6. The Golden Gate Bridge was built at a height of 300 m over San Francisco
Bay. From the ends of the bridge, the angles of depression to a tugboat in
the middle of the river are 25° and 32° respectively.
Find the length of the bridge.

The result of simplifying the number 4¹/₂ . 4¹/₂  using the properties of exponents is 4

To solve this exercise we have to resolve algebraic operations following the exponent properties.

4¹/₂ . 4¹/₂

Base= 4

Using the product rule that indicates that: the exponent result will be the addition of these exponents, we have:

(4) ⁽¹/₂ ⁺ ¹/₂⁾

(4) ⁽¹⁺¹⁾/₂

(4) ⁽²/₂⁾

(4)⁽¹⁾

4

In mathematics an exponent is the number of time that a number, called (base) is multiplied by itself. It is also called, power or index.

Example: 3² = 3*3 = 9

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[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-2}~,~\stackrel{y_1}{3})\qquad (\stackrel{x_2}{-7}~,~\stackrel{y_2}{-7})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(~~-7 - (-2)~~)^2 + (~~-7 - 3~~)^2} \implies d=\sqrt{(-7 +2)^2 + (-7 -3)^2} \\\\\\ d=\sqrt{( -5 )^2 + ( -10 )^2} \implies d=\sqrt{ 25 + 100 } \implies d=\sqrt{ 125 }\implies d\approx 11.18[/tex]

Exact Distance = [tex]\boldsymbol{5\sqrt{5}}[/tex] units

Approximate Distance = 11.1803 units

===================================================

Work Shown:

[tex](x_1,y_1) = (-2,3) \text{ and } (x_2, y_2) = (-7,-7)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(-2-(-7))^2 + (3-(-7))^2}\\\\d = \sqrt{(-2+7)^2 + (3+7)^2}\\\\d = \sqrt{(5)^2 + (10)^2}\\\\d = \sqrt{25 + 100}\\\\d = \sqrt{125}\\\\d = \sqrt{25*5}\\\\d = \sqrt{25}*\sqrt{5}\\\\d = 5\sqrt{5}\\\\d \approx 11.1803\\\\[/tex]

The exact distance is [tex]5\sqrt{5}[/tex] units, which approximates to about 11.1803 units

I used the distance formula. Round the approximate value however your teacher instructs.

A slight alternative is to plot the two points to form a right triangle. The hypotenuse goes from (-2,3) to (-7,-7). Then use the pythagorean theorem.

You can use tools like WolframAlpha or GeoGebra to confirm the answer.

Answer:

$3.42 spent on folders, She will receive $6.58 back in change

Step-by-step explanation:

$0.57 times 6 = $3.42 spent on the folders. $10-$3.42=$6.58 she will receive back.

Part 1: She spent $3.42 on the folders.

This is calculated by multiplying 6 folders by $0.57 for each folder

($0.57 x 6 = $3.42).

Part 2: She will receive $6.58 in change. This is calculated by subtracting the amount spent on the folders from the amount of money Gi Gi had to begin with ($10 - $3.42 = $6.58).

The logarithm statement log_b x / log_b y = log_b x/y is a false statement as log_b x / log_b y = log_b x - log_b y.

We are given the logarithm as:

log_b x / log_b y = log_b x/y

We need to tell if it is true of not.

We get that, it is false statement.

We know that log a / log b = log a - log b

So, log_b x / log_b y = log_b x - log_b y

So, it is not equal to log_b x / y

Therefore, we get that, the logarithm statement log_b x / log_b y = log_b x/y is a false statement as log_b x / log_b y = log_b x - log_b y.

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A quadrilateral has four sides that are each an even number of meters long (in meters).Considering a 46 meter of perimeter, the value of the longest side length out of the four sides is 12.

Given that,

A quadrilateral has four sides that are each an even number of meters long (in meters).

Considering a 46 meter of perimeter.

We know that even numbers differ by 2 and are multiples of 2.

Thus, due to the four sides of quadrilaterals.

let those sides' lengths be

2x,2x+2,2x+4 and 2x+6

The perimeter of a quadrilateral's boundary as a whole is its perimeter. An irregular or regular four-sided polygon is referred to as a quadrilateral. A regular quadrilateral has sides that are all the same length and angles that are all the same measure; an irregular quadrilateral does not have equal sides and angles.

Perimeter of a quadrilaterals is

perimeter=sum of the 4 sides.

36=sum of the 4 sides.

Which we have taken 2x,2x+2,2x+4 and 2x+6.

