Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1,5,5).​

Answer:

2 √3

Step-by-step explanation:

The value of sin 90° is 1, so we can simplify the expression as:

16 sin 10° sin 30° sin 50° sin 70° sin 90° = 16 sin 10° sin 30° sin 50° sin 70°

We can use the identity sin (90 - x) = cos x to rewrite sin 70° as cos 20°:

16 sin 10° sin 30° sin 50° sin 70° = 16 sin 10° sin 30° sin 50° cos 20°

We can use the identity sin (180 - x) = sin x to rewrite sin 170° as sin 10°:

16 sin 10° sin 30° sin 50° cos 20° = 8 sin 10° [2 cos 20° sin 30° sin 50°]

We can use the identity sin (2x) = 2 sin x cos x to simplify the expression inside the brackets:

8 sin 10° [2 cos 20° sin 30° sin 50°] = 8 sin 10° [sin 100° - sin 20°]

We can use the identity sin (180 - x) = sin x to rewrite sin 20° as sin 160°:

8 sin 10° [sin 100° - sin 20°] = 8 sin 10° [sin 100° - sin 160°]

We can use the identity sin (x + y) = sin x cos y + cos x sin y to rewrite sin 100° as sin (60° + 40°):

8 sin 10° [sin 100° - sin 160°] = 8 sin 10° [sin 60° cos 40° + cos 60° sin 40° - sin 160°]

We can use the identity sin (180 - x) = sin x to rewrite sin 160° as sin 20°:

8 sin 10° [sin 60° cos 40° + cos 60° sin 40° - sin 20°] = 8 sin 10° [sin 60° cos 40° + cos 60° sin 40° - sin 20°]

Now, we can use the values of sin 60°, cos 40°, and sin 20°, which are:

sin 60° = √3/2

cos 40° = √[(1+cos 80°)/2] = √[(1+sin 10°)/4]

sin 20° = sin (60° - 40°) = sin 60° cos 40° - cos 60° sin 40° = (√3/2)(√[(1+sin 10°)/4]) - (1/2)sin 10°

Substituting these values into the expression, we get:

8 sin 10° [sin 60° cos 40° + cos 60° sin 40° - sin 20°]

= 8 sin 10° [√3/2 × √[(1+sin 10°)/4] + 1/2 × sin 10° - (√3/2) × √[(1+sin 10°)/4]]

= 2 √3 sin 10° √[(1+sin 10°)/2]

Therefore, the value of the expression is:

16 sin 10° sin 30° sin 50° sin 70° sin 90° = 2 √3

By answering the presented question, we may conclude that As a result, function the value of f'(-1) is -9.

Mathematicians examine numbers and their variations, equations and associated structures, forms and their locations, and prospective positions for these things. The term "function" refers to the relationship between a collection of inputs, each of which has a corresponding output. A function is a connection of inputs and outputs in which each input leads to a single, identifiable outcome. Each function is assigned a domain, a codomain, or a scope. The letter f is widely used to denote functions (x). The symbol for entry is an x. The four primary types of accessible functions are on functions, one-to-one capabilities, so multiple capabilities, in capabilities, and on functions.

We are given the function f(x) = -3x3 - 4 and asked to calculate the value of f' (-1).

To determine f'(-1), calculate the derivative of f(x) with respect to x and evaluate it at x = -1.

f(derivative )'s is provided by:

[tex][-3x3 - 4] f'(x) = d/dx = -9x^2[/tex]

We can now calculate f'(-1) by entering x = -1:

[tex]f'(-1) = -9(-1) (-1)\\a^2 = -9[/tex]

As a result, the value of f'(-1) is -9.

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Answer:

y = 2x - 5

Step-by-step explanation:

the equation of a line in slope- intercept form is

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

given m = 2 and point where line crosses y- axis (0, - 5 ) , then c = - 5

y = 2x - 5 ← equation of line

Answer:

3 hours

Step-by-step explanation:

65 - 85x = 320

85x = 320 - 65

85x = 255

x = 255 / 85

x = 3

Answer:

360 in^2

Step-by-step explanation:

Area of bottom rectangle : A = 7.5 x 17 = 127.5

Area of front rectangle: A = 6 x 17 = 102

Area of back rectangle: A = 4.5 x 17 = 76.5

Area of 2 triangles: 2(4.5 x 6) = 54

Total surface area = 127.5 + 102 + 76.5 + 54 = 360 in^2

Using tautology, we can conclude that the argument here is invalid.

A compound statement known as a tautology is one that is true regardless of whether the individual statements inside it are true or false.

