Solve each equation. Show your thinking.

a. 5x-10=30

b. 3x+8=21

Answer:

a) $26

b) 24.07%

c) $94.3

Step-by-step explanation:

Given that:

Before tax:

Normal price of dress = $62

Discounted price of dress = $48

Normal price of a pair of shoes = $46

Discounted price of a pair of shoes = $34

a) Before the sales tax, total savings = ($64 - $48) + ($46 - $34) = $26

b) Total percentage discount on total sales.

Total bill for original price = $62 + $46 = $108

Percentage discount can be found by the formula:

\(\dfrac{\text{Total discount}}{\text{Total bill for original price}}\times 100\\\Rightarrow \dfrac{26}{108} \times 100 = 24.07\%\)

c) Total amount paid if the sales tax is 15%.

Amount paid with tax = Amount after discount + 15% of amount after discount

Amount after discount = $48 + $34 = $82

15% of $82 = $18.29

Amount paid with sales tax = $82 + $12.3 = $94.3

Answer:

1.45 pounds

Step-by-step explanation:

6.6 divided by 4

743, 437, 473, 347 are all the 3-digit odd numbers that can be created using those tiles

Answer:

The slope is 3.

The matrix representation of T relative to the basis β = {1, t, \(t^{2}\)} is:

\(\left[\begin{array}{ccc}3&0&0\\5&0&0\\0&4&1\end{array}\right]\)

To find the matrix representation of the linear transformation T relative to the basis β = {1, t, t^2}, we need to determine the images of the basis vectors under T and express those images as linear combinations of the basis vectors. The coefficients of these linear combinations will form the columns of the matrix representation.

Let's calculate the images of the basis vectors:

T(1) = 3\(a_{0}\) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

= (3\(a_{0}\)) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

T(t) = 3\(a_{0}\) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

= (3\(a_{0}\)) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

T(\(t^{2}\)) = 3\(a_{0}\) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

= (3\(a_{0}\)) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\)+ \(a_{2}\))\(t^{2}\)

Now we express these images as linear combinations of the basis vectors:

T(1) = (3\(a_{0}\)) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

= 3(1) + 5(1)t + 4(0)\(t^{2}\) + 0(1)\(t^{2}\)

= 3 + 5t

T(t) = (3\(a_{0}\)) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

= 3(0) + 5(0)t + 4(1)\(t^{2}\) + 1(0)\(t^{2}\)

= 4\(t^{2}\)

T(\(t^{2}\)) = (3\(a_{0}\)) + (5\(a_{0}\) - 2\(a_{1}\))t + (4\(a_{1}\) + \(a_{2}\))\(t^{2}\)

= 3(0) + 5(0)t + 4(0)\(t^{2}\) + 1(1)\(t^{2}\)

= \(t^{2}\)

Now we can form the matrix representation of T using the coefficients:

\(\left[\begin{array}{ccc}3&0&0\\5&0&0\\0&4&1\end{array}\right]\)

The matrix representation of T relative to the basis β = {1, t, \(t^{2}\)} is:

\(\left[\begin{array}{ccc}3&0&0\\5&0&0\\0&4&1\end{array}\right]\)

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The length of side b is around 7.30 if m(A) = 42°, m(B) = 113°, and C = 5 using the sine laws.

To calculate for the side b, we can use the law of sines:

a/sin(A) = b/sin(B) = c/sin(C)

Let us assume that the values a, b, and c are the sides of the triangle, and A, B, and C are the opposite angles, respectively.

Using the law of sines, the equation will be:

b/sin(B) = c/sin(C)

Substituting the above-given values, we get:

b/sin(113°) = 5/sin(42°)

b = \((5*sin(113°))/sin(42°)\)

b = 7.30

Therefore, we can conclude the length of side b is 7.30m.

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Answer:

x = 1

Step-by-step explanation:

Axis of symmetry is a line which divides an object into two equal halves giving it exactly like a mirror shape.

Here a line of x = 1 can be drawn to separate the figure into equal halves.

Answer:

x = 1

Step-by-step explanation:

I think that it is 1.

Answer:

17/6

Step-by-step explanation:

22*3=66+1=67

6*4=24+3=27

67/3-27/4 lowest common multiple of 3 and 4 is 12

67*4=268

27*3=81

268-81/12=17/6

The first four terms of the sequence an = (-1)^(n+1) + 13n - 1/(2n + 1) are:

a1 = -13/3   , a2 = 27/5   ,   a3 = -37/7,   a4 = 49/9

To find the first four terms of the sequence, we need to substitute n = 1, 2, 3, and 4 into the given formula for an.

