Find an equation of the tangent line to the curve at the given point. y = x^3 ? 3x + 2, (4, 54) Please show work

Applying the tangent ratio, the measure of angle B in the image given is calculated as: A. 39.8°

In order to find the measure of angle B, we will need to apply the tangent ratio which is expressed as the formula below:

tan ∅ = length of opposite side / length of adjacent side

From the image given, we have:

∅ = angle B

Length of opposite side = 10 cm

Length of adjacent side = 12 cm

tan B = 10/12

B = tan^(-1)(10/12)

Measure of angle B = 39.8°

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The Boolean duality principle is a fundamental concept in Boolean algebra which states that any statement or expression in the algebra can be transformed into an equivalent form by interchanging the roles of AND and OR operators, as well as 0's and 1's.

The Boolean duality principle, also known as De Morgan's laws, is a fundamental concept in Boolean algebra that states that every Boolean expression remains valid if you perform the following steps:
1. Swap AND (⋅) operators with OR (+) operators and vice versa.
2. Replace all 1's with 0's and all 0's with 1's.
3. Complement (invert) all variables.
In other words, the Boolean duality principle allows us to derive a dual expression from an original Boolean expression by applying these transformations. This principle is useful in simplifying Boolean expressions and designing digital circuits.

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The values of x, y and z from the triangle are z = 12√2, x = 6 and y = 12

Trigonometrical ratio of angles is used to find the sine, cosine, and tangent of angles in right triangles

The ratios are denoted by SOHCAHTOA

Tan A = Oppo/Adj

Tan 60 = 6√3/x

√3/ 1 = 6√3/x

Cross and multiply to have

x√3/3 = 6√3/√3

x = 6

To find y

First from pyth. rule

6² + (6√3)² = m²

36 + 36*3 = m²

144 = m²

m =√144 = 12

Using tan 45 = 1/1

Tan 45 = y/12

1/1 = y /12

y = 12

To find x w use sine

Sin 45 = 1/√2

1/√2 = 12/x

x = 12√2

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(a) To approximate P(15 < ΣXi < 22), we can use the Central Limit Theorem (CLT) since we have a large enough sample size (n = 30) and the mean and variance of the Poisson distribution are both finite.

First, we need to find the mean and variance of the sample mean, which is also the mean and variance of the Poisson distribution:

μ = λ = 2/3

σ^2 = λ = 2/3

Next, we can standardize the random variable Z = (ΣXi - nμ) / sqrt(nσ^2) to have a standard normal distribution:

Z = (ΣXi - 30(2/3)) / sqrt(30(2/3)) = (ΣXi - 20) / sqrt(20)

Then, we can use a standard normal table or calculator to find the probability:

P(15 < ΣXi < 22) ≈ P(-2.74 < Z < -1.77) ≈ 0.038

Therefore, the approximate probability is 0.038.

(b) To approximate P(21 < ΣXi < 27), we can use the same method with the CLT.

First, we need to find the mean and variance of the sample mean:

μ = λ = 2/3

σ^2 = λ = 2/3

Next, we can standardize the random variable Z = (ΣXi - nμ) / sqrt(nσ^2) to have a standard normal distribution:

Z = (ΣXi - 30(2/3)) / sqrt(30(2/3)) = (ΣXi - 20) / sqrt(20)

Then, we can use a standard normal table or calculator to find the probability:

P(21 < ΣXi < 27) ≈ P(-0.68 < Z < 0.68) ≈ 0.495

Therefore, the approximate probability is 0.495.

Independent-samples t test. This test is used to compare the means of two independent groups. The correct answer is a.

In this case, the average number of matching characteristics reported by the fingerprint analysts and the average number of matching characteristics calculated by the defense attorney. The standard deviation of s = 1.2 would be used in the calculation of the t-statistic.
To determine whether the average number of matching characteristics the defense attorney observed (M = 9.8) differs significantly from that reported by fingerprint analysts (16 matching characteristics), she should use a "Single-sample t test".
Here's why:
- The defense attorney is comparing a sample (her 30 cases) to a known population mean (16 matching characteristics).
- The standard deviation of the sample is provided (s = 1.2), but the population standard deviation is not known.
- An Independent-samples t test would be used if comparing two independent groups, which is not the case here.
So, the correct answer is c. Single-sample t test.

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The number of chicken and the rabbits in the farm are 8 and 16 respectively.

Given that Fred has 24 heads and 80 feet in his farm we need to find the number of chicken and the rabbits in the farm,

Let the number of chicken and the rabbits in the farm be c and r respectively.