2x+2x+2+2x+4+2x+6=36

8x+12=36

8x=36-12

8x=24

x=[tex]\frac{24}{8}[/tex]

x=3.

Now, we can find the sides by the x value which is 3.

1st side=2x=2[tex]\times[/tex]3=6

2st side=2x+2=2[tex]\times[/tex]3+2=8

3st side=2x+4=2[tex]\times[/tex]3+4=10

4st side=2x+6=2[tex]\times[/tex]3+6=12

Therefore, the value of the longest side length out of the four sides is 12.

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Answer:13

Step-by-step explanation:

x+x+1+x+2+x+3=46

4x+6=46

-6 -6

4x=40

4x/4=40/4

x=10

Longest side: x+3

=10+3

=13

Answer:

[tex]y=\frac{ac}{a-b-c}[/tex]

Step-by-step explanation:

[tex]y(a-b)=c(y+a) \\ \\ y(a-b)=cy+ac \\ \\ y(a-b)-cy=ac \\ \\ y(a-b-c)=ac \\ \\ y=\frac{ac}{a-b-c}[/tex]

Answer:

74

Step-by-step explanation:

[tex]5(2(10)-5)-1 \\ \\ =5(20-5)-1 \\ \\ =5(15)-1 \\ \\ =75-1 \\ \\ =74[/tex]

Answer:

154

Step-by-step explanation:

Use the correct order of operations.

(120 - 14) + 4² × 3 =

First, do the parentheses and the exponent.

= 106 + 16 × 3

Now do the multiplication.

= 106 + 48

Finally, add.

= 154

Answer in decimal form = -26.25

Answer as an improper fraction = -105/4

========================================================

Work Shown:

1/2 = 0.5

Replace every copy of x with 0.5 in the f(x) function.

f(x) = x-8

f(0.5) = 0.5-8

f(0.5) = -7.5

Repeat the same idea for the g(x) function

g(x) = x+3

g(0.5) = 0.5+3

g(0.5) = 3.5

Then multiply the results

f(0.5)*g(0.5) = (-7.5)*(3.5) = -26.25

Then optionally convert to an improper fraction following the steps shown below.

-26.25 = -(26.25)

-26.25 = -(26+0.25)

-26.25 = -(26+1/4)

-26.25 = -(26*(4/4)+1/4)

-26.25 = -(104/4+1/4)

-26.25 = -105/4

Answer:

b = [tex]\frac{3}{4}[/tex]

Step-by-step explanation:

[tex]\frac{3}{8}[/tex] = [tex]\frac{b}{2}[/tex] ( cross- multiply )

8b = 6 ( divide both sides by 8 )

b = [tex]\frac{6}{8}[/tex] = [tex]\frac{3}{4}[/tex]

Answer:

b=3/4

Step-by-step explanation:

3/8=b/2

First you will need to cross multiply because apparently that's how we do things. To cross multiply, you would just have to take the bottom of the fraction from the LHS (left hand side) and times it with the top part of the fraction in RHS (right hand side). You do the same thing to the other side (take the top part of LHS and multiply it with the bottom of RHS). Make sure to keep the = sign in the middle.

8 x b = 3 x 2 (8 x b= 8b)

8b=6

Then, you would have to make the b on the LHS alone so you would have to divide the RHS by 8. (To move from one side to another, you would have to change it to the opposite of the sign like if it's -9, then to the other side it would be +9 and if it's x8 then the other side would be /8 and vice versa).

b=6/8

You will have to simplify unless the question asks you not to but I'll do it anyways. 2 goes into 6 and 8 as a common factor and so the answer should look like this:

b=3/4

If you have any further questions please ask me!

The constant of variation is 1/4 .

The inverse variation property states that , whenever the product of values of two quantities is a constant value then there is opposite variation between these two corresponding values. Or we can also say that if the productive nature of two corresponding values is constant, then one quantity is inversely varies with second quantity. That means if one value increases than another one will automatically decrease and vice versa.

Let say if x and y are two values and k is a constant value, then

According to inverse variation,

k = xy or y = k/x

In the given question,

Take the order pair  [tex](\frac{3}{8} ,\frac{2}{3} )[/tex]

As it is an inverse variation, we can substitute it

y=k/x  or k = xy

put the values from order pair

k = [tex]\frac{3}{8}[/tex] × [tex]\frac{2}{3}[/tex]

k = [tex]\frac{1}{4}[/tex]

Therefore, the constant of variation for order pair  [tex](\frac{3}{8} ,\frac{2}{3} )[/tex] is 1/4 .

