Another equation question?

The area of the rectangle is 35.07 square feet which can be rounded to 35.1 square feet.

Given that:

Here ABDC is a rectangle and AD is diagonal. Hence, Δ ABC is a right triangle with AD = 9 feet and ∠CAD = 60°.

In ΔCAD,

sin ∠CAD = sin 60°= Perpendicular/Hypotenuse

sin 60° = AC/AD

√3/2 = AC/9

=> AC = 9√3/2 ft

cos ∠CAD = cos 60° = Base/ Hypotenuse

cos 60° = CD/9

1/2 = CD/9

=> CD = 4.5 ft

So, calculating the area of the rectangle = length x breadth

                                                                    = AC x CD

                                                                    = 9√3/2 x 9/2

                                                                     = 81√3/4 ft²

The area of the rectangle = 35.07 ft²  which can be rounded to 35.1 square feet.

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An equation of the function in standard form for the given table of values is 3x² + 24x + 50 = 0.

First, let us understand the standard form of a quadratic equation:

In Mathematics, the standard form of a quadratic equation is given by;

y = g(x) = ax² + bx + c = 0

To write a quadratic function equation in standard form, we would construct the following system of equations from the data in the given table:

5 = a(-5)² + b(-5) + c  ⇒ 5 = 25a - 5b + c     .......equation 1.

2 = a(-4)² + b(-4) + c  ⇒ 2 = 16a - 4b + c      .......equation 2.

5 = a(-3)² + b(-3) + c  ⇒ 5 = 9a - 3b + c       .......equation 3.

14 = a(-2)² + b(-2) + c  ⇒ 14 = 4a - 2b + c    .......equation 4.

From equation 1 and equation 3, we derive:

25a - 5b + c = 9a - 3b + c

⇒ 25a - 9a = -3b + 5b

⇒ 16a = 2b

⇒ b = 8a

Substituting the value of b into equation 2 and 4, we derive the following equations:

2 = 16a - 4 * 8a + c     ⇒ 2 = - 16a + c

14 = 4a - 2 * 8a + c     ⇒ 14 = -12a  + c

By using the elimination method, the value of a is given by:

2 - 14 = (-16a + 12a) + (b - c)

-12 = - 4a

a = 3

Next, we would determine the value of b as follows:

b = 8a

b = 8 * 3

b = 24

For the value of c, we have:

2 = - 16a + c

c = 16 * (3)+ 2

c = 48 + 2

c = 50.

Substituting the respective values of a, b and c into the standard form of a quadratic equation:

ax² + bx + c = 0

3x² + 24x + 50 = 0

Thus, an equation of the function in standard form for the given table of values is 3x² + 24x + 50 = 0.

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The solution of the inequality -1 > 6x + 9 is x < -1.67, therefore -1, -4 and -6 are the solutions of inequality

The inequality is

-1 > 6x + 9

The inequality is the mathematical statement that shows the relationship between two expression with and inequality sign. The inequality relationships are less than, less than or equal, greater than, greater than or equal etc. The equal sign will not be a part of inequality

The inequality is

-1 > 6x + 9

Rearrange the terms

6x + 9 < -1

Subtract both sides by 9

6x + 9 - 9 < -1 - 9

6x < -10

Divide both sides by 6

6x /6 < -10/6

x  < -5/3

x < -1.67

Hence, the solution of the inequality -1 > 6x + 9 is x < -1.67, therefore -1, -4 and -6 are the solutions of inequality

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Answer:

Step-by-step explanation:

360

The length of the border around the cake is the same as the perimeter of the cake, which is D. 44 inches.

The perimeter is the sum of the lengths of the four sides of the rectangular cake.

In other words, the perimeter is the total lengths of the boundary.  The result of the perimeter is always a linear measure in units.

We can compute the perimeter of a rectangle by adding up its four dimensions as follows:

Length = 13 inches

Width = 9 inches

Perimeter = 2(L + W)

= 2(13 + 9)

= 44 inches

Thus, this cake has Option D as the length of the border around it.