Establishing the system of equations,

c + r = 24

c = 24 - r.........(i)

2c + 4r = 80

c + 2r = 40

c = 40 - 2r..........(ii)

Equating the equation, since their LHS are equal,

24 - r = 40 - 2r

r = 16

Put r = 16 in eq(i)

c = 40 - 32

c = 8

Hence, the number of chicken and the rabbits in the farm are 8 and 16 respectively.

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Answer:

Step-by-step explanation: I might not be exact, cause im still a beginner, but, maybe halve of 72?

False. The slope of the line is actually -0.5.

Let's determine the slope of the line that passes through the points (20, 5) and (10, 10). To find the slope, we can use the formula:

Slope (m) = (y2 - y1) / (x2 - x1)

Plugging in the coordinates of the two points, we get:

m = (10 - 5) / (10 - 20) = (5) / (-10) = -1/2

The slope of the line is -1/2. Therefore, the statement "If a line passes through the points (20, 5) and (10, 10), then the slope of the line is -2" is FALSE.

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Answer:

divide and 16 per pound...

Answer:

divide, 16

Step-by-step explanation:

To find the weight, we have to divide the total number of ounces, 96, by the unit rate, which is the baseline for the conversion, 16 ounces.

Hope this helps! :)

The graph of the absolute value function f(x) = |x - 2| is given by the image presented at the end of the answer.

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The parent absolute value function is given as follows:

f(x) = |x|.

Which has the format of the V-graph, with vertex at the origin.

f(x) = |x - 2| is a translation right two units of f(x) = |x|, hence the vertex will be at the point (2,0).

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A line, ray, or segment that divides a line segment into two equal parts. It also makes a right angle with the line segment. The given statement is false.

A perpendicular bisector is a line, ray, or segment that intersects a line segment at its midpoint and creates two equal parts. It is called "perpendicular" because it always intersects the line segment at a 90-degree angle, but it does not necessarily make a right angle with the line segment. A line, ray, or segment that divides a line segment into two equal parts is called the perpendicular bisector.

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The area of the sector as shown in the diagram is 13.1 cm².

A sector is said to be a part of a circle made of the arc of the circle along with its two radii.

To calculate the area of the sector, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

Hence, the area of the sector is 13.1 cm².

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The requried coordinate locations of the respective checkpoints at the Reflections of You obstacle course are A' (0,5), B' (9,5), and C' (4,5).

To find the reflected coordinates of each checkpoint in the y-axis, we need to negate the x-coordinate while keeping the y-coordinate the same.

For checkpoint A(0,5), the reflected coordinates A' will be (-0,5) or just (0,5) since reflecting a point over the y-axis does not change its y-coordinate.

For checkpoint B(-9,5), the reflected coordinates B' will be (9,5) since negating -9 gives us 9 and the y-coordinate remains the same.

For checkpoint C(-4,5), the reflected coordinates C' will be (4,5) since negating -4 gives us 4 and the y-coordinate remains the same.

Therefore, the coordinate locations of the respective checkpoints at the Reflections of You obstacle course are A' (0,5), B' (9,5), and C' (4,5).

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The range for this population is 12, which is calculated by subtracting the smallest score from the largest score (20-8 = 12). Since the variable is discrete, only whole numbers are possible scores, and the score that is not possible is not relevant to the range calculation.

To find the range for this population, you'll need to use the largest and smallest scores.

Given:
- Population (N) = 10
- Discrete variable
- Smallest score (X) = 8
- Largest score (X) = 20

To find the range, simply subtract the smallest score from the largest score:

Range = Largest score - Smallest score
Range = 20 - 8
Range = 12

So, the range for this population is 12.

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Using the properties of exponents we can write the given expression as [tex]x^\frac{3}{2} y^\frac{5}{2}z^\frac{-2}{3}[/tex]. Therefore, option E is the correct answer.

The given expression is [tex]\frac{\sqrt{x^3} \sqrt{y^5} }{\sqrt[3]{z^2} }[/tex].

We can square root as 1/2.

So the expression becomes [tex]\frac{x^\frac{3}{2} y^\frac{5}{2} }{z^\frac{2}{3} }[/tex]

=  [tex]x^\frac{3}{2} y^\frac{5}{2}z^\frac{-2}{3}[/tex]

Therefore, option E is the correct answer.

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Answer:

Step-by-step explanation:

The surface area of a rectangular prism of dimensions 2 in, 5.2 in and 4 in is given as follows:

S = 78.4 in².

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

The dimensions for this problem are given as follows:

2 in, 5.2 in and 4 in.

Hence the surface area of the prism is given as follows:

S = 2 x (2 x 5.2 + 2 x 4 + 5.2 x 4)

S = 78.4 in².