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The factors are (x + 4), (x + 5) & (x - 4) and the solution is x = -4,  = -5 and x = 4 and the value of the maximum profit is 680

The given parameters are

f(x) = x^3 + 5x^2 - 16x - 80

x = -4 is a root

To set up the synthetic division, we start by writing out the coefficient of each term

So, we have

-4 |  1    5    -16    -80

Bring down the first coefficient

-4 |  1    5    -16    -80

      1

Multiply by -4

So, we have

-4 |  1    5    -16    -80

           (-4)

      1

Add

-4 |  1    5    -16    -80

           (-4)

      1     1

Multiply by -4

So, we have

-4 |  1    5    -16    -80

           (-4)  (-4)

      1     1

Add

So, we have

-4 |  1    5    -16    -80

           (-4)  (-4)

      1     1     -20

Multiply by -4

-4 |  1    5    -16    -80

           (-4)  (-4)    80

      1     1     -20

Add

So, we have

Multiply by -4

-4 |  1    5    -16    -80

           (-4)  (-4)    80

      1     1     -20    0

This means that,

Quotient = x^2 + x - 20

Remainder = 0

Expand x^2 + x - 20

x^2 + x - 20 =x^2 + 5x - 4x - 20

Factorize

x^2 + x - 20 = (x + 5)(x - 4)

Recall that the initial root is

x = -4

Rewrite as

x + 4 = 0

So, we have the complete factor to be

f(x) = (x + 4)(x + 5)(x - 4)

Set to 0

(x + 4)(x + 5)(x - 4) = 0

Solve for x

x = -4,  = -5 and x = 4

Hence, the factors are (x + 4), (x + 5) & (x - 4) and the solution is x = -4,  = -5 and x = 4

The equation of the profit function is given as

P(x) = 230 + 30x - 0.5x^2

Differentiate the function

So, we have

P'(x) = 0 + 30 - x

Evaluate the like terms in the function

So, we have

P'(x) = 30 - x

Set the function to 0

So, we have

30 - x = 0

Evaluate

x = 30

Substitute x = 30 in the equation P(x) = 230 + 30x - 0.5x^2

So, we have

P(30) = 230 + 30 x 30 - 0.5 x 30^2

Evaluate

P(30) = 680

Rewrite as

Pmax = 680

Hence, the value of the maximum profit is 680

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Answer:

QR = 3

Step-by-step explanation:

given Q is the midpoint of PR , then

PQ = QR , that is

3x = x + 2 ( subtract x from both sides )

2x = 2 ( divide both sides by 2 )

x = 1

Then

QR = x + 2 = 1 + 2 = 3

Answer:

QR = 3

Step-by-step explanation:

since Q is the midpoint, then PQ is the same as QR. so we equate the two as follows:

3x = x+2

then find x as shown below.

3x-x = 2 ( I grouped the like terms)

2x = 2. now divide by 2 to find x

x= 1,

since QR is x+2, now substitute x for 1

1+2 =QR

3 is the answer.

.thanks

Answer:

36 × 12 = 432

hopefully this helps :)

Answer:

12

Step-by-step explanation:

In this problem, we need to arrange 150 chairs in rows for a school play. You want each row to have the same number of chairs.

We can find the prime factorization of 150 as follows :

150 = 2, 3, 5, 5

150 is not a perfect square. 144 is a perfect square and 12² = 144. So, there are 12 possible arrangements.

Only the third statement (c)"To square a number multiply the number by itself" is true.

The square of any number is defined as the product of the number and itself .

Let us consider a number a , then the square of a is denoted by a² and  a² = a × a .

Let us take each statement and verify if they are true or false.

a)"To square a number, multiply the number by 2"

this is false as to square we have to multiply the number with itself.

example: 3²=9 but 3 × 2 = 6

b)The second statement is also false because square-root of a number is different than dividing a number by 2.

Example: √16=4 but 16÷2=8

c) "To square a number, multiply the number by itself."

This statement is true .

d) a perfect square is defined as a number whose square-root is an integer (both odd and even)

9 is a perfect square and √9=3 which is odd

Therefore the fourth statement is also false.

Hence only statement (c) is correct .