The Greek term "tautology," which means "same" and "logy," is where the word "tautology" comes from.

We need to build a truth-table and examine the truth value in the last column in order to determine whether a particular statement is a tautology.

It is a tautology if all of the values are true.

In the given case:

p is TRUE

and

q is FALSE

In this case:

p→q : is FALSE (the assumption “TRUE implies FALSE” is FALSE)

So, here:

p → (p→q) is equal to as p → FALSE

But p is TRUE so in that case it’s TRUE→ FALSE, which is in fact FALSE.

Since there a case where the expression is not true, then it’s not valid.

It’s invalid.

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The graph of the system of linear inequality is attached below

a. Graphing the system of linear inequality, the solutions are (0.7, -3.35)

b. To prove that the inequality remains true at any given point, we can use a test point. We will choose the point (0,0) as our test point.

For the first inequality, plugging in (0,0) gives:

4(0) + 8(0) ≤ -24

0 ≤ -24

This is clearly false, so (0,0) is not a solution to the first inequality. This means that any point below the line 4x + 8y = -24 will not satisfy the inequality.

For the second inequality, plugging in (0,0) gives:

12(0) - 4(0) < 5

0 < 5

This is true, so (0,0) is a solution to the second inequality. This means that any point to the left of the line 12x - 4y = 5 will satisfy the inequality.

Therefore, the system of inequalities will only be satisfied by points that are both below the line 4x + 8y = -24 and to the left of the line 12x - 4y = 5. We can verify this visually by graphing the two lines and shading in the appropriate regions. Any point in the shaded region will satisfy both inequalities, and any point outside of the shaded region will not satisfy at least one of the inequalities.

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Answer: It will cost $58.80 to buy enough stain for the whole porch.

Step-by-step explanation: The area of Carmen's front porch is:

Area = length x width

Substituting the given values, we get:

Area = 12 ft x 7 ft

Area = 84 square feet

To find the cost of the stain, we need to multiply the area of the porch by the cost per square foot:

Cost of stain = Area x Cost per square foot

Cost of stain = 84 square feet x $0.70/square foot

Cost of stain = $58.80

54,000 rounded to the nearest ten is 54,000  because the zeros after the 4 are insignificant

Rounding off in math is the process of approximating a number to a certain degree of accuracy. This involves changing the value of a number to one that is simpler or easier to work with, while still maintaining an acceptable level of accuracy.

For example, rounding off 3.75 to the nearest whole number would give you 4, while rounding off 3.75 to the nearest tenth would give you 3.8.

Rounding off 54000  to the nearest ten is still 54000 because the zeros after the 4 are insignificant

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Answer:

6 pencils

Step-by-step explanation:

120*4=480, 480/80=6

Based on the information provided, the volume of the pyramid would be  5308.33 cubic centimeters.

The volume or the space occupied by the pyramid can be calculated using the following formula

V=(1/3)Ah

In this formula,  A is the area of the base and h is the height.

Area of the base: 6.5 x 3.5 = 22.75 squared centimeters.

Now, let's calculate the volume:

V = (1/3)*22.75 x 700 cm = 5308.33 cm^3 (Cubic centimeters)

Here 1 meter = 100 cm, then 7 m = 700 cm

Note: This question is incomplete; here is the missing information:

The base of a pyramid is 6.5 cm by 3.5cm and its height is 7m. What is its volume?

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The unit rate of 24t = 98 is A. 24 feet per year.

Unit rate is the rate of increase or decrease of a unit of item

Now, since a 984-foot tree has grown at a constant rate each year. Since we have the equation 24t = 984, wheret is the age of the tree in years.

To determine the unit rate, we compare it with the function y = mx where m = gradient

Now, the gradient is the rate of change of y per unit of x.

Now, comparing y = mx with 24t = 984,

Since m = 24 which is the gradient which is also the unit rate.

So, the unit rate of 24t = 98 is A. 24 feet per year.

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Answer: We can start by noticing that the total number of small cubes used is 64, which is the same as the total number of unit squares on the surface of the large cube. To maximize the white surface area, we want to arrange the cubes in a way that maximizes the number of white faces on the surface.

Each small cube has 6 faces, so there are a total of 6 x 64 = 384 faces. Half of these faces are white and half are black, since we have an equal number of white and black cubes. Therefore, there are 192 white faces and 192 black faces.

Now, let's consider how the small cubes can be arranged to maximize the white surface area. If we arrange all the white cubes together in a 4 x 4 x 4 block, then all the faces of this block that are adjacent to a black cube will be black, and all the faces that are adjacent to another white cube will not contribute to the surface area. This leaves only 16 white faces that contribute to the surface area.