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Answer:

$240

Step-by-step explanation:

We Know

Karma earns $36 in 3 hours.

$36 in 3 hours = $12 in 1 hour

At this rate, how many dollars will he earn in 20 hours?

We Take

12 x 20 = $240

So, he earns $240 in 20 hours.

Answer: 240 dollars

Step-by-step explanation:

divide 36 dollars by 3 (=12)to know how much he earns per hour

then multiply the quotient by 20 which is 240

Step-by-step explanation:

8y-9=14

bring the 9 to the other side because of which the operator has to change to a plus sign.

8y=14+9=23

23÷8=

[tex] \frac{23}{8} [/tex]

or

[tex]2.875[/tex]

There are 19 ounces

There are 107 visitors per day

He needs 63 boxes

1) We have to convert the money to cents thus we have;

$10.45 = 1045 cents

Then Number of ounces = 1045/ 55

= 19 ounces

2) The total number of days the museum was open is 365 - 4 = 361 days

Thus we have that the number of visitors in a day is; 38627/361

= 107 visitors per day

3) If 1 box holds 45 books

x boxes will hold 2835 books

x = 2835/45

x = 63 boxes as shown

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Two different ways we can measure exactly 1 Liter are:

Use of the 500 mL cup two times

Use the 300 mL cup 3 times in total and the 100 mL cup once

This question we want to know the different combination of the given cups that can lead to us getting a volume of 1 liter.

Now, we know that

1 L = 1000 mL

This means that we can make use of the 500 mL cup two times in total to get:

500 mL + 500 mL= 1000 mL = 1 L

Another way we can do the 1 liter is that we can use the 300 mL cup 3 times in total and the 100 mL cup once to get:

300 mL + 300 mL + 300 mL + 100 mL = 1000 mL = 1 L

Other ways are:

- use of the 500 mL cup two timese the 100 mL cup

- Take 10 times the 100-mL cup

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Complete question is:

Suppose you have a 100-mL cup,a 300-mL cup, and a 500-mL cup. list two different ways you can measure exactly 1 L

We can easily identify the dependent and independent (free) variables by looking at the matrix after converting augmented matrix.

When we convert the augmented matrix of a system of equations into reduced row-echelon form, we can easily identify the dependent and independent (free) variables by looking at the matrix. The columns that contain leading 1's correspond to the variables that are dependent on the others. The columns that do not contain leading 1's correspond to the variables that are independent or free.

For example, if we have a matrix in reduced row-echelon form that looks like this:

1 0 2 | 3
0 1 -1 | 2
0 0 0 | 0

We can see that the first and second columns have leading 1's, so the variables corresponding to those columns (in this case, x1 and x2) are dependent on each other. The third column does not have a leading 1, so the variable corresponding to that column (in this case, x3) is independent or free.

In general, we can say that the number of independent variables is equal to the number of columns without leading 1's, and the number of dependent variables is equal to the number of columns with leading 1's. By identifying the dependent and independent variables, we can then use this information to solve the system of equations using substitution or elimination.

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Option B. Theresa's work is correct its shows that Matrix multiplication is not commutative.

Clearly, it is true that AB and BA are distinct matrices, so matrix multiplication is not generally commutable.

Nevertheless, there are special conditions where the product of matrices could be commutable, such as when one of them is an id matrix or diagonal matrix with equivalent diagonal entries.

Hence we can say that Theresa's work is correct its shows that Matrix multiplication is not commutative. since the results are different

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Given the following 2nd-order differential equation (Cauchy-Euler equation), [tex]x^2y''+4xy'-4y=0[/tex], find the general solution.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A differential equation in the form [tex]ax^2y''+bxy'+cy=0[/tex] , which is a Cauchy-Euler equation, can be solved in the following manner.

Where the characteristic equation => [tex]am^2+(b-a)m+c=0[/tex]

After solving for "m" the possible solutions are...

[tex]\bold{Real, \ Distinct \ Roots} \Rightarrow y=c_1x^{m_1}+c_2x^{m_2}\\\bold{ Repea ted \ Roots} \Rightarrow y=c_1x^{m}+c_2x^{m}ln(x)\\\bold{Complex \ Roots} \Rightarrow y=c_1x^{\alpha}cos(ln(x^\beta ))+c_2x^{\alpha}sin(ln(x^\beta )); \ m= \alpha \pm \beta i[/tex]

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

[tex]\Longrightarrow x^2y''+4xy'-4y=0 \ where \ a=1, \ b=4, \ and \ c=-4[/tex]

The characteristic equation:

[tex]\Longrightarrow (1)m^2+(4-1)m+(-4)=0 \Longrightarrow \boxed{m^2+3m-4=0}[/tex]

Solve for "m."