For n = 1, we have a1 = (-1)^(1+1) + 13(1) - 1/(2(1) + 1) = -1 + 13 - 1/3 = -13/3.

For n = 2, we have a2 = (-1)^(2+1) + 13(2) - 1/(2(2) + 1) = 1 + 26 - 1/5 = 27/5.

For n = 3, we have a3 = (-1)^(3+1) + 13(3) - 1/(2(3) + 1) = -1 + 39 - 1/7 = -37/7.

For n = 4, we have a4 = (-1)^(4+1) + 13(4) - 1/(2(4) + 1) = 1 + 52 - 1/9 = 49/9.

Therefore, the first four terms of the sequence are a1 = -13/3, a2 = 27/5, a3 = -37/7, and a4 = 49/9.

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In conclusion: (a) The linear transformation f: M₂x₂ → R₃ given by f(a b; c d) = (b+d, a+b, d-a) is injective (one-to-one). (b) The linear transformation f is surjective (onto) if and only if every value of z can be expressed as the difference d - a for some real numbers d and a.

To determine whether the linear transformation f: M₂x₂ → R₃ is injective (one-to-one) and surjective (onto), we need to analyze its properties and conditions.

Let's define the linear transformation f as:

f(a b; c d) = (b+d, a+b, d-a)

(a) Injective (One-to-One):

A linear transformation f is injective if every distinct input vector in the domain corresponds to a distinct output vector in the codomain. In other words, if f(a₁ b₁; c₁ d₁) = f(a₂ b₂; c₂ d₂), then (a₁ b₁; c₁ d₁) = (a₂ b₂; c₂ d₂).

To test injectivity, we need to compare the outputs of f for two different input matrices and see if they are equal.

Let's assume two different input matrices: A₁ = (a₁ b₁; c₁ d₁) and A₂ = (a₂ b₂; c₂ d₂).

If f(A₁) = f(A₂), then we have:

(b₁+d₁, a₁+b₁, d₁-a₁) = (b₂+d₂, a₂+b₂, d₂-a₂)

Comparing the corresponding elements, we get the following system of equations:

b₁ + d₁ = b₂ + d₂ (1)

a₁ + b₁ = a₂ + b₂ (2)

d₁ - a₁ = d₂ - a₂ (3)

From equation (1), we can deduce that b₁ - b₂ = d₂ - d₁. Let's call this equation (4).

Similarly, equation (2) can be rewritten as a₁ - a₂ = b₂ - b₁. Let's call this equation (5).

Now, subtracting equation (3) from equation (4), we have:

(b₁ - b₂) - (d₁ - d₂) = (d₂ - d₁) - (a₂ - a₁)

(b₁ - b₂) - (d₁ - d₂) = (d₂ - d₁) - (b₂ - b₁)

Simplifying further, we get:

2(b₁ - b₂) = 2(d₂ - d₁)

b₁ - b₂ = d₂ - d₁

Using equation (5), we can substitute b₁ - b₂ = d₂ - d₁:

a₁ - a₂ = b₂ - b₁ = d₂ - d₁

This implies that a₁ = a₂, b₁ = b₂, and d₁ = d₂.

Therefore, we have shown that if f(A₁) = f(A₂), then A₁ = A₂. This confirms that f is an injective (one-to-one) linear transformation.

(b) Surjective (Onto):

A linear transformation f is surjective if every vector in the codomain has at least one corresponding input vector in the domain. In other words, for every vector (x, y, z) in the codomain R₃, there exists an input matrix A = (a b; c d) such that f(A) = (x, y, z).

To test surjectivity, we need to check if every vector (x, y, z) in R₃ can be expressed as f(A) for some matrix A = (a b; c d).

The codomain R₃ consists of 3-dimensional vectors, and the range of f is determined by the values of b, d, and the differences between b and d (b - d).

From the transformation equation f(a b; c d) = (b+d, a+b, d-a), we can observe that the third component z in R₃ is given by z = d - a. Therefore, any vector in R₃ can be expressed as f(A) if and only if z = d - a.

Since a and d are the diagonal elements of the input matrix A, we can conclude that for every vector (x, y, z) in R₃, there exists a matrix A = (a b; c d) such that f(A) = (x, y, z) if and only if z = d - a.

Therefore, f is surjective (onto) if and only if every value of z can be expressed as the difference d - a for some real numbers d and a.

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Answer:

It's B

Step-by-step explanation:

Answer:

B.

