The sales tax rate for a local county is 8.7%. If an item costs $184 (before tax), how much sales tax will be due?

Final answer:

To convert z = -14 + 8i to polar form, calculate the magnitude as √(260)≅17, and the argument by adding π to the arctan(8/-14), which accounts for the angle being in the second quadrant. Express the number in polar form using these values.

Explanation:

To convert the complex number z = -14 + 8i from rectangular form to polar form, we start by finding the magnitude (r) and the argument (θ) of the complex number. The magnitude is given by √(x² + y²), which in our case is √((-14)² + 8²) = √(196 + 64) = √(260)≅17. The argument θ is the angle with respect to the positive x-axis which can be found using the arctan function: θ = arctan(y/x). However, since our x is negative and y is positive, our complex number is in the second quadrant, so we need to add π radians (or 180 degrees) to the arctan(y/x) value to obtain the correct argument in the complex plane.

θ = arctan(8/-14) + π ≅ arctan(-4/7) + π. We then get the polar form: z = r(cos(θ) + i sin(θ)) or z = r eˣiθ. Substituting the magnitude and argument into this formula, we find the polar form of the complex number z.

Final answer:

To calculate the length of the shadow cast by a 28-foot tree, we use proportional relationships, which yield a shadow length of approximately 11.2 feet.

Explanation:

Shadow Length Calculation

To find the length of the shadow cast by a 28-foot tree, we can use the concept of similar triangles. The relationship between the height of an object and the length of its shadow is proportional provided the sun's rays are coming at the same angle.

In this case, a 30-foot tree casts a 12-foot shadow.

We can set up a proportion to compare the two trees to find the unknown shadow length for the 28-foot tree.

Let's set up the proportion: Since the shadows are proportional to their respective tree heights, we can express this as follows:

(Shadow length of 28-foot tree) / 28 feet = 12 feet / 30 feet

By cross-multiplying and solving for the unknown shadow length, we would get:

Performing this calculation:

So, a 28-foot tree will cast a shadow that is approximately 11.2 feet long.

The correct answer is that the length of the shadow cast by a 28-foot tree is 11.2 feet.

To solve this problem, we can use the concept of similar triangles. The tree and its shadow form a right-angled triangle with the ground, and the angle of elevation of the sun is the same for both trees. Therefore, the triangles formed by the trees and their respective shadows are similar.

Let's denote the height of the first tree as [tex]\( h_1 = 30 \)[/tex] feet, the length of its shadow as [tex]\( s_1 = 12 \)[/tex] feet, the height of the second tree as [tex]\( h_2 = 28 \)[/tex] feet, and the length of its shadow as [tex]\( s_2 \)[/tex], which we want to find.

Since the triangles are similar, the ratios of the corresponding sides are equal:

[tex]\[ \frac{h_1}{s_1} = \frac{h_2}{s_2} \][/tex]

Substituting the known values:

[tex]\[ \frac{30}{12} = \frac{28}{s_2} \][/tex]

Now, we solve for [tex]\( s_2 \)[/tex] :

[tex]\[ s_2 = \frac{28 \times 12}{30} \][/tex]

[tex]\[ s_2 = \frac{336}{30} \][/tex]

[tex]\[ s_2 = 11.2 \][/tex]

Answer:

30

Step-by-step explanation:

-10 = -b/4

to get b multiply both sides by 4

-10 *4 = -40

since b already has the negative sign the 40 would be positive

 so b = 40

Final answer:

After converting Abeer's scores to percentages, it is clear that she did better in the German test with a score of 70%, as opposed to the French test where she scored 68%.

Explanation:

To find out in which test Abeer did better, we need to compare her test scores in percentages. For the French test, Abeer scored 34 out of 50, which we can convert to a percentage by dividing 34 by 50 and then multiplying by 100:

34 ÷ 50 = 0.68 × 100 = 68%

For the German test, she scored 14 out of 20, which we also convert to a percentage:

14 ÷ 20 = 0.70 × 100 = 70%

By comparing these percentages, we can see that Abeer did better in the German test where she scored 70% compared to the French test where she scored 68%.

The total number of ways can first, second and third place can be assigned are:

                          6840

The total number of people that compete are: 20

Out of these 20 people we have to chose 3 people and also arrange them according to their ranks.

We know that when we have to choose and  arrange r items out of total n items then we need to use the method of permutation.

Hence, the formula is given by:

[tex]n_P_r=\dfrac{n!}{(n-r)!}[/tex]

Here we have:

n=20 and r=3

Hence, the number of ways first, second and third place can be assigned are:

[tex]{20}_P_3=\dfrac{20!}{(20-3)!}\\\\\\{20}_P_3=\dfrac{20!}{17!}\\\\\\{20}_P_3=\dfrac{20\times 19\times 18\times 17!}{17!}\\\\\\{20}_P_3=20\times 19\times 18\\\\\\{20}_P_3=6840[/tex]

         Hence, the answer is:

                 6840

Answer:

The correct answer is

1. 6,840

2. 342

3. 18

At a gymnastics meet, twenty gymnasts compete for first, second, and third place.