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For the given equation -2(5x + 8) = 14 + 6x, the statements for the process used to achieve steps 1 - 4 should be completed as follows:

In Mathematics, an equation can be defined as a mathematical expression which shows that two (2) or more thing are equal.

In this exercise, you're required to describe the steps that should be taken to rearrange and solve for x. Therefore, the appropriate and required steps that were used to achieve steps 1 - 4 include the following:

Multiplying 5x and 8 by -2, we have the following:

-2(5x + 8) = 14 + 6x

-10x - 16 = 14 + 6x

Subtracting 6x from both sides of the equation, we have the following:

-10x - 16 - 6x = 14 + 6x - 6x

-16x - 16 = 14

Next, we would add sixteen (16) to both sides of the equation as follows:

-16x - 16 + 16 = 14 + 16

-16x = 30

Dividing both sides of the equation by -16, we have the following:

x = -30/16

x = -15/8.

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Answer:

Step-by-step explanation:

Ordered pairs

x = 1                        Given

y = 1 - 6 = - 5          Put the given into the equation

Ordered pair: (1,-5)

x = 4

y = 4 - 6

y = - 2

Ordered pair:(4,-2)

Graph

The value of tan(m - p), where the value of sin(m) = -(4/5), and the value of cos(p) = -(15/17) using trigonometric identities is; [tex]tan(m - p) = \dfrac{36}{77}[/tex]

Trigonometric identities are equations that involve trigonometric functions which are true for all values of the input variables.

The information in the question are;

Therefore;

[tex]cos(m) = \sqrt{1- \left(-\dfrac{4}{5}\right)^2} =\pm\dfrac{3}{5}[/tex]

The location of m is in Quadrant III, therefore;

[tex]cos(m) =-\dfrac{3}{5}[/tex]

[tex]sin(p) = \sqrt{1- \left(-\dfrac{15}{17}\right)^2} =\pm\dfrac{8}{17}[/tex]

The angle p is located in Quadrant III, therefore;

[tex]sin(p) = -\dfrac{8}{17}[/tex]

180° ≤ Angle ∠m ≤ 270°; definition of angles in Quadrant III

180° ≤ Angle ∠p ≤ 270°; definition of angles in Quadrant III

Therefore;

270° - 270° ≤ |∠m - ∠p| ≤ 270° - 180°
0° ≤ |∠m - ∠p| ≤ 90°

The trigonometric identity for tan(m - p) is presented as follows;

[tex]tan(m - p) = \dfrac{tan(m) -tan(p)}{1+tan(m)\cdot tan(p)}[/tex]

[tex]tan(m - p) = \dfrac{\dfrac{sin(m) }{cos(m) } -\dfrac{sin(p) }{cos(p) } }{1+\dfrac{sin(m) }{cos(m) } \cdot \dfrac{sin(p) }{cos(p) } }[/tex]

Therefore;

[tex]tan(m - p) = \dfrac{\left(\dfrac{ -\dfrac{4}{5} }{-\dfrac{3}{5} }\right) -\left(\dfrac{ -\dfrac{8}{17} }{ -\dfrac{15}{17}}\right) }{1+\left(\dfrac{ -\dfrac{4}{5} }{-\dfrac{3}{5} }\right) \times \left(\dfrac{ -\dfrac{8}{17} }{ -\dfrac{15}{17}}\right) } }= \dfrac{36}{77}[/tex]

[tex]tan(m - p) = \dfrac{36}{77}[/tex]

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Answer:

-1/3

Step-by-step explanation:

Substituting into the slope formula, the slope is [tex]\frac{-6-(-5)}{-6-(-9)}=-\frac{1}{3}[/tex]

The coordinates along the directed line segment is (-4, -6)

The points are given as:

A  = (-7, -4)

B = (2, -10)

The ratio on the segment can be represented as;

m : n = 1 : 2

So, we have the coordinates of the point that lies along the directed line segment to be

Point = 1/(m + n) * (mx₂ + nx₁, my₂ + ny₁)

So, we have:

Point = 1/(1 + 2) * (1 * 2 + 2 * -7, 1 * -10 + 2 * -4)

Evaluate

Point = (-4, -6)

Hence, the coordinates of the point is (-4, -6)

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The trigonometric function is useful to find the sides of a right-angle triangle. The value of x using the trigonometric function sin will be 7.884.