Step-by-step explanation:

Answer: 7/4

Step-by-step explanation:

Answer:

Exact Form:

7/4

Decimal Form:

1.75

Mixed Number Form:

1 3/4

Step-by-step explanation:

The dimensions of the poster with the smallest area are as follows: The width is 20 cm, and the height is 16 cm.

To determine these dimensions, we need to consider the area of the printed material on the poster, which is fixed at 388 square centimeters. Let's assume the width of the printed material is x cm.

Since the side margins are each 2 cm, the total width of the poster (including the margins) becomes x + 2 cm + 2 cm = x + 4 cm.

Similarly, the total height of the poster is x + 4 cm as well because the top and bottom margins are both 4 cm.

To calculate the area of the entire poster, we multiply the width by the height: (x + 4 cm) * (x + 4 cm) = (x + 4)^2 cm^2.

According to the problem, the area of the printed material is 388 square centimeters. Therefore, we have the equation (x + 4)^2 cm^2 = 388 cm^2.

Solving this equation, we find x + 4 = 19 cm, which means x = 15 cm.

Hence, the dimensions of the poster, width is 15 cm + 4 cm = 19 cm, and the height is also 15 cm + 4 cm = 19 cm.

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dellana company tested 50 products for 75 hours each. during this time, it experienced four breakdowns. compute the number of failures per hour, 937.50 hours is the mean time between failures.

The span of time before something occurs or before a given time frame expires. We can utilise the current computers while waiting for the new ones to arrive next week.

Given that,

dellana company tested 50 products

time =75 hours

Thus, total products in hours

= 50 products × 75 hours each

= 3750 product hours

Also said that, 4 breakdowns in 3750 product hours,

so, (4/3750) = 1.067 × 10⁻³ = 0.001067 breakdowns per hour.

Also, an average of 0.001067 breakdowns per hour = λ

therefore, mean time between failures:

= 1/ λ

= 1/0.001067

= 937.50 hours

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Answer:

16-4[3+2÷(9-7)]= 0

Simplify the equation

Hope this helps!!

Answer:

8-(-2)

minus,minus means magiging plus so,

8+2=10

Step-by-step explanation:

Mark me please

the average height of the arch above the ground is `4a/π`.

The given function is

y = 6 sin(x), where 0 ≤ x ≤ π.

We need to find the average height of the arch above the ground.Answer: The average height of the arch above the ground is `4a/π`.Explanation:

To find the average height of the arch above the ground, we have to use the following formula:average height

= `(1/π) ∫_0^(π) y dx

`Here, y = 6 sin

xdx = dx (integral of dx is x)

Hence, the average height is:`

(1/π) ∫_0^(π) 6 sin x dx``

= (6/π) ∫_0^(π) sin x dx``

= (6/π) [- cos x]_0^(π)`

`= (6/π) [- cos π + cos 0]`

`= (6/π) [1 + 1]``= (6/π) (2)``

= 12/π``

= 3.8197 (approx)``≈ 4a/π`

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Alessia spent $5,408 on entertainment in 2021.

To determine how much Alessia spent on entertainment in 2021, we need to work backward from the given information about her salary in 2022.

Let's assume Alessia's salary in 2021 was the same as in 2022, which is $26,000 after tax.

Since Alessia spends 35% of her after-tax salary on rent, we can calculate her rent expenditure:

Rent = 35% × $26,000

= $9,100

The remaining amount is split between food, petrol, and entertainment in the ratio 6:11:8.

To determine the ratio's total parts, we sum the individual parts:

Total parts = 6 + 11 + 8 = 25

Next, we calculate the value of one part of the ratio:

One part = ($26,000 - $9,100) / 25

= $16,900 / 25

= $676

Now, we can calculate how much Alessia spent on entertainment in 2021, which is 8 parts of the ratio:

Entertainment expenditure = 8 × $676

= $5,408

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Ash trees are key to the health of certain forest ecosystem

because they: d. ) support many other species

The name ash comes from the word "spear" and may refer to the spear-shaped leaves, or even the fact that ancient people used this tree for the manufacture of weapons. It is also associated with many legends.

Ash trees are special in the fact that they can restore natural systems. They prefer to settle in riparian areas where their roots help stabilize riparian banks, their leaves support both aquatic and terrestrial ecosystems, and their branches provide shade and nesting sites for many animals.  

Therefore from the above definitions, we can conclude that the correct option is Option D: because support many other species

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Answer:

C. 9

Step-by-step explanation:

3n - 18 = 9

3n (- 18 + 18) = 9 + 18

3n = 27

3n/3 = 27/3

n = 9

Answer:

The magnitude is sqrt((-4)^2 + (-9)^2) = 9.85. The angle is atan(-9/-4) = 180 deg + 66 deg = 246 deg = -114 deg.

