

[tex]s(6)=\dfrac{(6)^3}{3}-\dfrac{9(6)^2}{2}+18(6)+1=19[/tex]

[tex]s(8)=\dfrac{(8)^3}{3}-\dfrac{9(8)^2}{2}+18(8)+1=\dfrac{83}{3}\approx27.67[/tex]

Therefore, in the interval 0 ≤ t < 3, the particle travels:

[tex]|s(3)-s(0)|=|23.5-1|=22.5\; \sf meters\;(to\;the\;right)[/tex]

In the interval  3 < t < 6, it travels:

[tex]|s(6)-s(3)|=|19-23.5|=4.5\; \sf meters\;(to\;the\;left)[/tex]

In the interval 6 < t ≤ 8, it travels:

[tex]|s(8)-s(6)|=|27.67-19|=8.67\; \sf meters\;(to\;the\;right)[/tex]

So the total distance the particle has traveled between 0 and 8 seconds is:

[tex]\textsf{Total distance}=22.5+4.5+8.67=35.67\; \sf meters[/tex]

we know that

the perimeter of a polygon is the sum of the length sides

in this problem we have a triangle

so

the polygon has three sides

Let

[tex]A(-5,4)\\B(1,4)\\C(3,-4)[/tex]

the perimeter is equal to

[tex]P=AB+BC+AC[/tex]

The formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

Step 1

Find the distance AB

[tex]A(-5,4)\\B(1,4)[/tex]

substitutes the values in the formula

[tex]d=\sqrt{(4-4)^{2}+(1+5)^{2}}[/tex]

[tex]d=\sqrt{(0)^{2}+(6)^{2}}[/tex]

[tex]dAB=6\ units[/tex]

Step 2

Find the distance BC

[tex]B(1,4)\\C(3,-4)[/tex]

substitutes the values in the formula

[tex]d=\sqrt{(-4-4)^{2}+(3-1)^{2}}[/tex]

[tex]d=\sqrt{(-8)^{2}+(2)^{2}}[/tex]

[tex]d=\sqrt{68}[/tex]

[tex]dBC=8.25\ units[/tex]

Step 3

Find the distance AC

[tex]A(-5,4)\\C(3,-4)[/tex]

substitutes the values in the formula

[tex]d=\sqrt{(-4-4)^{2}+(3+5)^{2}}[/tex]

[tex]d=\sqrt{(-8)^{2}+(8)^{2}}[/tex]

[tex]d=\sqrt{128}[/tex]

[tex]dAC=11.31\ units[/tex]

Step 4

Find the perimeter

the perimeter is equal to

[tex]P=AB+BC+AC[/tex]

substitutes the values

[tex]P=6+8.25+11.31=25.56\ units=25.6\ units[/tex]

therefore

the answer is

[tex]25.6\ units[/tex]

Answer:

It must be greater than 1/2XY.

Step-by-step explanation:

We draw an arc from point X below the arc that passes through X and Y.  Keeping our compass the same width, we will draw another arc from point Y below, intersecting the first arc.

If the width of the compass is not set to more than 1/2XY, then the two arcs will not intersect and we will not complete our construction.

Perpendicular lines are lines that meet at 90 degrees.

The true statement about the width of the compass is that: (d) It must be greater than 1/2XY.

From the question, we understand that he has drawn arc XY already.

The next step is to draw an arc less than the width of XY, but greater than half width XY.

This will ensure that the arcs bisect one another.

Hence, the true option is (d).

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The required polynomial function is x⁴ - 67x² + 1450x - 4984 = 0 with 4 degrees with real coefficients.

A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers. A polynomial can have more than one term.

The zeros are given in the question as 4, -14, and 5 + 8i

The required polynomial function of minimum degree with real coefficients whose zeros include those listed above.

