Solve the trigonometric equation in the interval [0, 2π). give the exact value, if possible; otherwise, round your answer to two decimal places. (enter your answers as a comma-separated list.) 2 cos2(x) + cos(2x) = 0 x =

Answer:

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

The two smaller right triangles are similar by AA property.

Use ratios of corresponding sides to get:

Simplify and solve for y:

To select all the tables that show quadratic functions, look for tables with a second-degree polynomial equation in the form of "y = ax² + bx + c".

Tables that show quadratic functions:

(a).

x y

-2 8

-1 3

0 0

1 1

2 4

This table shows a quadratic function in the form of y = x² - 2x.

(b)

x y

-3 0

-2 1

-1 4

0 9

1 16

2 25

3 36

This table shows a quadratic function in the form of y = x².

A quadratic function is a second-degree polynomial function that can be expressed in the general form of "y = ax² + bx + c", where a, b, and c are constants.

In this form, the variable "x" is squared, and the coefficient "a" determines whether the parabola opens upward or downward.

To identify tables that show quadratic functions, we need to look for tables that display data points that follow a quadratic pattern.

That is, the dependent variable (y) changes in a way that corresponds to a quadratic equation.

In the first table, the values of y correspond to the quadratic equation y = x² - 2x. The second table shows a set of data points that corresponds to the quadratic function y = x².

Therefore, these two tables show quadratic functions.

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Answer:

5x=380

x = 76

Carla made 76 cookies each hours

Step-by-step explanation:

Just make an equation, so the total number of cookies is 380 and she works for 5 hours, so it is just 380/5.

Prerequisites for the MVT are not met due to the discontinuity and non-differentiability of f(x) at x = 0 within the interval [-1, 1].

Let's first understand the Mean Value Theorem (MVT). The MVT states that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a)

Now, consider the function f(x) = 1/x^2. This function is continuous and differentiable for all x ≠ 0. However, in the interval [-1, 1], the function is not continuous nor differentiable at x = 0. Therefore, the Mean Value Theorem does not apply to this interval.

Since the MVT does not apply, we cannot say there exists a c in the interval (-1, 1) such that f'(c) = (f(1) - f(-1)) / (1 - (-1)). This is because the prerequisites for the MVT are not met due to the discontinuity and non-differentiability of f(x) at x = 0 within the interval [-1, 1].

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the evaluated double integral is approximately 14.25.

To evaluate the given double integral, we need to first understand the problem properly. We have the function f(x, y) = y + xy, and the region R is described by the inequalities: 0 < x < y^2, and 1 < y < 2.

Now we can set up the double integral:

∬(y + xy) dA over the region R.

Since we are given that 0 < x < y^2 and 1 < y < 2, we can set up the integral using the given limits of integration:

∫(from y = 1 to 2) ∫(from x = 0 to y^2) (y + xy) dx dy.

Now, we can start by integrating the inner integral with respect to x:

∫(from y = 1 to 2) [(yx + (1/2)x^2*y) evaluated from x = 0 to x = y^2] dy.

After evaluating the inner integral, we have:

∫(from y = 1 to 2) (y^3 + (1/2)(y^2)^2*y) dy.

Now, we can integrate the outer integral with respect to y:

[((1/4)y^4 + (1/6)y^6) evaluated from y = 1 to y = 2].

After evaluating the outer integral, we get:

[(1/4)(2^4) + (1/6)(2^6)] - [(1/4)(1^4) + (1/6)(1^6)].

Calculating the final result:

(4 + 10.6667) - (0.25 + 0.1667) = 14.6667 - 0.4167 ≈ 14.25.

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According to given question Mike has 25 candies in the beginning.

Let's assume that Mike had "x" candies in the beginning.

After giving 15 candies to his friend, he would have had (x - 15) candies left.

His mom then bought him twice as many candies as he had in the beginning, which would be 2x candies.

So, the total number of candies Mike has now is (x - 15) + 2x = 60.

Combining like terms, we get 3x - 15 = 60.

Adding 15 to both sides, we have 3x = 75.

Finally, dividing both sides by 3, we find that x = 25.

Therefore, Mike had 25 candies in the beginning.

