Given z1 = 5(cos 240° + isin 240°) and z2 = 15(cos 135° + isin 135°), what is the product of z1 and z2?

Solution

We want to solve

The zero product property states that if a⋅b=0 then either a or b equal zero. This basic property helps us solve equations like (x+2)(x-5)=0.

For the question,

The answer is x = 0 and x = 1/3

To write this in standard form, we need to look at the power of 10. In this case, it's a 7. Then, we need to shift the decimal point 7 places to the right:

We can see that behind the 1 (where was the decimal point before) now there are 7 numbers, then what we do is correct.

Question:

Find the length of AC.

Solution:

Applying the Pythagorean Theorem, we get:

this is equivalent to:

then, we can conclude that the length of AC side is:

The value of the given expression x^6 -5x^4 - 5 + x simplified as;  x^4(x^2 - 5) - (5 + x)

The statement is given as ;

Factor: [tex]x^6 -5x^4 - 5 + x[/tex]

From the given expression, we have

[tex]x^6 -5x^4 - 5 + x[/tex]

Group the expression in two part;

So, we have;

[tex]x^6 -5x^4 - 5 + x = (x^6 - 5x^4) - (5 + x)[/tex]

Now Factorize each group of the expression;

[tex]x^6 -5x^4 - 5 + x = x^4(x^2 - 5) - (5 + x)[/tex]

The given expression cannot be further simplified.

Hence, the value of [tex]x^6 -5x^4 - 5 + x is x^4(x^2 - 5) - (5 + x)[/tex]

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We know that

• The earnings are $7,592.

• The time is 16 years.

• The annual interest rate is 9.04%.

This problem is about simple interest, its formula is

Where A = 7,592, r = 9.04, t = 16, and P is the principal.

Let's replace each value.

Notice that 9.04% is equivalent to 0.0904.

Now, we solve it for P.

a) The inverse of a statement : If If two angles are not supplementary then the angles are not a linear pair. FALSE

b) The converse of a statement : If the angles are a linear pair, then the two angles are supplementary. TRUE

c) The contrapositive of a statement : If the angles aren't supplementary, then they aren't linear pair . FALSE

The given statement is : If two angles are supplementary then the angles are a linear pair .

(a) The inverse of a statement : If If two angles are not supplementary then the angles are not a linear pair.

This statement is FALSE because supplementary angles do not have to be adjacent, whereas a linear pair must be adjacent and create a straight line. So, no, supplementary angles are not always linear pairs. However, linear pairs are always supplementary.

(b) The converse of a statement : If the angles are a linear pair, then the two angles are supplementary.

This statement is TRUE because in a linear pair, if the two angles have a common vertex and a common arm, then the non-common side makes a straight line and the sum of the measure of angles is 180°. Linear pairs are always supplementary.

(c) The contrapositive of a statement : If the angles aren't supplementary, then they aren't linear pair .

This statement is FALSE because if they are adjacent and share a vertex and one side. See the first picture below. They might not form a linear pair, like in a parallelogram.

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Answer

The exact value of the lateral surface area = 395 cm²

The exact value of the total surface area = 572 cm²

Explanation

The solid shape is a cone with a height of 15 cm and the diameter of the base of the solid shape is 15 cm.

The Lateral Surface Area of the Solid Shape:

The formula to calculate the lateral surface area (LSA) of a cone is given by:

The radius will be = diameter/2 = 15cm/2 = 7.5 cm

To find legth, we use Pythagoras rule:

Put π = 3.14, r = 7.5 cm, and l = 16.77 cm into the lateral surface area formula:

Therefore, The exact value of the lateral surface area = 395 cm²

Total Surface Area of the Solid Shape:

To find the exact value for the total surface area of the solid shape, we use the total surface area (TSA) formula of a cone which is:

put π = 3.14, r = 7.5 cm, and l = 16.77 cm

The exact value of the total surface area = 572 cm²

Answer:

5/10

15/30

Step-by-step explanation:

The triangles are similar, so the lengths of corresponding sides are in the same ratio.

One pair of corresponding sides is 5 and 10.

10 is twice 5.

5/10

The other pair of corresponding sides is 15 and ?.

? must be twice 15, so ? is 30.

15/30

SOLUTION :

Step 1 :

In this question, we have that the recipe for trail mix requires

5/6 cup of raisins for 1 batch.

Then, Alaina uses

Step 2 :

To get the number of batches of trial mix Alaina makes, we have that :

CONCLUSION :

There would be 2 batches .

