What is the simplest form of

x^2+ 5x-36/
X^2-16


?

The equivalent trigonometric expression to sin(2x)cos(x) is given as follows:

sin(2x)cos(x) = 2sin(x)cos²(x).

The expression for this problem is defined as follows:

sin(2x)cos(x).

The trigonometric identity used to simplify sin(2x) is given as follows:

sin(2x) = 2sin(x)cos(x).

Hence the expression can be written as follows:

sin(2x)cos(x) = 2sin(x)cos(x)cos(x)

sin(2x)cos(x) = 2sin(x)cos²(x).

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Answer:

Let's call the number of cubic yards of mix A that the farmer buys "x", and the number of cubic yards of mix B that the farmer buys "y".

We can write the following constraints based on the minimum monthly requirements:

20x + 10y >= 460 (for acid)

30x + 30y >= 939 (for nitrogen)

5x + 10y >= 200 (for potash)

We also want to minimize the cost, which is given by 25x + 49y.

This is a linear programming problem, which can be solved using simplex method or the graphical method.

By using the graphical method, we find that the optimal solution occurs at x = 11.75 and y = 4.5. This means that the farmer should buy 11.75 cubic yards of mix A and 4.5 cubic yards of mix B to meet the minimum monthly requirements at a minimum cost.

So, the answer to the question is: If mix A costs $25 per cubic yard and mix B costs $40 per cubic yard, the farmer should blend 11.75 yd of mix A & 4.5 yd of mix B to meet the minimum monthly requirements at a minimum cost.

Answer:

=17 + 6n.

Step-by-step explanation:

[tex]4 \times 5 + 6n - 3 \\ = 20 + 6n - 3 \\ = 20 - 3 + 6n \\ = 17 + 6n[/tex]

7.21 × 10⁻⁷

Step-by-step explanation:

(6.94 × 10⁻⁷) + (2.7 × 10⁻⁸) = 7.21 × 10⁻⁷

To find the distance from a point P to a plane defined by three points Q, R, and S, we can use the following formula:

distance = |(A * x + B * y + C * z + D) / √(A^2 + B^2 + C^2)|

where (A, B, C) are the coefficients of the normal vector of the plane, (x, y, z) are the coordinates of the point P, and D is the constant in the equation of the plane.

To find the coefficients of the normal vector, we can use the cross product of two vectors lying in the plane, say, QR and RS.

Let's first find the vectors QR and RS:

QR = (5 - 0, 0 - (-1), 3 - 7) = (5, 1, -4)

RS = (7 - 0, 5 - (-1), 0 - 7) = (7, 6, -7)

Next, we find the cross product of QR and RS:

cross product of QR and RS = (B, C, A) = (7, -30, 15)

So, the normal vector of the plane is (15, -30, 7).

Now we substitute the values for the coefficients and the coordinates of the point P into the formula:

distance = |(15 * 3 + (-30) * 1 + 7 * 7 + D) / √(15^2 + (-30)^2 + 7^2)|

To find the value of D, we can use any of the three points on the plane and substitute their coordinates into the equation of the plane in the form Ax + By + Cz + D = 0.

Let's use the point Q:

15 * 5 + (-30) * 0 + 7 * 3 + D = 0

Expanding and solving for D, we get:

75 + 21 + D = 0

D = -96

Finally, substituting the values back into the formula for distance, we get:

distance = |(15 * 3 + (-30) * 1 + 7 * 7 + -96) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-24 + -30 + 49 - 96) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-101) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-101) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-101) / √(225 + 900 + 49)|

distance = |(-101) / √1274|

distance = |(-101) / 35.6|

distance = 2.85

So, the distance between the point P and the plane is approximately 2.85.