The fundamental 6 functions of trigonometry have a range of numbers as their result and a domain input value that is the angle of a right triangle.

The trigonometric function is only valid for the right angle triangle and it is 6 functions which are given as sin cos tan cosec sec cot.

Given right angle triangle,

Use sin trigonometric function.

Sin21° = x=22

x = 22 × 0.358

x = 7.884

Hence "The trigonometric function is useful to find the sides of a right-angle triangle. The value of x using the trigonometric function sin will be 7.884".

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The value of car depreciates by $310 per month.

What is Depreciation?

The reduction in the value of an asset due to wear and tear is termed as Depreciation.

Solution:

Value of Car after 7 months = $26,930

Value of Car after 30 months = $19,800

So, it can be said that in 23 months the value of car depreciated by $7,130.

Since, straight line depreciation method is been followed,

Therefore, monthly depreciation will be calculated by dividing $7,130 to 23

= $7,130/23

= $310

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The solution to the trigonometric equation 5cos(5x) = 4 is given as follows:

x = 7.37º.

The definition of the trigonometric equation is presented as follows:

5cos(5x) = 4

The first step towards solving the trigonometric equation is isolating the trigonometric variable, hence:

cos(5x) = 4/5.

Then we have to isolate the variable x, which is done applying inverse trigonometric functions, as follows:

arccos(cos(5x)) = arccos(4/5)

5x = arccos(4/5)

Using a trigonometric calculator to obtain the arc cosine of four fifths, for the smallest positive integer, we have that:

5x = 36.86º.

Now we only apply the division to isolate the variable x and obtain the solution, as follows:

x = 36.86º/5

x = 7.37º.

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The number of tiles need to cover the closet floor is 180 tiles

The length of the side of the tile = 1/3 feet

The area of the square = Side × Side

The area of the tile = (1/3) × (1/3)

= 1/9 square feet

The width of the closet = [tex]3\frac{1}{3}[/tex] feet

Convert the mixed fraction to simple fraction

[tex]3\frac{1}{3}[/tex] feet = 10/3 feet

The length of the closet = 6 feet

Total area of the closet = 10/3 × 6

= 20 square feet

Number of tiles needed = The area of the closet /  The area of the tile

Substitute the values in the equation

= 20 / (1/9)

= 180 tiles

Hence, the number of tiles need to cover the closet floor is 180 tiles

The complete question is

Vanessa wants to cover her closet floor with SRB tiles that are 1/3 foot on each side. The closet is   [tex]3\frac{1}{3}[/tex] feet wide and 6feet deep, How many tiles will Vanessa need to cover the closet floor?

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Answer: 1

Step-by-step explanation:

Answer:

The correct answer is 1.

Step-by-step explanation:

The greatest common factor of two non-zero integers, x(7) and y(8), is the greatest positive integer m(1) that divides both x(7) and y(8) without any remainder. Therefore, the greatest common multiple of 7 and 8 is 1.

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The initial height of the rocket is 264 feet

From the given graph

The height of the rocket is h is feet and time taken t is in seconds

The graph is defined as the pictorial representation of the mathematical equation or function.  

The x axis of the graphs represents the Time in seconds

The y axis of the graph represents the height of the rocket h in feet

From the graph we can say that

At t = 0, the value of h = 264 feet

That means the height of the rocket is 264 feet when the time t = 0

Hence, the initial height of the rocket is 264 feet

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The object had an initial velocity of 0m/s

Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only.