We have the following equation

Now we solve for "x", we must use the logarithmic properties

Point = (8,-4)

When coordinate points (x,y) are rotated by 90° the image became (y,-x)

So, for this case:

Image: (-4,-8)

We are asked to determine the equation of a line that is parallel to:

Two lines are parallel if their slopes are in the following relationship:

Therefore, if m1 is the slope of the given line and m2 is the slope of the parallel line we may determine the value of the slope of the parallel line by replacing it in the relationship. Let's remember that when a line is written in the form:

Where "m" is the slope. Therefore, m1 is -2/3. Replacing in the relationship we get:

Now, we go back to the general form of a line equation and replace the value of the new slope:

The value "b" is the y-intercept and can be found using the point through which the line passes:

Now we solve the operations:

Subtracting 21/2 from both sides:

Solving the operations:

Now we replace in the line equation:

And thus we get the equation of the parallel line.

Answer:

Step-by-step explanation:

The centre of mass of 40g, (1,3); 30g, (2,-1); 70g, (0,0) and 50g (0,-2) is located at (10/19,-1/19).

The formula to find the coordinates (X,Y) of centre of mass of a discrete particles system is,

X = (m₁x₁+m₂x₂+m₃x₃+m₄x₄)/(m₁+m₂+m₃+m₄)

Y = (m₁y₁+m₂y₂+m₃y₃+m₄y₄)/(m₁+m₂+m₃+m₄)

As we know,

m = 40

m = 30

m = 70

m = 50

Putting all the values in the formula,

for X coordinate,

X = [(40×1)+(30×2)+(70×0)+(50×0)]/(40+30+70+50)

X =(40+60+0+0)/(190)

X = 100/190

X = 10/19

For Y coordinate,

Y = [(40×3)+(30×-1)+(70×0)+(50×-2)]/(40+30+70+50)

Y= (120-30+0-100)/(190)

Y = -10/190

Y = -1/19

So, the centre of mass of the discrete particle system is at (10/19,-1/19)

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Andrew will have $9,450 in his savings account that earns 5% interest per year.

According to the question,

We have the following information:

Principal amount in Andrew's savings account = $9,000

Interest rate = 5% per year

Time = 1 year

We know that we use the following formula to find the simple interest on any amount:

Simple interest = (principal*rate*time)/100

Simple interest = (9000*1*5)/100

Simple interest = $450

Now, the total amount in his savings account will be the sum of the amount earned from the interest and the principal amount submitted by him.

Total amount = interest+principal

Total amount = 450+9000

Total amount = $9,450

Hence, he will have $9,450 in his savings account.

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Answer:

4 and 1/2.

Step-by-step explanation:

There are 9 oranges, so there are not enough for both to get a whole amount.

They will split the last one into half, so they has 4 and 1/2 oranges each.

Answer: 4 and 1/2

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Answer:

(0,-4) and (-1,-6) or B

Step-by-step explanation:

Notice that equation 1 and equation 2 are both equal to y.

Substitute equation 2 into equation 1:

[tex]2x-4=x^2+3x-4[/tex]

Simplify:

[tex]x^2+x=0[/tex]

Factor:

[tex]x(x+1)=0[/tex]

Notice that the zeros for x are x = 0 and x = -1.

Now plug both values into either equation 1 or 2 to find y:

[tex]y=0^2+3(0)-4 = -4[/tex]

[tex]y=(-1)^2+3(-1)-4=-6[/tex]

Therefore, our values are (0,-4) and (-1,-6) or option B

ANSWER

B. 25

EXPLANATION

We want to identify the number of students that live between 1 and 1.9 miles from school.

To do this, we have check the frequency corresponding to the bar for 1 - 1.9 miles on the frequency axis.

From the histogram, we see that the number of students that live between 1 and 1.9 miles from the school is 25.

The correct answer is option B.

Answer:

I really don't understand much about math sometimes I need help

Given

a) The car has its original (initial) value when no time has passed since it was bought; in other words t=0. Then,

The initial value is 19900.

b) After 11 years, t=11; then,

Rounded to the nearest tenth, the second answer is 2924

From the fact that the sum of the angle measures of a triangle must be equal to 180º and that two angles have measures of 40º and 70º, we can deduce that the third angle must have a measure of 70º.

Therefore, this was a deductive argument.

Given: A system of linear equations in three variables x, y, and z as follows-

Required: To solve the system using elimination.