Answer:

See below for solution steps for parts (a) and (c)

Step-by-step explanation:

The trick here is to get all the roots to have the same radical. Each of the lowest radical is the square root pf a prime so cannot be reduced further

For example, in a, the lowest is [tex]\sqrt{3}[/tex]

So get all the others to this level a[tex]\sqrt{3}[/tex]  and then you can just add up the non-radicals and use the radical as the common expression

I will solve a couple and that will give you a general idea of solving the rest

Let's do a.
[tex]5\sqrt{3} , \sqrt{3}[/tex] are already at the lowest radical level of [tex]\sqrt{3}[/tex] so let's reduce 108 to have [tex]\sqrt{108}[/tex] to have [tex]\sqrt{3}[/tex] as one of its components

We can factor 108 as follows;

108/3 = 36

So 108 = 36 x 3

√108 = √36 × √3 = 6√3

Therefore the original expression becomes

[tex]5\sqrt{3} - 6\sqrt{3} + \sqrt{3} = (5 - 6 + 1)\sqrt{3} = 0 \sqrt{3} = 0[/tex]

[tex]-------------------------------[/tex]

c.
[tex]2\sqrt{20} - 7\sqrt{5} + \sqrt{45}[/tex]

Lowest radical is [tex]\sqrt{5}[/tex] whose coefficient is 7

Factor 20 = 4 x 5

[tex]2\sqrt{20} = 2\sqrt{4 \times 5} = 2\sqrt{4} \times \sqrt{5} = 2\times 2 \times \sqrt{5}\\= \bold{4\sqrt{5}}[/tex]

Factor 45:
45 = 9 x 5

[tex]\sqrt{45} = \sqrt{9} \times \sqrt{5} = \bold{3\sqrt{5}}[/tex]

Original expression becomes:
[tex]4\sqrt{5} - 7\sqrt{5} + 3\sqrt{5}\\\\=(4 - 7 + 3)\sqrt{5}\\\\= 0 \sqrt{5}\\\\= 0\\----------------------------[/tex]

I am sure you can do the rest. If not post them as another question. Maybe someone else will answer a few of the ones I did not.

All the best

Answer:

x = 81

Step-by-step explanation:

If a is directly proportional to y²:

[tex]\boxed{a \propto y^2 \implies a=ky^2}[/tex]

If b is inversely proportional to y:

[tex]\boxed{b \propto \dfrac{1}{y} \implies b=\dfrac{m}{y}}[/tex]

Given equation:

[tex]x=a+b+4[/tex]

Substitute the derived expressions for a and b into the given equation:

[tex]\implies x=ky^2+\dfrac{m}{y}+4[/tex]

Substitute the given values of x and y into the equation to create two equations in terms of k and m.

Given that x = -3 when y = 1:

[tex]\implies -3=k+m+4[/tex]

Given that x = 18 when y = 2:

[tex]\implies 18=4k+\dfrac{m}{2}+4[/tex]

Rewrite the first equation to isolate k:

[tex]\implies k=-m-7[/tex]

Substitute the expression for k into the second equation and solve for m:

[tex]\implies 18=4(-m-7)+\dfrac{m}{2}+4[/tex]

[tex]\implies 18=-4m-28+\dfrac{m}{2}+4[/tex]

[tex]\implies -4m+\dfrac{m}{2}=42[/tex]

[tex]\implies -8m+m=84[/tex]

[tex]\implies -7m=84[/tex]

[tex]\implies m=-12[/tex]

Substitute the found value of m into the expression for k and solve for k:

[tex]\implies k=-(-12)-7[/tex]

[tex]\implies k=5[/tex]

Substitute the found values of k and m into the equation to create an equation for x in terms of y:

[tex]\boxed{x=5y^2-\dfrac{12}{y}+4}[/tex]

Finally, to find the value of x when y = 4, substitute y = 4 into the equation and solve for x:

[tex]\implies x=5(4)^2-\dfrac{12}{4}+4[/tex]

[tex]\implies x=5(16)-3+4[/tex]

[tex]\implies x=80-3+4[/tex]

[tex]\implies x=81[/tex]

Answer:

Step-by-step explanation:

heyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy

The equation for the graphed quadratic function is:

y = -1*(x - 3)^2 + 2

Here we have the graph of a quadratic equation. Remember that if the leading coefficient is a and the vertex is (h, k) then we can write:

y = a*(x - h)^2 + k

Here the vertex is at (3, 2) then we can write:

y = a*(x - 3)^2 + 2

We can see that it also passes through the point (1, -2), replacing these values we will get:

-2 = a*(1 - 3)^2 + 2

-2 = a*4 + 2

-2 - 2 = a*4

-4/4 = a = -1

Then the quadratic equation is:

y = -1*(x - 3)^2 + 2

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Answer:

661,068.8j

Step-by-step explanation:

The car has 661068.8 J of potential energy at a height of 62 m having a mass of 1088 grams.

Potential energy is a form of stored energy that is dependent on the relationship between different system components. When a spring is compressed or stretched, its potential energy increases.

We know, The gravitational constant is 9.8 m/s².

From the given information, m = 1088 grams, h = 62 meters.

Therefore, The amount of potential energy the car has is,

PE = mgh.

PE = 1088×9.8×62 J.

PE = 661068.8 J.

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The equation of the inner boundary of the circular walkway around the monument is x² + y² = 900.

The equation of the outer boundary of the circular walkway around the monument is x² + y² = 1369.

A circle is a conic section whenever the plane cuts the cone parallel to its base. Every point of the circumference lies at the same distance from its center. The equation of a circle is (x-h)² + (y-k)² = r², where (h, k) is the center and r is the radius.

a.

According to the diagram, the radius of the inner boundary is 30 feet and the center is (0, 0).

Substitute h = 0, k = 0, and r = 30 into the equation of circle (x-h)² + (y-k)² = r².

(x-0)² + (y-0)² = 30²

x² + y² = 900

Therefore, the obtained answer is x² + y² = 900.

b.

According to the diagram, the radius of the outer boundary is 37 feet (30 feet + 7 feet) and the center is (0, 0).

Substitute h = 0, k = 0, and r = 37 into the equation of circle (x-h)² + (y-k)² = r².

(x-0)² + (y-0)² = 37²

x² + y² = 1369

Therefore, the obtained answer is x² + y² = 1369.

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The complete question is mentioned in the image below.

Answer: To find the percent error, we can use the formula:

percent error = (estimated value - actual value) / actual value * 100

Plugging in the values:

percent error = (76 - 59) / 59 * 100 = 29.49%

Rounding to the nearest tenth of a percent:

percent error = 29.5%

So the percent error is 29.5%.

Step-by-step explanation:

The short-run total cost is TC is wL + rK and short-run average cost curve is TC/Q. The SMC is 4.01 and the SMC curve intersects the SAC curve at its lowest or minimum points because the average cost is a weighted average of the marginal cost and the fixed cost.

A. To calculate the firm’s short-run total cost curve, we need to calculate the total cost at each level of output. The short-run total cost can be represented as follows:

TC = wL + rK,

where w is the wage rate, L is the labor used, r is the interest rate, and K is the capital used. In this case, w = $4 and r = $1.

To calculate the short-run average cost curve, we need to divide the total cost by the level of output:

SAC = TC / Q

Where Q is the level of output.

B. The short-run marginal cost function can be calculated as the change in total cost for a small change in the level of output:

SMC = dTC / dQ

At a level of output of 25, the labor used is L = 25 / 2 = 12.5. The total cost can be calculated as follows:

TC = 4 * 12.5 + 1 * 100 = 50 + 100 = 150

So, the short-run average cost is SAC = 150 / 25 = $6 and the short-run marginal cost is SMC = dTC / dQ. To calculate the derivative, we need to use the production function given:

Q = 2 * sqrt(KL)

Taking the derivative with respect to L, we get:

dQ / dL = (1 / (2 * sqrt(KL))) * 2K = K / (sqrt(KL))

Substituting the values, we get:

SMC = 4 + (1 / (sqrt(100 * 12.5)))