A projectile is an object upon which the only force acting is gravity. There are a variety of examples of projectiles. An object dropped from rest is a projectile (provided that the influence of air resistance is negligible). An object that is thrown vertically upward is also a projectile (provided that the influence of air resistance is negligible). And an object which is thrown upward at an angle to the horizontal is also a projectile (provided that the influence of air resistance is negligible). A projectile is any object that once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

Using equation of motion, we can find it's initial velocity.

v² = u² - 2gh

substituting the values and solving for u

But we don't know h; we can use the acceleration to find that

a = v / t

a = 18/4

a = 4.5 m/s²

v = u + at

18 = u + 4.5 * 4

u = 0

The initial velocity was 0m/s

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Taking into consideration the upstream and downstream speed, and the equal time taken to travel upstream and downstream distance, the speed of the boat in the still water is found out to be 10mph.

It is given to us that -

The speed of the river current = 2mph

Time taken for the boat to travel 30 miles downstream = Time taken for the boat to travel 20 miles upstream --- (1)

We have to find out the speed of the boat in still water.

Let us say that the speed of the boat in still water is x mph.

We know that -

Speed = Distance/Time

=> Time = Distance/Speed ----- (2)

When travelling upstream, the boat is slower and thus, we subtract speed of the current from the speed of the boat.

When travelling downstream, the boat is faster as it goes with the current and thus, we add the speed of the current with the speed of the boat.

From equation (1), we have

Time to travel 30 miles downstream = Time to travel 20 miles upstream

[tex]= > \frac{30}{x-2} =\frac{20}{x+2} \\= > 30(x+2)=20(x-2)\\= > 30x+60=20x-40\\= > 10x=100\\= > x=10[/tex]

Thus, taking into consideration the upstream and downstream speed, the speed of the boat in the still water is 10mph.

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Answer:

The two graphs have the same shape, but the second graph shifted 3 units left.

Answer:

[tex]y = 4(1.4)^x[/tex]

[tex]\text{at x = 5, }y=21.513[/tex]

Step-by-step explanation:

If you think about an exponential function in the form: [tex]f(x) = a(b)^x[/tex]

you'll realize a pattern, which is by definition.

Evaluating the function at x = 1: [tex]f(1) = a(b)^1 \implies a(b)[/tex]

Evaluating the function at x = 2: [tex]f(2) = a(b)^2 \implies a(b * b)[/tex]

Evaluating the function at x = 3: [tex]f(3) = a(b)^3\implies a(b * b * b)[/tex]

Each term just has an additional "b" that it's being multiplied by. This is by definition of an exponential function, repeated multiplication.

You'll also notice something else interesting, we can find this "b" value by dividing any term by it's previous term.

Take for example f(1) and f(2): [tex]\frac{a(b* b)}{a(b)}\to b[/tex]

As mentioned before, as x increases by one, we just have another term to multiply by. So if we divide by the previous term (where x is one less), then we should just have this "b" value, which is the generally expressed in either a growth or decay rate.

So now that we know that, we can divide f(2) by f(1) using the values shown on the graph:

given:

[tex]f(1) = 5.6\\f(2) = 7.84[/tex]

Let's divide the f(2) by f(1) to get: [tex]\frac{7.84}{5.6} \to1.4[/tex]

Lastly to find our "a" value, we just need to find the y-intercept, since when x = 0, the b will be equal to one (since anything raised to the power of zero is one), we can just look at the graph to see the y-intercept.

Looking at the graph, we can see the y-intercept is four, so a = 4.

Another way to do this algebraically rather than visually, would be to use our knowledge of exponentials to realize that as x increases by one (when we're going right), the y-value is just being multiplied by "b" as mentioned before.

So as x decreases by one (when we're going left) the y-value is just being divided by "b".

So f(0) should be equal to f(1) / b, and we know both values! Which are going to substitute into: [tex]\frac{5.6}{1.4}\to 4[/tex]

Anyways, once we plug in "a" and "b" into the standard form of an exponential function, we get: [tex]y = 4(1.4)^x[/tex]

We can now use this equation to find y-values that are not shown in the graph.

