A ladder is propped against the side of a building making a 30 degree
angle with the building.
If the base of the ladder is 14 feet from the base of the building, how
long is the ladder, in feet? Round your answer to the nearest whole
number.

The rewrite value as a sum or difference is cos 2x cos 4x = (1/2)[cos(6x) + cos(2x)].

We are given the expression cos 2x cos 4x, and we need to rewrite it as a sum or difference.

The following formula can be used to write the product of two trigonometric functions as a sum or difference:

cos A cos B = (1/2)[cos(A + B) + cos(A - B)]sin A sin B = (1/2)[cos(A - B) - cos(A + B)]sin A cos B

= (1/2)[sin(A + B) + sin(A - B)]cos A sin B = (1/2)[sin(A + B) - sin(A - B)]

Here, we have cos 2x cos 4x, so we can use the first formula with A = 2x and B = 4x.cos 2x cos 4x

= (1/2)[cos(2x + 4x) + cos(2x - 4x)]cos 2x cos 4x = (1/2)[cos(6x) + cos(-2x)]

We can simplify further by using the fact that cos(-θ) = cos(θ).cos 2x cos 4x = (1/2)[cos(6x) + cos(2x)]

So, we have rewritten cos 2x cos 4x as the sum of two cosine functions.

The first term has an argument of 6x, and the second term has an argument of 2x.

Summary: To rewrite cos 2x cos 4x as a sum or difference, we can use the formula cos A cos B = (1/2)[cos(A + B) + cos(A - B)].

Using this formula, we get cos 2x cos 4x = (1/2)[cos(6x) + cos(2x)].

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The minimum value of V occurs at x ≈ 0.93, which means that the volume of the box is smallest when the height is about 0.93 units.

To find the extreme values of V, we need to take the derivative of V and set it equal to zero. So, let's begin:

\(V(x) = x(10-2x)(16-2x)\)
Taking the derivative with respect to x:
\(V'(x) = 10x - 4x^2 - 32x + 12x^2 + 320 - 48x\)
Setting V'(x) = 0 and solving for x:
\(10x - 4x^2 - 32x + 12x^2 + 320 - 48x = 0\\8x^2 - 30x + 320 = 0\)
Solving for x using the quadratic formula:
\(x = (30 ± \sqrt{(30^2 - 4(8)(320))) / (2(8))\\x = (30 ± \sqrt{(1680)) / 16\\x = 0.93 or x =5.07\)
So, the extreme values of V occur at x ≈ 0.93 and x ≈ 5.07. To determine whether these are maximum or minimum values, we need to examine the second derivative of V. If the second derivative is positive, then the function has a minimum at that point. If the second derivative is negative, then the function has a maximum at that point. If the second derivative is zero, then we need to use a different method to determine whether it's a maximum or minimum.

Taking the second derivative of V:
V''(x) = 10 - 8x - 24x + 24x + 96
V''(x) = -8x + 106

Plugging in x = 0.93 and x = 5.07:
V''(0.93) ≈ 98.36 > 0, so V has a minimum at x ≈ 0.93.
V''(5.07) ≈ -56.56 < 0, so V has a maximum at x ≈ 5.07.

Now, to interpret these values in terms of the volume of the box, we need to remember that V(x) represents the volume of a box with length 2x, width 2x, and height x. So, the maximum value of V occurs at x ≈ 5.07, which means that the volume of the box is greatest when the height is about 5.07 units. The minimum value of V occurs at x ≈ 0.93, which means that the volume of the box is smallest when the height is about 0.93 units.

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a) The extreme values of V are:

Minimum value: V(0) = 0

Relative maximum value: V(3) = 216

Absolute maximum value: V(4) = 128

b) The absolute maximum value of V at x = 4 represents the case where the box has a square base of side length 4 units, height 2 units, and width 8 units, which has a volume of 128 cubic units.

a) To find the extreme values of V, we first need to find the critical points of the function. This means we need to find where the derivative of the function equals zero or is undefined.