[tex]\Longrightarrow m^2+3m-4=0 \Longrightarrow (m-1)(m+4)=0 \Longrightarrow \boxed{m=1,-4}[/tex]

Note that m's are distinct and real.

[tex]Thus, \ \boxed{y=c_1x+c_2x^{-4}} \therefore Sol.[/tex]

Where the arbitrary constants c_1 and c_2 can be solved for given an initial condition.

The correct answer is option (e) [tex]$y = c_1 x + c_2 x^2$[/tex].

We can use the Cauchy-Euler equation to find the general solution of the given differential equation. The Cauchy-Euler equation is given by:

[tex]$a_n x^n y^{(n)}+a_{n-1} x^{n-1} y^{(n-1)}+\cdots+a_1 x y^{\prime}+a_0 y=0$[/tex]

where [tex]$a_n, a_{n-1}, \ldots, a_1, a_0$[/tex] are constants and [tex]$y^{(n)}, y^{(n-1)}, \ldots, y'$[/tex] denote the [tex]$n$[/tex]th, [tex]$(n-1)$[/tex]th, [tex]$\ldots$[/tex], first derivatives of y with respect to x.

In the given differential equation, we have [tex]$a_2 = 1$[/tex], [tex]$a_1 = 4$[/tex], and [tex]$a_0 = -4$[/tex]. Therefore, the Cauchy-Euler equation becomes:

[tex]$x^2 y^{\prime \prime}+4 x y^{\prime}-4 y=0$[/tex]

We assume a solution of the form [tex]$y=x^r$[/tex]. Substituting this into the differential equation, we get:

[tex]$$x^2 r(r-1) x^{r-2}+4 x r^1 x^{r-1}-4 x^r=0$$[/tex]

Simplifying, we get:

[tex]$$\begin{aligned}& r(r-1)+4 r-4=0 \\& r^2+3 r-4=0 \\& (r+4)(r-1)=0\end{aligned}$$[/tex]

So, we have two roots[tex]$r_1 = -4$[/tex] and [tex]$r_2 = 1$[/tex]. Therefore, the general solution of the differential equation is:

[tex]$y=c_1 x^{-4}+c_2 x^1$[/tex]

Thus, the correct answer is option (e) [tex]$y = c_1 x + c_2 x^2$[/tex].

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As per the mathematical operation both Jillian and Samuel are correct.

Given expression = 30+40+70,

The methodology by Jillian = added 30 and 40 and then 70

The methodology by Samuel = added 30 and 70 and then 40.

Determining the result of the given equation:

30 + 40 + 70

= 140

Reviewing the operations of both Jillian and Samuel

Jillian

He added 30 and 40 and then 70:

30 + 40 = 70

70 + 70 = 140

Samuel

He added 30 and 70 and then 40

30 + 70 = 100

100 + 40 = 140

The result of both mathematical operations is 140. Thus, it can be stated that both are correct.

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The possible number of times the binary search algorithm splits the list when searching for a term in the list varies depending on the length of the list and the position of the term within the list.

However, in general, the number of times the list is split during the binary search algorithm is proportional to the logarithm of the length of the list. So, for example, if the list has 8 items, the binary search algorithm may split the list up to 3 times (log base 2 of 8 is 3), while if the list has 1024 items, the binary search algorithm may split the list up to 10 times (log base 2 of 1024 is 10).

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This suggests that there is enough evidence to support the alternative hypothesis, indicating that the average fuel economy today is greater than 21.17 MPG.

Based on the given information, the null hypothesis (H₀) states that the average fuel economy is less than or equal to 21.17 MPG (μ ≤ 21.17), while the alternative hypothesis (H₁) states that the average fuel economy is greater than 21.17 MPG (μ > 21.17). You conducted a one-sample mean hypothesis test and observed a p-value of 0.0427.
To conclude at the 5% level of significance (α = 0.05), you compare the p-value to the significance level. Since the p-value (0.0427) is less than the significance level (0.05), you reject the null hypothesis.

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The domain will be all real numbers and the range will be all real numbers greater than or equal to 1.

The domain of a function is the set of all possible input values that can be used as arguments for the function. In this case, there are no restrictions on the input values of x, so the domain is all real numbers.

Domain: All real numbers

Range:

To find the range of the function, we can complete the square to rewrite the function in vertex form:

f(x) = x^2 + 10x + 26

f(x) = (x + 5)^2 + 1

Since the square of a real number is always nonnegative, the minimum value of the function is 1, which occurs when x = -5. Therefore, the range of the function is all real numbers greater than or equal to 1.

Range: All real numbers greater than or equal to 1.

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