Answer:

3 years and 3 months.

Step-by-step explanation:

Since, the amount formula in compound interest,

[tex]A= P(1+\frac{r}{100})^t[/tex]

Where,

P = principal amount,

r = rate per period,

t = number of periods,

Here, A = $ 8,000, r = 6.9%, P = $6,449.82,

By substituting the values,

[tex]8000 = 6449.82(1+\frac{6.9}{100})^t[/tex]

[tex]\frac{8000}{6449.82}=(1+\frac{6.9}{100})^t[/tex]

[tex]\ln (\frac{8000}{6449.82})=t\ln(1.069)[/tex]

[tex]\implies t = \frac{\ln (\frac{8000}{6449.82})}{\ln(1.069)}=3.228[/tex]

Hence, it would be take 3.228 years or 3 years and 3 months.

Final answer:

The sum of -4 and 3 is -1, while the sum of -3 and 5 is 2. The sum of two numbers depends on their signs and the rules of adding negative and positive numbers.

Explanation:

The sum of -4 and 3 is -1, while the sum of -3 and 5 is 2. The reason why the sum of -4 and 3 is a smaller number than the sum of -3 and 5 is because of the rules of adding positive and negative numbers:

In this case, -3 is larger than -4, so when adding 5 to -3, we retain the negative sign and get -1. However, when adding 3 to -4, we retain the negative sign and get -1, which is smaller than 2.

This will be the same area as that inside a circle with a radius of 5

So....the area will  be  ≈  pi * 5^2  ≈    25 pi  units^2

The area of the region of the complex plane defined by |z|<5 is 78.5 square units.

The region of the complex plane defined by |z|<5 is the inside of the circle with radius 5 centered at the origin. To find the area of this region, we can use the formula for the area of a circle, which is A = πr². In this case, the radius is 5, so the area is A = 3.14 × 5² = 78.5 square units.

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The operations on geometric shapes that involves moving, flipping, change of shape, reflection and new shape creation through operations on another shape is known as transformations

The reason the above values are correct is as follows:

The given transformation sequence is T₍₋₂, ₄₎ [tex]\circ[/tex] R₀ 180°

Where;

T₍₋₂, ₄₎ = A translation two (2) units to the left and four (4) units upwards

R₀ 180° = A rotation of 180° about the origin

The image and preimage of a 180° rotation transformation about the origin is as follows;

Preimage (x, y) [tex]\underset \longrightarrow {R_o \ 180 ^{\circ} }[/tex]Image(-x, -y)

It is to be noted that the first transformation to be performed in a composite transformation is the transformation to the right

The composite transformation is therefore presented as follows;

Combined; (x, y) [tex]\underset \longrightarrow {T_{-2, \ 4} \circ R_o \ 180 ^{\circ} }[/tex] (-x - 2, -y + 4 )

The coordinates of the vertices points on triangle ΔXYZ are X(2, 5), Y(3, 2), Z(4, 3)

Therefore;

The coordinates of the vertices points on triangle ΔX'Y'Z' after the composite transformations are;

Which gives that the coordinates of the triangle ΔX'Y'Z' after the transformation, T₍₋₂, ₄₎ [tex]\circ[/tex] R₀ 180° are;

X'(-4, -1), Y'(-5, 2), and Z'(-6, 1)

Learn more about composite transformations here:

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The only graph with a slope of 4/5 is;

Graph 2

The slope of the graph is simply rise divided by run or defined as the change in y-coordinates divided by the corresponding change in x-coordinate.

Thus; m = (y₂ - y₁)/(x₂ - x₁)

Graph 1; Let us pick two coordinates;

(1, 2) and (2, 3)

m = (3 - 2)/(2 - 1)

m = 1

Graph 2; Let us pick two coordinates;

(-1, 1 ) and (4, 5)

m = (5 - 1)/(4 - (-1))

m = 4/5

Graph 3; Let us pick two coordinates;

(-2,0) and (3, 2)

m = (2 - 0)/(3 - (-2))

m = 2/5

Graph 4; Let us pick two coordinates;

(1, 1) and (4, 5)

m = (5 - 1)/(4 - 1)

m = 4/3

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The problem is an algebraic translation of a real-life situation using inequalities. The number of loaves, for both banana and zucchini bread, must be between 4 and 12, with banana bread needing to be twice as many as zucchini.

This question deals with the concept of inequality and translating word problems into algebraic expressions. In this question, if x represents the number of zucchini bread loaves you bake and y represents the number of banana bread loaves, then the inequalities can be written as:

These inequalities mean that the total number of loaves of bread, including both banana and zucchini, should be no fewer than 4 and no more than 12, and the number of banana bread loaves should be at least twice the number of zucchini bread loaves.

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