On the other hand, if we alternate white and black cubes in each layer of the cube, then all the faces that are adjacent to a black cube will be white, and all the faces that are adjacent to another white cube will be black. This maximizes the white surface area. Specifically, there are 6 layers of the cube, and each layer has 16 white faces (4 along the top and bottom and 4 along each side). Therefore, there are a total of 6 x 16 = 96 white faces that contribute to the surface area.

So the fraction of the surface that is white is:

96 white faces / 384 total faces = 1/4

Therefore, one quarter of the surface area of the cube is white.

Step-by-step explanation:

Step-by-step explanation:

the total cost is $200.

the given function calculates the total costs.

so, all we need to do is set the function result and calculate backwards to the necessary value of m :

200 = 0.75m + 50

150 = 0.75m

m = 150/0.75 = 200

the van must have been rented for 200 miles to have cost $200.

Answer: 4

See the graph attached

Hope it helps

Answer:

Amount of benzoic acid in the solvent phase = 0.303 kg/L * 100 L/min * 20 min = 606 g.

Step-by-step explanation:

To determine the amount of benzoic acid that has passed into the solvent phase after 20 minutes, we need to use the given simplifications to set up a mass balance equation.

Let's express the stream-flow rates on a solute-free basis, which means that we consider only the flow rate of the solvent (water) and the flow rate of the toluene (ignoring the mass of benzoic acid in each stream). Let Fw be the flow rate of water and Ft be the flow rate of toluene.

Since the toluene and water are immiscible, we can assume that the volume of the two phases leaving the tank is the same as the volume of the mixture entering the tank (which is 100 L/min). Let V be this volume, which is also equal to the volume of the tank (120 L).

Let x be the concentration of benzoic acid leaving the stage in the organic phase (i.e., the solvent phase), and y be the mass concentration of benzoic acid in the aqueous phase.

From the given simplification, we have:

y = 5x

Let's use the mass balance equation to relate the flow rate of benzoic acid entering the tank to the flow rate of benzoic acid leaving the tank:

FtCt,in = VxCt,out + (FwCw,in - Fw*Cw,out)*y

where Ct,in and Cw,in are the concentrations of benzoic acid in the toluene and water entering the tank, respectively, and Ct,out and Cw,out are the concentrations of benzoic acid in the toluene and water leaving the tank, respectively.

We can assume that the feed concentration is constant, so Ct,in = 0.12 kg/L. We also know that the flow rate of the toluene and water entering the tank are 100 L/min and 110 L/min, respectively.

The concentration of benzoic acid in the water leaving the tank is given by:

Cw,out = y/5 = x/5

Since the flow rate of each phase is steady, we can assume that the concentration of benzoic acid in the toluene leaving the tank is the same as the concentration of benzoic acid in the toluene entering the tank, which is also 0.12 kg/L.

Substituting these values into the mass balance equation, we get:

1000.12 = 120x*0.12 + (110 - 120)*y

Simplifying, we get:

12 = 14.6x + 10y

Substituting y = 5x, we get:

12 = 39.6x

x = 0.303 kg/L

Therefore, the concentration of benzoic acid leaving the tank in the organic phase is 0.303 kg/L.

To find the amount of benzoic acid that has passed into the solvent phase after 20 minutes, we need to multiply the concentration by the flow rate of the toluene:

Amount of benzoic acid in the solvent phase = 0.303 kg/L * 100 L/min * 20 min = 606 g.

We can use the mass balance equation to solve this problem. Let x be the concentration of benzoic acid leaving the stage in the organic phase (in kg/L) and y be the concentration of benzoic acid in the aqueous phase (in kg/L). Then we have:

Mass of benzoic acid entering the stage per minute = Mass of benzoic acid leaving the stage per minute

The mass of benzoic acid entering the stage per minute is:

100 L/min x 0.12 kg/L = 12 kg/min

The mass of benzoic acid leaving the stage per minute can be expressed in terms of x and y as:

100 L/min x (0.12 kg/L - x) + 110 L/min x y = (100 + 110) L/min x m x x

Simplifying and using the given linear relationship between y and x, we get:

12 - 100x + 110(5x) = 210x

Solving for x, we get:

x = 0.034 kg/L

Therefore, the concentration of benzoic acid in the organic phase after 20 minutes is 0.034 kg/L.

Answer:

6h65

Step-by-step explanation:

Add up all the values:

1h+2h+3h= 6

20+40+05= 65

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The time taken, in hours, for the bacteria to double in size, found using the given function is 26.25 hours.