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The complete question is  Mike had some candies. He gave 15 of them to his friend. After that, his mom bought him twice as many candies as he had in the beginning. How many candies did Mike have in the beginning if he now has a total of 60 candies?

To evaluate the indefinite integral of 3x^2 – 3x +1/ x^3 + 2, we can use partial fraction decomposition.

First, we factor the denominator: x^3 + 2 = (x + ∛2)(x^2 – ∛2x + 2).

Next, we can write the fraction as:

3x^2 – 3x +1/ x^3 + 2 = A/x + B(x^2 – ∛2x + 2) + C(x + ∛2)

Multiplying both sides by the denominator, we get:

3x^2 – 3x + 1 = A(x^2 – ∛2x + 2)(x + ∛2) + Bx(x + ∛2) + C(x^2 – ∛2x + 2)

To solve for A, B, and C, we can plug in specific values of x. For example, if we plug in x = -∛2, we get:

-2√2 + 1 = A(4√2) + C(0)

Therefore, A = (2 – √2)/8.

If we plug in x = 0, we get:

1 = A(2√2) + B(0) + C(√2)

Therefore, C = 1/√2.

Finally, if we plug in x = 1, we get:

1 = A(3√2) + B(1 – √2) + C(1 + √2)

Therefore, B = (-1 + √2)/4.

Now that we have A, B, and C, we can write the original fraction as:

3x^2 – 3x +1/ x^3 + 2 = (2 – √2)/8 * 1/x + (-1 + √2)/4 * (x^2 – ∛2x + 2) + 1/√2 * (x + ∛2)

Using this partial fraction decomposition, we can now integrate each term separately.

Integrating the first term, we get:

∫(2 – √2)/8 * 1/x dx = (1/8)(2ln|x| – √2 ln|x^2 + 2|) + C

Integrating the second term, we can complete the square to get:

∫(-1 + √2)/4 * (x^2 – ∛2x + 2) dx = (-1 + √2)/4 * ∫(x – ∛1/2)^2 + 3/2 dx = (-1 + √2)/4 * ((x – ∛1/2)^2 + 3/2) + C

Integrating the third term, we get:

∫1/√2 * (x + ∛2) dx = (1/2√2) * (x^2/2 + ∛2x) + C

Putting it all together, we have:

∫(3x^2 – 3x +1)/ (x^3 + 2) dx = (1/8)(2ln|x| – √2 ln|x^2 + 2|) + (-1 + √2)/4 * ((x – ∛1/2)^2 + 3/2) + (1/2√2) * (x^2/2 + ∛2x) + C

where C is the constant of integration.

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The solution to the differential equation is f(a) = 1/√(12a⁴/125 - 769/5000).

To find f(a), we need to solve the differential equation:

dy/dx = 24yx³

Separating variables, we get:

dy/y³ = 24x³ dx

Integrating both sides, we get:

-1/(2y²) = 6x⁴ + C

where C is the constant of integration.

To find the value of C, we use the fact that the y-intercept of the curve y = f(2) is 5. This means that when x = 2, y = 5. Substituting these values into the equation above, we get:

-1/(2(5)²) = 6(2)⁴ + C

Simplifying and solving for C, we get:

C = -1/(2(5)²) - 6(2)⁴

C = -769/125

So the solution to the differential equation is:

-1/(2y²) = 6x⁴ - 769/125

Solving for y, we get:

y = 1/√(12x⁴/125 - 769/5000)

Therefore, f(a) = 1/√(12a⁴/125 - 769/5000).

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m∠f to the nearest degree is 83°.

The Law of Cosines is used to find an angle when all triangle sides are known.

f² = d² +e² -2de cos(F)

To find angle ∠F, we need to find the value of cos(∠F), which we can do by rearranging the Law of Cosines as follows:

cos(F) = (d² +e² -f²) / (2de)

cos(F) = (20² + 25² - 30²) / (2 × 20 × 25)

cos(F) = (400 + 625 - 900) / (1000)

cos(F) = 125/1000

∠F = arccos(1/8)

∠F = 82.8°

Rounding to the nearest degree

∠F = 83°

Hence, m∠f to the nearest degree is 83°.