For the zero 4, you can write (x-4)

For the zero -14, you can write (x+14)

For the zero 5+8i since it is complex it will be accompanied by its conjugate 5-8i

So, you can write (x-(5+8i) and (x-(5-8i)) =(x²-10x+89)

(x - 4)(x + 14)(x² - 10x + 89)

Expanding the expression, we get

x⁴ - 67x² + 1450x - 4984 = 0

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Answer:

622,080

Step-by-step explanation:

thanks me later

Answer:

[tex]x=1[/tex]

Step-by-step explanation:

We have been given an equation [tex]\sqrt{x+3}+4=6[/tex]. We are asked to find the solution of our given equation.

[tex]\sqrt{x+3}+4-4=6-4[/tex]

[tex]\sqrt{x+3}=2[/tex]

Now, we will square both sides of our given equation.

[tex]x+3=2^2[/tex]

[tex]x+3=4[/tex]

[tex]x+3-3=4-3[/tex]

[tex]x=1[/tex]

To see whether [tex]x=1[/tex] is an extraneous solution or not, we will substitute [tex]x=1[/tex] in our given equation as:

[tex]\sqrt{1+3}+4=6[/tex]

[tex]\sqrt{4}+4=6[/tex]

[tex]2+4=6[/tex]

[tex]6=6[/tex]

Since both sides of our given equation are equal, therefore, [tex]x=1[/tex] is a solution for our given equation.

Answer:

Helen can make 2 loaves of bread and 4 batches of muffins.

Step-by-step explanation:

Let x be the number of loaves of bread

Let y be the number of batches of muffins

As per the given requirement of flour, the equation becomes:

[tex]3.5x+2.5y=17[/tex]   .......(1)

As per the given requirement of sugar, the equation becomes:

[tex]0.75x+0.75y=4.5[/tex]  .....(2)

Multiplying equation (1) with 0.3 and subtracting (2) from (1)

[tex]1.05x+0.75y=5.1[/tex] now subtracting (2) from this we get

=> [tex]0.3x=0.6[/tex]

So, x = 2

And as [tex]3.5x+2.5y=17[/tex] ; so substituting x = 2 here we get

[tex]3.5(2)+2.5y=17[/tex]

[tex]7+2.5y=17[/tex]

[tex]2.5y=17-7[/tex]

[tex]2.5y=10[/tex]

So, y = 4

Hence, there will be 2 loaves of bread and 4 batches of muffins.

we know that

Vertical angles are a pair of opposite and congruent angles formed by intersecting lines

In this problem

m∠JNM=m∠KNL -------> by vertical angles

so

[tex](4x+6)\°=(7x-21)\°[/tex]

Solve for x

[tex]7x-4x=6+21\\3x=27\\x=9\°[/tex]

Find the value of m∠JNM

m∠JNM=[tex](4x+6)\°[/tex]

substitute the value of x

m∠JNM=[tex](4*9+6)\°[/tex]

m∠JNM=[tex]42\°[/tex]

therefore

the answer is

m∠JNM=[tex]42\°[/tex]

To prove that (x-s-t) is a factor of the polynomial $x^3 - s^3 - t^3 -3st(s+t)$, we applied the factor theorem which states that (x-c) will be a factor of a polynomial if f(c) equals zero. By substituting x with (s+t) into the polynomial, we got a value of zero, confirming that (x-s-t) is indeed a factor.

To prove that (x-s-t) is a factor of $x^3 - s^3 -t^3 -3st(s+t)$, we can use the factor theorem. According to the factor theorem, a polynomial f(x) has a factor (x-c) if and only if f(c) equals zero.

Given the polynomial $x^3 - s^3 - t^3 -3st(s+t)$, we substitute (x = s + t) into the polynomial, thus:

$f(s + t) = (s + t)^3 -s^3 -t^3 - 3st(s + t)$.

After simplifying the equation, we obtain:

$s^3 + 3s^2t +3st^2  + t^3 - s^3 - t^3 - 3st^2 - 3s^2t$.

When we cancel out the like terms, the result is 0. Therefore,

$(x - s - t)$ is a factor of the given polynomial $x^3 - s^3 -t^3 -3st(s+t)$. This proves our result according to the Factor theorem.

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89.87 is 215% of approximately 41.8. This is found by converting 215% to a decimal (2.15) and dividing the given number (89.87) by this decimal.