What is a function?

A relation between a set of inputs and a set of allowable outputs is called a function. It has the property that every input is associated with exactly one output. A function from one set X to another allocates precisely one element of the other set Y to each element of X. Both the set X and the set Y are referred to as the function's domain and codomain, respectively. The basic conception of functions was the relationship between fluctuating quantities and other variables.

The given function is:

[tex]y = y_0e^{(0.0264t)}[/tex]

We are asked to find the time taken for the bacteria to double.

y = a number of bacteria at present time t.

y₀ = initial number of bacteria

When y = 2y₀

The function becomes,

[tex]2y_0 = y_0e^{(0.0264t)}\\\\2 = e^{(0.0264t)}\\\\ln (2) = 0.0264t\\\\0.693 = 0.0264t\\\\t = 26.25[/tex]

Therefore the time taken, in hours, for the bacteria to double in size, found using the given function is 26.25 hours.

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Answer:

which of the following set of nos could not be the length of sides of aright angled triangles.

The measure of angle ABC is 40 degrees.

What is inscribed angle theorem ?

The inscribed angle theorem, also known as the central angle theorem, states that an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc, or in other words, the angle that connects the endpoints of the arc.

In other words, if we draw a chord of a circle and an angle with its vertex on the circumference of the circle and its sides passing through the endpoints of the chord, then the measure of the inscribed angle is half the measure of the central angle that subtends the same arc as the inscribed angle.

According to the question:

By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the arc it intercepts.

Since chord AB is congruent to chord CB, we can conclude that angle A is congruent to angle C, because they both intercept the same arc AC. Thus, we have:

angle BAC + angle ABC + angle ACB = 180 degrees (by the angle sum property of triangles

Substituting the measure of arc AC, we get:

angle BAC + angle ABC + 70 degrees = 180 degrees

Simplifying, we get:

angle BAC + angle ABC = 110 degrees

Since angles A and C are congruent, we have:

2 x angle BAC + angle ABC = 180 degrees

Substituting angle BAC + angle ABC = 110 degrees, we get:

2 x 110 degrees - angle ABC = 180 degrees

Solving for angle ABC, we get:

angle ABC = 40 degrees

Therefore, the measure of angle ABC is 40 degrees.

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Answer:

18.2 square miles

Step-by-step explanation:

r = radius of circle

 = 4 miles

Area of shaded region = Area of sector

                                        = [tex]\frac{AngleOfSector}{360}[/tex] ×[tex]\pi r^{2}[/tex]

                                       = [tex]\frac{130}{360}[/tex] × [tex]\pi (4)^{2}[/tex]

                                       = 18.2 [tex]mi^{2}[/tex] (Rounded to 3 significant figures)

If you own 4 stocks, there are 81 possibilities that could occur, according to the provided assertion.

In most cases, the probability is described as the ratio of favourable outcomes to all results in the survey region. Probability of an occurrence P(E) = (Number of positive events) is how it is stated (Sample space).

For each of the 4 stocks, there are 3 possibilities - up, down, or unchanged. Therefore, the total number of possibilities for owning 4 stocks is 3 x 3 x 3 x 3, which is equal to 81.

So there are 81 possible scenarios if you own 4 stocks.

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Answer:

Para resolver este problema, primero debemos encontrar la cantidad total de raciones que tienen los 50 hombres para 20 días a razón de tres raciones diarias:

50 hombres x 3 raciones por día x 20 días = 3000 raciones

Esto significa que tienen un total de 3000 raciones de alimentos para los 50 hombres durante 20 días.

Luego, podemos calcular la cantidad de raciones diarias después de la disminución en 1/3:

3 raciones diarias - (1/3)(3 raciones diarias) = 2 raciones diarias

Esto significa que cada hombre ahora recibe 2 raciones diarias en lugar de 3.

Después de aumentar el número de hombres en 10, tenemos un total de 60 hombres:

50 hombres + 10 hombres = 60 hombres

Podemos calcular la cantidad total de raciones diarias para los 60 hombres durante los días que durarán las provisiones:

60 hombres x 2 raciones por día = 120 raciones diarias

Finalmente, podemos calcular la cantidad de días que durarán los víveres:

3000 raciones totales / 120 raciones diarias = 25 días

Por lo tanto, los víveres durarán 25 días después de disminuir las raciones en 1/3 y aumentar el número de hombres en 10.

Step-by-step explanation:

The possible calculated length of the segment AC is a length of any value that exceeds around 7.15 feet.

If triangle ABC is an isosceles triangle, then the length of AC must be equal to the length of BC.