Taking the derivative of V(x), we get:

\(V'(x) = 48x - 36x^2 - 4x^3\)

Setting this equal to zero and solving for x, we get:

\(48x - 36x^2 - 4x^3 = 0\)
4x(4-x)(3-x) = 0

So the critical points are x = 0, x = 4, and x = 3.

We now need to test these critical points to see which ones correspond to maximum or minimum values of V.

We can use the second derivative test to do this. Taking the derivative of V'(x), we get:

\(V''(x) = 48 - 72x - 12x^2\)

Plugging in the critical points, we get:

V''(0) = 48 > 0 (so x = 0 corresponds to a minimum value of V)
V''(4) = -48 < 0 (so x = 4 corresponds to a maximum value of V)
V''(3) = 0 (so we need to do further testing to see what this critical point corresponds to)

To test the critical point x = 3, we can simply plug it into V(x) and compare it to the values at x = 0 and x = 4:

V(0) = 0
V(3) = 216
V(4) = 128

So x = 3 corresponds to a relative maximum value of V.
b) In terms of the volume of the box, the function V(x) represents the volume of a rectangular box with a square base of side length x and height (10-2x) and width (16-2x).
The minimum value of V at x = 0 represents the case where the box has no dimensions (i.e. it's a point), so the volume is zero.
The relative maximum value of V at x = 3 represents the case where the box is a cube with side length 3 units, which has a volume of 216 cubic units.
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An eigenvalue of a matrix is a scalar value that satisfies the equation ax = λx for some vector x. If a is an n × n matrix, then if ax = λx for some vector x, λ is an eigenvalue of a

An eigenvalue of a matrix is a scalar value that satisfies the equation ax = λx for some vector x. If a matrix a is an n × n matrix, then the equation ax = λx can be used to find the eigenvalues associated with the matrix. By taking the equation ax = λx and rearranging it, we can isolate the eigenvalue λ on the left side of the equation, leaving the vector x on the right side. λ is then the eigenvalue of the matrix a. For example, if we have a 2 x 2 matrix a, and we have an equation of the form ax = λx, where x is a 2-dimensional vector, then by rearranging the equation we can find the eigenvalue λ associated with the matrix a. In summary, if a is an n × n matrix and ax = λx for some vector x, then λ is an eigenvalue of a.

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Answer:

\(\boxed{70 \ \text{units}^{2}}\)

Step-by-step explanation:

I assume that the polygon is made up of a rectangle and two triangles. Thus, the area of the polygon is the area of the triangles and the rectangle.

≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡

\(\text{Area of polygon} = \text{Area of rectangle} + \text{Area of triangle}_{1} } + \text{Area of triangle}_{2}\)

➡ \(\text{Area of polygon}= 49 + 7 + 14\)

➡ \({\bold{Area \ of\ polygon = 49 + 21 = \underline{70 \ units}^{2}}\)

≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡

Given the expression: (3.2+0.2)2+12​

To get the value of the expression, do the following:

1. add 3.2 and 0.2

2. multiply 3.4 by 2

3. add 6.8 and 12

So, the value of the expression = 18.8

A pharmacist puts 32 pills into a prescription order that calls for 30 pills: The absolute error, in this case, would be 2 pills.

To find the absolute error in this situation, we'll follow these steps:

1. Determine the actual number of pills the pharmacist put in the prescription order: 32 pills
2. Determine the number of pills the order calls for 30 pills
3. Calculate the absolute error by subtracting the expected number of pills from the actual number: |32 - 30|

The absolute error, in this case, is |32 - 30| = 2 pills.

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Answer:

answer is option A

Step-by-step explanation:

Sure hope this helps uou :)

Answer: 5 to the power of 2

Step-by-step explanation:

Answer:

\(5^{2}\)

Step-by-step explanation:

\(\frac{5^4 x 5^3}{5^5} = \frac{5^7}{5^5} = 5^2\)

The total surface area of the resulting solid is \(6+96=\boxed{102}\) square inches.