At a level of output of 200, the labor used is L = 200 / 2 = 100. The total cost can be calculated as follows:

TC = 4 * 100 + 1 * 100 = 400

So, the short-run average cost is SAC = 400 / 200 = $2 and the short-run marginal cost is SMC = dTC / dQ. To calculate the derivative, we need to use the production function given:

Q = 2 * sqrt(KL)

Taking the derivative with respect to L, we get:

dQ / dL = (1 / (2 * sqrt(KL))) * 2K = K / (sqrt(KL))

Substituting the values, we get:

SMC = 4 + (1 / (sqrt(100 * 100)))

SMC = 4.01

C. To draw the graph for the SAC and SMC, we need to calculate the SAC and SMC for different levels of output. We can then plot the SAC and SMC against the level of output to obtain the graphs.

D. The SMC curve intersects the SAC curve at its lowest or minimum points because the average cost is a weighted average of the marginal cost and the fixed cost. When the level of output is low, the marginal cost is high and the fixed cost is a large portion of the total cost. As the level of output increases, the marginal cost decreases and the fixed cost becomes a smaller portion of the total cost. As a result, the average cost decreases and reaches a minimum at the point where the marginal cost intersects the average cost.

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The value of x that will make L||m is 9.

Geometry is a branch of mathematics that deals with shapes, angles, dimensions and sizes of different things we see in everyday life

The angle (4x²- 324)° is equal to 90°. That is:

(4x²- 324)° = 90°

4x²- 324 = 90

4x² = 90 + 324

4x² = 324

x² = 324/4

x² = 81

x = √81

x = 9

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The algebraic equation for the statement is 21 = 2 x - 5

Algebraic equations can be written using mathematical symbols and operations to represent relationships between variables or quantities. Variables are the quantities that the equation is trying to represent or solve for.

There is only one variable here and we can call it x.

The number is doubled so the representation would be :

= 2 x

This is then decreased by 5 and becomes :

= 2 x - 5

This is equated to 21 to become:

21 = 2 x - 5

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The values of x and y would be   [tex]x=\frac{20 \sqrt{3}}{3}[/tex] and [tex]y=\frac{40\sqrt{3}}{3}[/tex].

What are trigonometric ratios?

Trigonometric ratios are mathematical functions that describe the relationship between the angles and sides of a right triangle. The three main trigonometric ratios are the sine, cosine, and tangent.

In the given figure, by using trigonometric ratios, we can find x

tan60° = 20/x

√3 = 20/x

x = 20/√3

=> [tex]x=\frac{20 \sqrt{3}}{3}[/tex]

Now by using the ratio sin60°

sin60° = 20/y

√3/2  = 20/y

y = 40/√3

[tex]y=\frac{40\sqrt{3}}{3}[/tex]

Hence, the values of x and y would be   [tex]x=\frac{20 \sqrt{3}}{3}[/tex] and [tex]y=\frac{40\sqrt{3}}{3}[/tex].

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Answer:

slope

Step-by-step explanation:

Step-by-step explanation:

always find the quantity of 100%.

then

1% = 100%/100

then calculate the absolute quantity change.

then divide the quantity change by 1% to see how many % fit into it.

and that is your answer.

so, if the original quantity is q1, the new quantity is q2, then the % of quantity change is

|q1 - q2| / (q1/100)

or

100 × |q1 - q2| / q1

where || means the absolute value.

The future value of the investment when interest is compounded continuously is; $3408.29

Compound interest is a method of calculating the interest charge. In other words, it is the addition of interest on interest.

Given Data

P= $1500, r = 3.95%, t = 5 yr

To Calculate the number of periods: in this case we have 5*12= 60 months.

- The monthly interest rate, we have to divide the annual nominal rate by 12 (the number of periods in the year).

Then, we can calculate the future value as:

A = P (1 + r/n)^nt

A = 1500(1 + 3.95/365)(365)^(5)

P = 1500/ (1 + 0.00010137)^(5)

P = $3408.29

The future value when compounded monthly is $3408.29.