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The number of pairs of parallel sides the rotated figure have is 2

Parallel sides or lengths are line segments or lines in the same plane that extend on both sides without intersecting

The shape is given as:

Shape = Rhombus

Rotation= 90 degrees counterclockwise rotation about the vertex G

The rotation transformation is a rigid transformation

This means that when a shape is rotated by 90 degrees counterclockwise about a vertex (in this case, the vertex G), the image of the rotation would have the same properties as the preimage

A rhombus has 2 pairs of parallel sides

This means that the number of pairs of parallel sides the rotated figure have will also be 2 (same as the original shape)

Hence, the number of pairs of parallel sides the rotated figure have is 2

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The value of x when r(x) = -4 is 20/3.

Given, the function is r(x) = ₋2/3x ₊ 4

substitute r(x) = ₋4 in the value of r(x)

r(₋4) = ₋2/3(₋4) ₊ 4

r(₋4) = 8/3 ₊ 4

r(₋4) = 8 ₊ 12/3

r(₋4) = 20/3

hence we get the value as 20/3

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Answer:

7, 5

Step-by-step explanation:

Change the denominator to make the end result 0. Anything with a 0 denominator results in it being undefined.

The expression (10-t)/(t-7) is undefined when t=7.

The expression 10/(3t-15) is undefined when t=5.

To find the values of t for which the expressions are undefined, we need to look for any values of t that make the denominators equal to zero.

we know that dividing by zero is undefined.

For the expression (10-t)/(t-7),  the denominator is t-7.

So, this expression is undefined when t-7=0

To find the value of t that makes t−7=0, solve the equation:

t-7=0

t=7

So, the expression (10-t)/(t-7) is undefined when t=7.

For the expression 10/(3t-15), the denominator is t-7.

So, this expression is undefined when 3t-15=0

To find the value of t that makes 3t-15=0, solve the equation:

3t-15=0

Add 15 on both sides:

3t=15

Divide both sides by 3:

t=5

So, the expression 10/(3t-15) is undefined when t=5.

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Answer:

x = 4

Step-by-step explanation:

[tex]\frac{x}{7}[/tex] = [tex]\frac{28}{49}[/tex] ( cross- multiply )

49x = 7 × 28 = 196 ( divide both sides by 49 )

x = 4

Answer: 0.004 = 4.0 x [tex]10^{-3}[/tex]

Step-by-step explanation:

Rules guiding exponential notation :

Writing numbers in exponential notations have some guiding rules. These include:

The base is always 10

The exponent cannot be zero and must be an integer. It can be positive or negative.

The coefficient can take any form - negative or positive, integer or decimal. All significant digits must be included after the decimal.

Following these rules :

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A parabola's maximum value is the y-coordinate of its vertex when it opens downward.

Therefore, a parabola's maximum value is the y-coordinate of its vertex when it opens downward.

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Answer:

2 and - [tex]\frac{1}{2}[/tex]

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the gradient and c the y- intercept )

f(x) = 2x - 1 ← is in slope- intercept form

with gradient m = 2

• Parallel lines have equal gradients , then

the gradient of a parallel line is also 2

given a line with gradient m then the gradient of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{2}[/tex]

Answer:

5000

Step-by-step explanation:

5x1000=5000

The radical expression 3√0.008 y^3 x^6 can be simplified into 0.2 y x^2.

Radical expression is an expression containing a √ symbol. It shows the nth root of a number or variable. This number or variable inside a radical expression is the nth power of another number. Radical expression can also be expressed in exponent notation such that the nth root of x is equal to x raised to 1 over n.

n √x = x^1/n

Expressing 3√0.008 y^3 x^6 into exponent notation:

3√0.008 y^3 x^6 = (0.008 y^3 x^6)^1/3

Applying the distributive property of exponents,

(0.008 y^3 x^6)^1/3 = 0.008^1/3 (y^3)^1/3 (x^6)^1/3

Using the Power Rule for Exponents which states that to raise a number with an exponent to a power, multiply the exponent times the power:

(a^m)^n = a^(m*n).

0.008^1/3 (y^3)^1/3 (x^6)^1/3 = 0.008^1/3 y x^2

Simplifying the constant,

0.008^1/3 y x^2 = 0.2 y x^2

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Answer: -13

Step-by-step explanation: so order of operations is also PEMDAS.

and u go from left to right so using this you would multiply 8 x 5 and then you get 35. after this subtract 23-36 and u get -13.

Answer:

[tex]48\sqrt 6[/tex]

Step-by-step explanation:

[tex]2\sqrt{8}=2(2\sqrt{2})=4\sqrt{2} \\ \\ 6\sqrt{12}=6(2\sqrt{3})=12\sqrt{3} \\ \\ (4\sqrt 2)(12\sqrt 3)=48\sqrt 6[/tex]