Let's denote the length of AC as "x".

To find the length of the diagonal line segment referred to as BD, which represents the height of the peaked roof, we can use the Pythagorean theorem:

BD² = AB² - (AC/2)²

BD² = 8² - (x/2)²

BD² = 64 - x²/4

In order for the roof to be peaked, BD must be less than x, otherwise the roof would be flat.

Therefore, we can set up the inequality:

x√(64 - x²/4) < x

64 - x²/4 < x²

Multiplying both sides by 4 gives:

256 - x² < 4x²

5x² > 256

Dividing both sides by 5 gives:

x² > 51.2

x > √(51.2)

x > 7.15 feet

As a result, any value greater than the square root of 51.2 (approximately 7.15 feet) is a possible length for AC.

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The possible lengths for AC are any values greater than 7.15 feet.

We are given that the triangle ABC is an isosceles triangle. This means that the length of segment AC must be equal to the length of segment BC.

Let x denote the length of AC.

With the aid of the Pythagorean theorem, the length of the diagonal line segment BD which denotes the height of the peaked roof is:

BD² = AB² - (AC/2)²

BD² = 8² - (x/2)²

BD² = 64 - x²/4

If the roof is to be peaked, then the length of segment BD must be less than x. Thus, we generate the inequality:

BD < x

√(64 - (x²/4)) < x

Square both sides to get:

64 - x²/4 < x²

256 - x² < 4x²

5x² > 256

x² > 51.2

x > √(51.2)

x > 7.15 ft

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Using the simple interest formula it is seen that Mrs Jo saves an amount of $180 when she chooses option B over option A.

What is Simple Interest?

Simple interest is a way to figure out how much interest will be charged on a sum of money at a specific rate and for a specific duration of time. Contrary to compound interest, where we add the interest of one year's principal to the next year's principal to compute interest, the principal amount under simple interest remains constant.

To determine which option is better, we need to compare the total amount of interest paid for each option.

Option A -

The principal amount is $900 and the interest rate is 8%. The term is 5 years.

Using the formula for simple interest -

Interest = Principal × Rate × Time

Interest = $900 × 0.08 × 5

Interest = $360

Total amount paid = Principal + Interest

Total amount paid = $900 + $360

Total amount paid = $1260

Option B -

The principal amount is $900 and the interest rate is 10%. The term is 2 years.

Using the formula for simple interest -

Interest = Principal × Rate × Time

Interest = $900 × 0.10 × 2 = $180

Total amount paid = Principal + Interest

Total amount paid = $900 + $180

Total amount paid = $1080

Therefore, option B is the better option because it has a lower total amount paid by choosing option B.

Mrs. Jo will save -

$1260 - $1080 = $180

Therefore, Mrs. Jo will save $180 by choosing option B instead of option A.

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Answer:763.02

Step-by-step explanation:

r=9

R=r+9=18

[tex]A=R^{2}\pi-r^{2}\pi = \pi (R^{2}-r^{2})=\pi =243\pi \\\pi =3.14\\243\pi=763.02[/tex]

Answer:

763.02

Step-by-step explanation:

Area of a circle is πr²

so the area of the whole circle is the sum of both radius

9+9 = 18

so Area = 3.14 × 18² = 1017.36

then we subtract the white part to leave the shaded part

area of the white circle = 3.14 × 9² = 254.34

then 1017.36 - 254.34 = 763.02

Answer:

Option C

Step-by-step explanation:

The function S (m) = 105m represents the amount of steps after m minutes.

What is a function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable) is known as a function.

We are given two functions.

Tim's function : S (m) = [tex]105^{m}[/tex]

This function cannot be used to find the amount of steps taken after m minutes.

Each minute, they take an additional 105 steps. Equal differences represent a linear relationship.

Kristin's function: S (m) = 105m

This function can be used to find the amount of steps taken after m minutes.

For every minute that passes, 105 more steps are take.

Hence, the function S (m) = 105m represents the amount of steps after m minutes.

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To estimate how many Boomers prefer to shop in-store, we can use the percentage given and apply it to the sample size of 75 people.

84% of 75 = 0.84 x 75 = 63

Therefore, we can estimate that approximately 63 Boomers out of a sample size of 75 prefer to shop in-store.

To represent this data visually, we can create a simple bar graph. The x-axis can represent the different generations and the y-axis can represent the percentage of people who prefer to shop in-store. We can then plot the data for Boomers as a bar that reaches 84% on the y-axis.

Answer:

Rational numbers, Real Numbers, Fractions, and Percentages.