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The point that lies in the solution to the system of inequalities is (5,-2). The correct answer would be an option (B).

An Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.

The system of inequalities is given in the question, as follows:

y ≤ x -5 and y ≥ -x-4

Since the required point is to satisfy the system of inequalities.

Then the graph attached below represents a solution to the system of inequalities.

Point A (-5,2) does not lie in the solution to the system of inequalities.

Point B (5,-2) lies in the solution to the system of inequalities.

Point C (-5,-2) does not lie in the solution to the system of inequalities.

Point D (5,2) does not lie in the solution to the system of inequalities.

Therefore, the point that lies in the solution to the system of inequalities is (5,-2).

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The vertex of the given function is (0,4)  

What is a quadratic equation?

A quadratic equation is a second-order polynomial equation with a single variable such as x, where ax²+bx+c=0. with a ≠ 0 . Given that it is a second-order polynomial equation, the algebraic fundamental theorem ensures that it has at least one solution. Real or complicated solutions are both possible.

identify the coefficients a and b of the quadratic function

y =ax² +bx+c-------(1)

a=2, b=0

x-coordinate of the vertex is x= -b/2a = -(0)/(2(2)) = 0

Substitute  x=0 into given equation  to get y-coordinate of the vertex

y= 2(0)² + 4 = 0+4 = 4

Vertex= (x,y) = (0,4)

The vertex of the given function is (0,4)

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Answer:

5(3-2)

Step-by-step explanation:

15-10

5×3-5×2

Answer:

C) 0.24 kg

----------------------------

Find the weight of two loaves of bread:

Find the difference of weights of 2 loaves of bread and 6 muffins:

The matching choice is C.

The difference of weights of 2 loaves of bread and 6 muffins:

= 0.24 kg

We have to given that,

One loaf of bread weighs 0.68 kilogram.

And, Six muffins weigh 1.12 kilograms.

Now, We can Find the weight of two loaves of bread:

Here, One loaf of bread weighs 0.68 kilogram.

Hence, the weight of two loaves of bread:

0.68 x 2 = 1.36 kg

Hence, the difference of weights of 2 loaves of bread and 6 muffins:

= 1.36 kg - 1.12 kg

= 0.24 kg

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The formula to calculate the nth term of the sequence is an = -3n+18.

The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same amount. It also goes by the name Arithmetic Sequence.

Given that the arithmetic series is 15,12,9,6.....

The nth term of the arithmetic sequence will be calculated as:-

an = a₁ + (n-1)d

The first term is 15 and the common difference for the series is 12-15 = -3.

The nth term will be calculated as:-

an = a₁ + (n-1)d

an = 15 + (n-1)-3

an = 15 -3n +3

an = -3n + 18

Therefore, an = -3n+18 is the formula to find the nth term in the sequence.

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According to the given information the lateral surface area of given cylinder is 725.34 in².

In mathematics, a cylindrical is indeed a three-dimensional solid that retains two parallel bases spaced apart by a curving surface. These bases frequently have a circular shape (like a circle), and also an axis links their respective centers.

A cylinder is a three-dimensional form having two parallel, round sides on either end, one side that curves, no vertices or edges.

As we know that the lateral surface area of cylinder (A) = 2πrh

Given

D = [tex]16\frac{1}{2}[/tex]in

h = 14in

r = D/2

 = 33/4 in

A = (2 × 3.14 × 33 × 14) / 4

   = 2901.36 /4

   = 725.34 in²

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Answer:32768/7

Step-by-step explanation:

double every term

6x8 is 48. You can't simplify 48/5, so that's your answer. If you wanted to create a mixed number, you could write 9 3/5

Answer:

Given that s = 1.5, the expressions that have a value of 13.5 are:

(1) 5s + 8 = 5 × 1.5 + 8 = 13.5

(2) 2s^2 + 9 = 2 × 1.5^2 + 9 = 13.5

Note: The value of any other expression will depend on the specific formula and the value of other variables used in the expression.