Step-by-step explanation:

hope it help

Answer:

9.85

Step-by-step explanation:

|v|= √9²+(-4)²

=√81+16

=√97

|v|= 9.85

The first thing that he has to do is take that "minus" sign through the parentheses containing the second polynomial. Some students find it helpful to put a " 1 " in front of the parentheses, to help them keep track of the minus sign. Here's what the subtraction looks like, when working horizontally/

Factors of the polynomial are: (x - 3)(x + 3)(x - 5).

What if factoring polynomial?

A polynomial with coefficients in a certain field or in integers is expressed as the product of irreducible factors with coefficients in the same domain by the process of factorization of polynomials, also known as polynomial factorization.

Given:   \(t(x) = x^3 - 5x^2 -9x+45\)

We have to show tat x - 5 is a factor of the polynomial and to find the factor of the polynomial.

First to show x - 5 is a factor of the given polynomial.

We know that,

If x - 5 is the factor of the given polynomial then by factor theorem x - 5 = 0.

So, x = 5

Plug x = 5 in t(x).

\(t(5) = 5^3 - 5(5)^2-9(5)+45 = 0\)

t(5) = 0

That means 5 is the zero of the polynomial t(x).

Therefore, x - 5 is the factor of the polynomial.

Now, to factor the polynomial.

\(x^3 -5x^2 -9x+45 = x^2(x-5) - 9(x-5) = (x^2-9)(x-5) = (x-3)(x+3)(x-5)\)

Therefore, after factoring the polynomial we get, \((x-3)(x+3)(x-5)\).

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The exact value of the trigonometric expression \(\frac{tan-(2\pi/3) }{sin(7\pi /4)} - sec(-\pi )\) in the simplified radical form is √6  +  1.

\(\frac{tan-(2\pi/3) }{sin(7\pi /4)} - sec(-\pi )\)

We have the angle,

-2π/3  =  -120° = (360 - 120)°  = 240° = 60°

Put the value of -2π/3 in tan -2π/3, we get

tan -2π/3 =  tan (60)  = √3

7π/4 × 180/π  =  7 × (180/4)  =  7 × 45  =  315° =  45°

Put the value of 7π/4 in sin 7π/4, and we get

sin 7π/4 = sin (45°) = √2/2

sec (-π)  =  sec (π)  = -1

Now, substitute all the values in the given equation,

= {√3/(√2/2)} - (-1)

= (2√3/√2) + 1

= (2√3 × √2/√2 × √2) +1

= (2√6/2) + 1

= √6 + 1

--The given question is incorrect, the correct question is

''What is the exact value of the trigonometric expression? \(\frac{tan-(2\pi/3) }{sin(7\pi /4)} - sec(-\pi )\) State your answer in simplified radical form."--

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To determine whether there is a decrease in voter support for the candidate after the scandal, we can test the hypothesis using the proportions of the two polls. The first poll showed 54% support among 630 randomly selected voters, while the second poll showed 51% support among 1010 voters. By conducting a hypothesis test, we can determine if the difference in proportions is statistically significant.

To test the hypothesis, we can use the two-sample z-test for proportions. The null hypothesis (H0) assumes no difference in voter support, while the alternative hypothesis (H1) assumes a decrease in support. We can calculate the test statistic using the formula:

z = (p1 - p2) / √[(p(1 - p) / n1) + (p(1 - p) / n2)]

where p1 and p2 are the proportions from the two samples, and n1 and n2 are the respective sample sizes. p is the pooled proportion, calculated as (x1 + x2) / (n1 + n2), where x1 and x2 are the respective number of successes (votes for the candidate) in each sample.

By calculating the z-value, we can compare it to the critical value for the desired significance level (e.g., α = 0.05). If the z-value exceeds the critical value, we reject the null hypothesis in favor of the alternative hypothesis, indicating a statistically significant decrease in voter support.

After performing the calculations, if the z-value exceeds the critical value, we can conclude that there is evidence to support the claim of a decrease in voter support for the candidate after the scandal. Conversely, if the z-value does not exceed the critical value, we fail to reject the null hypothesis, suggesting that there is insufficient evidence to conclude a significant decrease in support.

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Answer:

x= -1

Step-by-step explanation:

the answer is x= -1

hope it helps

The graph shows the  inequality x² ≤ 16

An inequality is a relationship that makes a non-equal comparison between two numbers or other mathematical expressions e.g. 2x > 4.

Inequalities are often used to describe conditions or constraints in real-world problems.