Explanation: Let the given system as-

We can solve the system by reducing the system to a system of 2 variables. Suppose we would like to remove the variable z.

Multiplying equation (1) by 2, adding to equation (2) as follows-

Now, add equations (1) and (3) as follows-

Now, equations (4) and (5) represent a system of linear equations in two variables. Subtracting the equations as follows-

Substituting x=3 in equation (4)-

Substituting x=3 and y=-1 in equation (1) as follows-

Final Answer: The solution to the system is-

Answer:

(3000/(5-3))(3+5)

= (3000/2) 8

= 12000

Step-by-step explanation:

Answer: x=107​=0.7

Step-by-step explanation:

1001​=70x​Multiply both sides by 70.1001​×70=xMultiply 1001​ and 70 to get 10070​.10070​=xReduce the fraction 10070​ to lowest terms by extracting and canceling out 10.107​=xSwap sides so that all variable terms are on the left hand side.x=107​

The y-intercept is found replacing x = 0 into the equation as follows:

5*0 + y = 5

y = 5

Then, the line intercepts the y-axis at (0, 5)

The x-intercept is found replacing y = 0 into the equation as follows:

5x + 0 = 5

5x = 5

x = 5/5

x = 1

Then, the line intercepts the x-axis at (1, 0)

The line is graphed connecting these two points

the area of the windshield, cleaned by the windshield wiper is mathematically given as 0.096inche^2

This is further explained below.

The term "acute area of the windshield glazing" refers to the section of the windshield that measures eight and one-half inches by eleven inches and is located immediately in front of the driver's line of sight, as seen in the image.

Then, let's call the part of the windshield labeled "ABC" A.

[tex]A=\frac{1}{2} r^2 \theta[/tex]

Now, substituting the given values we get,

[tex]\begin{aligned}&A=\frac{1}{2} r^2 \theta \\&A=\frac{1}{2} \times(7)^2 \times \frac{\pi}{180} \\\end{aligned}$$[/tex]

A=0.4276

Then let the area of the windshield, ADE, be denoted by the letter A',

[tex]A^{\prime}=\frac{1}{2} r^2 \theta[/tex]

Now, after replacing those values with the ones we were provided,

[tex]\begin{aligned}&A^{\prime}=\frac{1}{2} r^2 \theta \\&A^{\prime}=\frac{1}{2} \times(1)^2 \times 60 \times \frac{\pi}{180} \\&A^{\prime}=0.5236\end{aligned}[/tex]

In conclusion, In order to calculate the area of the windshield, you need to take the difference between A and A'.

0.5236-0.4276=0.096

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By using commutative property the result after solving expression (1/4 + 1/3 + 3/4) will be 7/4.

The commutative property of addition for three numbers is given by -

a + (b + c) = (a + b) + c

Given is the following expression -

1/4 + 1/3 + 3/4

We have the following expression -

1/4 + 1/3 + 3/4

Now, lets solve the expression by adding first two terms first and than add the third term to result of the addition of first two terms. Mathematically -

(1/4 + 1/3) + 3/4

Let (1/4 + 3/4) = K

Than the expression becomes -

K + 3/4

Now, first solve K, we will get -

K = 1/4 + 3/4

K = 4/4 = 1

Now, adding the resultant [K] to third term -

1 + 3/4

1/1 + 3/4

(4 + 3)/4

7/4

So, using commutative property, we have solved the expression (1/4 + 1/3 + 3/4) and the final value will be 7/4.

Therefore, by using commutative property the result after solving expression (1/4 + 1/3 + 3/4) will be 7/4.

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The graph of the function is

The domain of the function is determined as the x -value of the function that satisfy the given function.

We have the equation:

And we want to find the value n, correspondign to the number of years necessary in order to the substance to decay in half.

Let's say that we have 1 Kg of the substance, this is Wo, the initial weight. Since we want to find the the decay of half the substance we use W = 1/2

And write:

Now we can use a property of logarithms:

Thus applying natural log on both sides:

By the property:

We can divide on both sides by ln(1/2):

The number of years for the radioactive substance to decay to half its initial weigh are 100 years.

The step to get rid of the ln(1/2) is:

To find the number of term, we will use the formula;

when n= 1

for the second term

n = 2

= 3 + 5

=8

For n= 3

=3 + 2(5)

=3 + 10

=13

For n=4

=3+3(5)

= 3 + 15

= 18

For n= 5

= 3 + 4(5)

= 3 + 20

= 23

Therefore, the terms; 3, 8, 13, 18 and 23