The resulting solid after the described procedure consists of a central cube of side length 1 inch and 6 congruent smaller cubes, each of side length 2 inches. We can find the surface area of the resulting solid by adding up the surface area of each of these cubes.

The surface area of the central cube is simply \(6(1^2)=6\) square inches.

The surface area of each of the smaller cubes is \(6(2^2)=24\) square inches, but since each cube shares one face with the central cube and another face with its neighboring cube, we only count four of its faces, so the contribution of each smaller cube to the surface area of the resulting solid is 4(24/6)=16 square inches.

Since there are 6 smaller cubes in total, their contribution to the surface area of the resulting solid is \(6\times16=96\)square inches.

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Answer:

5%

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

Round up if the number is 5 or more

Answer:

x<-2

Step-by-step explanation:

17+4x<9

4x<-8

x<-2

The solution is:

Work/explanation:

Recall that the process for solving an inequality is the same as the process for solving an equation (a linear equation in one variable).

Make sure that all constants are on the right:

\(\bf{4x < 9-17}\)

\(\bf{4x < -8}\)

Divide each side by 4:

\(\bf{x < -2}\)

The expression that is equivalent to the function \(y=log_7x\) is \(x = 7^y\)

Given the logarithmic function \(y=log_7x\)

We can express this logarithm function in indices form. For instance;

If, \(y=log_ab\), then \(b=a^y\)

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Answer: No is  hasn't

Step-by-step explanation:

Answer:

yes

Step-by-step explanation:

Brainliest me pls

Have a nice day!

(a) Yes, there must be a sequence of consecutive games in which the Ants scored exactly 8 goals.

(b) No, there doesn't have to be a sequence of consecutive games in which the Bees scored exactly 8 goals.

(c) No, there doesn't have to be a sequence of consecutive games in which the Cicadas scored exactly 8 goals.

To show that there must be a sequence of consecutive games in which the Ants scored exactly 8 goals, we can use the Pigeonhole Principle. Since the Ants played 40 games and scored a total of 71 goals, their average goal per game is 71/40 = 1.775. This means that there must be at least one game where the Ants scored 2 goals.

Similarly, if we subtract 2 goals from the total and divide by 39 games remaining, we get an average of 1.692. This means there must be another game where the Ants scored at least 2 goals. By repeating this process, we can find a sequence of consecutive games where the Ants scored exactly 8 goals.

To show that there doesn't have to be a sequence of consecutive games in which the Bees scored exactly 8 goals, we can provide a counterexample. Let's say the Bees scored 40 goals in the first 20 games and 40 goals in the remaining 20 games. In this case, the Bees scored a total of 80 goals, but there is no sequence of consecutive games where they scored exactly 8 goals. This counterexample demonstrates that the claim is not universally true for all scenarios.

To prove or disprove whether there must be a sequence of consecutive games in which the Cicadas scored exactly 8 goals, we need more information. The given total number of goals (79) and the total number of games (40) do not provide enough data to reach a conclusion. Without additional details about the distribution of goals across the games, we cannot determine whether such a sequence exists or not.

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18. Perimeter of rectangle = 2(l + b)

l = 16m

b = 12m

Perimeter of rectangular park = 2 (12 + 16)

= 24 + 32

= 56m

Perimeter of square park = 56m

(given that, perimeter of rectangular and square park are equal)

Perimeter of square = 4a

4a = 56

a = 56/4

a = 14m

Area of square park = a²

= 14²

= 196m²

Answer:

A: -8-4

Step-by-step explanation:

Answer:

I think it is (-8,-4).

Step-by-step explanation:

I hope this helps.

Answer:

612½ pi units² or 1920 units²

Answer:

77 miles per hour

Step-by-step explanation:

You can make a ratio

\( \frac{miles}{hours} = \frac{462}{6} = \frac{x}{1} \)

Now we cross multiply

\(6x = 462\)

Lastly, divide by 6

\(x = 77\)

I'm not entirely sure what you mean but simplifying this expression results in 96x + 5 1/2.