You will notice that the values represented in the graph ranges from -4 to 4. Thus, solving x² ≤ 16 will produce these values. That is:

x² ≤ 16

x ≤ ±√16

x ≤ ±4

x ≤ -4 or x ≤ 4

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(a) The area under the standard normal distribution for P(Z < 0.32) is approximately 0.6255. (b) The area for P(Z > 1.37) is approximately 0.0853. (c) The area for P(-0.68 < Z < 1.73) is approximately 0.7085.


To find the area under the standard normal distribution represented by each probability, we can use the standard normal distribution table or a calculator.
(a) P(Z < 0.32):
Using the standard normal distribution table or a calculator, we find that P(Z < 0.32) is approximately 0.6255.

(b) P(Z > 1.37):
Since the standard normal distribution is symmetric, P(Z > 1.37) is equal to 1 – P(Z < 1.37). Using the standard normal distribution table or a calculator, we find that P(Z < 1.37) is approximately 0.9147. Therefore, P(Z > 1.37) is approximately 1 – 0.9147 = 0.0853.

(c) P(-0.68 < Z < 1.73):
To find P(-0.68 < Z < 1.73), we need to calculate the difference between the cumulative probabilities P(Z < 1.73) and P(Z < -0.68). Using the standard normal distribution table or a calculator, we find that P(Z < 1.73) is approximately 0.9571 and P(Z < -0.68) is approximately 0.2486. Therefore, P(-0.68 < Z < 1.73) is approximately 0.9571 – 0.2486 = 0.7085.

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The overall sample for the survey consists of 50 freshmen, 40 sophomores, 35 juniors, and 30 seniors, totaling 155 students.

To create an overall sample for the survey, the administrators selected a certain number of students from each class level. They randomly sampled 50 freshmen, 40 sophomores, 35 juniors, and 30 seniors. By combining these individual samples from each class, the administrators obtained an overall sample size of 50 + 40 + 35 + 30 = 155 students. This overall sample represents a portion of the student population from each class level and allows for a representation of students from different academic years in the survey.

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Construction steps for both parties will be different, these steps are explained below.

In Mathematics, a polygon can be defined as a two-dimensional geometric figure that consists of straight line segments and a finite number of sides. Additionally, some examples of a polygon include the following:

• Triangle

• Quadrilateral

• Pentagon

• Hexagon

• Heptagon

• Octagon

• Nonagon

Generally speaking, the measure of the angle at the center of a regular polygon is equal to 360 degrees. Therefore, the smallest angle of rotation that maps (carries) a regular polygon onto itself can be calculated by using this formula:

α = 360/n

The number of sides being produced is one significant distinction between Laura's construction steps for an inscribed square and Gavin's construction stages for an inscribed regular hexagon.

The primary distinction between the two projects is that Laura's entails building a square with four sides, whereas Gavin's entails building a hexagon with six sides. Additionally, Laura uses perpendicular lines while Gavin uses a rotating radius to create his shapes, which are both slightly distinct from one another.

Laura would begin by sketching a circle, then two perpendicular diameters that connect at the centre of the circle, to create an inscribed square. Then she would draw two more lines, each one parallel to one of the current diameters and cross through the centre of the circle. The four vertices of the square would be the spots where these lines cross the circle.

Gavin, on the other hand, would first draw a circle and then a radius from the circle's centre to a point on the circle in order to build an inscribed regular hexagon. Then, after rotating the radius around the circle's centre, he would mark six evenly spaced locations on the circle's perimeter.

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While a common rule of thumb is to have a minimum sample size of 100 for continuous data to be statistically significant, the actual appropriate sample size may vary depending on the specific study design and research question. Option(D)

In statistics, the term "statistical significance" refers to whether an observed effect or relationship in the data is likely to be real and not just due to random chance. To determine statistical significance, we often perform hypothesis testing.

The sample size is a crucial factor in hypothesis testing. A larger sample size generally provides more reliable and precise estimates of population parameters and increases the statistical power of the test. With a larger sample size, even smaller effects or differences between groups can become statistically significant.

While there is no hard and fast rule for the minimum sample size to achieve statistical significance, a common guideline is to aim for at least 30 samples. This guideline is often used in the context of the Central Limit Theorem, which states that the sampling distribution of the sample mean becomes approximately normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

In practice, the appropriate sample size depends on various factors, including the nature of the data, the effect size being studied, the desired level of confidence, and the statistical test used. Researchers often conduct sample size calculations based on these factors before conducting their studies to ensure they have an adequate sample size to achieve meaningful results and detect significant effects if they exist.

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