The student's question relates to an application of percentages. To work out this problem, we must first understand that in this context, 215% is equivalent to 2.15 when transformed into a decimal. We then divide the given number, 89.87, by 2.15 to find the original value. The calculation would be as follows: 89.87 ÷ 2.15, which equates to approximately 41.8. Therefore, 89.87 is 215% of the number 41.8.

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(a) The formula for the sum of the first n even positive integers is n(n+1) and (b) the formula has been proved with an example.

Arithmetic progression is the sequence of numbers that has a fixed common difference between any two consecutive numbers.

For the given situation,

Part (a):

The sum of even numbers formula is obtained by using the sum of terms in an arithmetic progression formula.

Sum of Even Numbers Formula = n(n+1),

where n is the number of terms in the series.

Part (b):

Let us consider the even positive numbers,

2,4,6,8,10,.......

Now take first 5 positive numbers 2,4,6,8,10

By using the formula, n=5

Sum of n even positive integers = [tex]n(n+1)[/tex]

⇒ [tex]5(5+1)[/tex]

⇒ [tex]5(6)[/tex]

⇒ [tex]30[/tex].

Now add the first 5 positive numbers without the formula,

⇒ [tex]2+4+6+8+10=30[/tex]

Hence we can conclude, that the formula for the sum of the first n even positive integers is n(n+1) and the formula has been proved with an example.

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The formula for the sum of the first n even positive integers is S = n(n+1). This was derived by recognizing the pattern of the series and using the formula for the sum of an arithmetic series. The proof by induction confirms the validity of our formula.

The problem asks us to find and prove a formula for the sum of the first n even positive integers. To find the formula, we recognize that the first n even integers can be represented as 2, 4, 6, ..., 2n. Hence, the sum S of the first n even integers can be written as S = 2 + 4 + 6 + ... + 2n. This is an arithmetic series with the first term a = 2 and the common difference d = 2. The sum of an arithmetic series can be given by the formula S = n/2 [2a + (n-1)d], substituting a = 2 and d = 2, we get S = n/2 [2*2 + (n-1)2] = n[2 + 2(n-1)], which simplifies to S = n(n+1).

To prove this formula, we use induction. For n=1, the sum is 2, which matches our formula 1(1+1) = 2. Assuming the formula holds for n = k, for n = k+1, the sum would include one more term, 2(k+1), so the new sum is S + 2(k+1) which upon simplification, verifies our formula for all n.

Answer:

His insurance company will pay for damages.

Step-by-step explanation:

An insurance is a form of contract that permits an individual to transfer the responsibilities of a financial loss to an insurance company. The company bears the risk of the financial loss. Small amount of money are collected from their clients and summed together to pay for losses that the client may encounter in the future. Insurance safeguards an individual and his property from losses, misfortune, hazards or theft. Covered losses are paid for by the insurance company thereby reducing the financial costs for the  individual.  Examples include auto insurance , health insurance, disability insurance, and life insurance.

The set of statements always have the same truth value will be Conditional and Contrapositive i.e. Option [tex](D)[/tex] .

Conditional and Contrapositive : If the conditional statement is “If P then Q.” Then converse of the conditional statement is “If Q then P.” The contrapositive of the conditional statement is “If not Q then not P.” The inverse of the conditional statement is “If not P then not Q.”

So,

As per given options,

Option [tex](B)[/tex] and [tex](C)[/tex] have inverse, so they have nothing to do with inverse.

Now,

So, according to the above mentioned definition,

Option [tex](D)[/tex] Conditional and Contrapositive is the correct option.

Hence, we can say that the set of statements always have the same truth value will be Conditional and Contrapositive Option [tex](D)[/tex] .

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Final answer:

To convert the equation 7x - 4y + 8 = 0 to slope-intercept form, solve for y to get y = (7/4)x + 2, with a slope of 7/4 and a y-intercept of 2.

Explanation:

To write the equation 7x − 4y + 8 = 0 in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we want to solve for y. The steps are as follows:

Thus, the equation of the line in slope-intercept form is y = (7/4)x + 2. Here, the slope is 7/4 and the y-intercept is 2.