Answer:

b = 125

Step-by-step explanation:

Given a varies directly as \(\sqrt[3]{b}\) then the equation relating them is

a = k\(\sqrt[3]{b}\) ← k is the constant of variation

To find k use the condition a = 3 , b = 64 , then

3 = k\(\sqrt[3]{64}\) = 4k ( divide both sides by 4 )

\(\frac{3}{4}\) = k

a = \(\frac{3}{4}\) \(\sqrt[3]{b}\) ← equation of variation

When a = \(\frac{15}{4}\) , then

\(\frac{15}{4}\) = \(\frac{3}{4}\) \(\sqrt[3]{b}\) ( multiply both sides by 4 to clear the fractions )

15 = 3\(\sqrt[3]{b}\) ( divide both sides by 3 )

5 = \(\sqrt[3]{b}\) , then

b = 5³ = 125

Answer:

Step-by-step explanation:

To convert square meters to square centimeters, we need to multiply by the conversion factor (100 cm / 1 m)^2.

So,

3.9 m² = 3.9 × (100 cm / 1 m)²

3.9 m² = 3.9 × 10,000 cm²

3.9 m² = 39,000 cm²

Therefore, 3.9 square meters is equal to 39,000 square centimeters.

There are 36 cards in the box.  In this scenario, we are given that the probability of choosing a red card from a box is one-third. We also know that there are 24 blue cards in the box. Our goal is to find the total number of cards in the box.

Let's use the concept of probability to solve this problem. The probability of an event is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, the probability of choosing a red card is given as 1/3, and the favorable outcome is selecting a red card. Let R be the number of red cards, and T be the total number of cards in the box. The probability formula can be represented as:

Probability of choosing a red card = R / T

We are given that the probability of choosing a red card is 1/3:

1/3 = R / T

Since there are 24 blue cards in the box, we can represent the total number of cards as the sum of red and blue cards:

T = R + 24

Now, we can solve the system of equations to find the values of R and T:

1. 1/3 = R / T
2. T = R + 24

From equation 1, we can express R as:

R = (1/3) * T

Substitute this expression for R in equation 2:

T = (1/3) * T + 24

Multiplying both sides by 3 to eliminate the fraction, we get:

3T = T + 72

Subtracting T from both sides:

2T = 72

Now, divide by 2 to find the total number of cards, T:

T = 36

So, there are 36 cards in the box.

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If two angles are complementary, then they cannot be congruent. This statement is true. Complementary angles are two angles whose sum is 90°. They do not have to be equal in measure.

On the other hand, congruent angles are angles with equal measure.What are complementary angles?Two angles are said to be complementary when they add up to 90 degrees. This means that the sum of the measures of these angles is 90 degrees.

The measures of complementary angles need not be equal.What are congruent angles? Congruent angles are two or more angles with the same measure. In other words, they have the same angle measurement.

Hence, if two angles are congruent, it means they are exactly equal in measure. If two angles are complementary, they cannot be congruent.

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Answer:

x = 0 or x = 1/4

Step-by-step explanation:

You are able to factor out 3x² from 12x³ - 3x² to get 3x²(4x-1).

Set everything to zero and solve.

Answer:

1/12 a cup

Step-by-step explanation:

1/3*1/4=1/12

Answer:

1/12

Step-by-step explanation:

1/3*1/4=1/12

Answer:

brainliest???

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and insta:   nbsbellaaaaaa

there are 6 a's in it ^

aweee thank youu

i hope you have an amazing day and much love!!!!  mwah

The Riemann zeta-function is defined for all complex numbers x with real part greater than 1, that is, the domain of ζ is {x ∈ C : Re(x) > 1}.

However, the zeta function can be analytically extended to a meromorphic function on the whole complex plane except for a simple pole at x = 1, where it has a limit of infinity.

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The general equation of a circle is

Here, (h,k) is the center of the circle and r is the radius of the circle.

change the given equation into the general form as follows:

So, the center is (0,0) and the radius is r = 5.