To find the x-value at x = 5, simply substitute in "5" as x

[tex]y = 4(1.4)^x\\\\\text{substitute 5 as x}\\\\y=4(1.4)^5\\\\y=4(5.37824)\\\\y\approx 21.513[/tex]

According to the given temperature function, the substance will not be in liquid state in range is written as the inequality -30.874 < x < 566.121.  

Function:

Function refers the special relationship where each input has a single output. It is often written as "f(x)" where x is the input value.

Given,

Matter in a liquid state when it’s temperature is between it’s melting point and boiling point. Suppose that some substance has a melting point of -34.93 degrees Celsius and a boiling point of -332.29 degrees Celsius.

Here we need to find the range of temperatures in degrees Fahrenheit for which this substance is not in liquid state.

Here we have the following details

Melting point = -34.93C°

Boiling point = -332.29C°

Function = C = 5/9 (F - 32)

n order to find the inequality of the function we have to apply the value of  boiling and melting point in it,

First, we have to apply the value of melting point, then we get,

=> -34.93 = 5/9 (F - 32)   Distribute

=> -34. 93 = 5/9 F - 160/9    Multiply both sides by 9

=> -314.37 = 5F - 160   Add 160 on both sides

=> -154.37 = 5F     Divide both sides by 5

=> -30.874= F

Therefore, the melting point in F= -30.874.

Similarly, for the boiling point it can be calculated as,

=> -332.29 = 5/9 (F - 32)   Distribute

=> -332.29 = 5/9 F - 160/9    Multiply both sides by 9

=> -2990.61 = 5F - 160   Add 160 on both sides

=> -2830.61 = 5F     Divide both sides by 5

=> -566.121= F

Therefore, the boiling point in F= -566.121

So, the resulting inequality is -30.874 < x < 566.121.

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Answer:2,300

Step-by-step explanation:

Answer:

Step-by-step explanation:

after 9 months, the man will have received $836. It would take 4 years to get $992 with a 6% simple interest annually

From the calculation, the two fractional parts are; 6/14.

We know that a fraction is composed of a numerator and a denominator. A numerator is the number that is found at the top while the denominator is the number that is found below. We are told that we have the 6/7 and we are told to express it as the sum of two equal fractional parts;

Hence let the fractional parts be x. We know that the fractional parts are equal hence;

2x = 6/7

x = 6/7 * 1/2

x = 6/14

To check our working;

6/14 + 6/14 = 12/14 = 6/7

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Using the linear function equation that models the situation given, the amount of credit left after 24 minutes of calls is: $25.2.

One of the ways to write a linear function equation is to express it in slope-intercept form as y = mx + b, where the slope is represented as m and the y-intercept is represented as b.

From the information given, we can deduce the following:

y = credit card remaining in dollars

x = minutes

Slope (m) = -0.13

A point = (x, y) = (32, 24.16)

Substitute m = -0.13, x = 32, and y = 24.16 into y = mx + b to find the value of b:

24.16 = -0.13(32) + b

24.16 = -4.16 + b

24.16 + 4.16 = b

28.32 = b

b = 28.32

Write the equation of the linear function by substituting b = 28.32 and m = -0.13 into y = mx + b:

y = -0.13x + 28.32

To find how much credit there was after 24 minutes of calls, substitute x = 24 into y = -0.13x + 28.32:

y = -0.13(24) + 28.32

y = 25.2

$25.2 credit was left.

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The equation of the line in slope-intercept form is y = 2x-8.

According to the question,

We have the following information:

Slope of the line = 2

Points through which the line is passing = (3,-2)

We know that the following formula is used to find the equation of the line passing through a point:

y-y' = m(x-x') where m is the slope of the line

In this case, we have the followings values:

m = 2

x' = 3 and y' =-2

y-(-2) = 2(x-3)

y+2 = 2x-6

Subtracting 2 from both sides of the equation:

y = 2x-6-2

y = 2x-8

Hence, the equation of the line in slope-intercept form is y = 2x-8.

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