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a. The smallest value in the data set is 4 miles per hour. b. The largest value in the data set is 48 miles per hour. c. The median is 26 miles per hour.

a. The smallest value in the data set is 4 miles per hour.

b. The largest value in the data set is 48 miles per hour.

c. To find the median, we need to arrange the values in order from smallest to largest:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

The median is the middle value in this list. Since there are an even number of values, we take the average of the two middle values:

Median = (24 + 28) / 2 = 26

Therefore, the median speed of the 100 cars as they passed through the busy intersection is 26 miles per hour.

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The seconds, is it between the time plunk passes Ms. q and the time that plunk finishes the race is 7.5 seconds..

Flow Distance In the Mathematics or Quants part of any competitive test, time is one of the most well-liked and significant topics. For inquiries about a variety of subjects, including motion in a straight line, circular motion, boats and streams, races, clocks, etc.

The notion of Speed, Time, and Distance is frequently employed. Candidates should make an effort to comprehend how the variables of speed, distance, and time interact.

Ms. q being slow will get a head start for the race so,

3 second head start = 3 x 5 = 15 meters

There difference in speed is 8- 5 = 3 m/s          

Time required for the Plunk to catch up to Ms. q is:

15 / 3 = 5 seconds when P catches Q  

(this is 8 seconds after Q starts the race)

In 5 seconds Plunk runs   5 x 8 = 40 meters   this is when they are at the same point that is at time 8 seconds.

60 meters left in the race will take  Plunk :

60 m / 8 m/s = 7.5 seconds to finish the race.

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Answer:

x/4 + 3/28 = y

Step-by-step explanation:

To find the inverse, switch the x's and y's (note that f(x) is y) and solve for y:

x = 4y - 3/7

x + 3/7 = 4y

x/4 + 3/28 = y

Answer:

−1(x) = 3√2(x+7) 2 f - 1 (x) = 2 (x + 7) 3 2 is the inverse of f (x) = 4x3 − 7 f (x) = 4 x 3 - 7.

The sum of vectors u  = <4,8>,and v = <-2, 6>  i.e. (u+v) is <2, 14>

To find the sum of vectors u and v (u+v), you need to perform the following steps:

1. Identify the components of vectors u and v: u = <4, 8> and v = <-2, 6>.

2. Add the corresponding components of both vectors: To find the sum (u+v), add the x-components (4 and -2) and the y-components (8 and 6) separately.

3. Calculate the sum of the x-components: 4 + (-2) = 2.

4. Calculate the sum of the y-components: 8 + 6 = 14.

5. Combine the results to form the new vector (u+v): <2, 14>.

So, the sum of vectors u and v (u+v) is <2, 14>.

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Answer:

x = -9

Step-by-step explanation:

As this is an absolute value function, the line of symmetry is the x-value of the maximum/minimum point. An absolute value function can be denoted as y = |x - h| + k, where (h, k) is the maximum/minimum point. We only need the x-value of the maximum/minimum point, so we only have to look at "h". Now, we can use y = |x - h| + k and turn it into g(x):

y = |x - h| + k

g(x) = |x - -9| + -11   --> this means h = -9, and the line of symmetry is at x = -9

g(x) = |x + 9| - 11

Answer:

I think x equals --9

The expression that is equivalent to 3v+2v is:

D. v+v+v+v+v

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we substitute the same value(s) for the variable(s).

To find the expressions that are equivalent to 3v+2v, we need to find the expression which when simplified will give the same expression as 3v+2v. That is: 3v + 2v = 5v

v*5 = 5v

v+5 = v + 5

v+5v = 6v

v+v+v+v+v = 5v

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To determine whether a critical point is a relative minimum or maximum using concavity, we need to examine the second derivative of the function at the critical point.

If the second derivative is positive, then the function is concave up, meaning it is shaped like a bowl opening upwards. At a critical point where the first derivative is zero, this indicates a relative minimum, as the function is increasing on either side of the critical point.

On the other hand, if the second derivative is negative, then the function is concave down, meaning it is shaped like a bowl opening downwards. At a critical point where the first derivative is zero, this indicates a relative maximum, as the function is decreasing on either side of the critical point.

If the second derivative is zero, then the test is inconclusive and further analysis is needed, such as examining higher order derivatives or using other methods such as the first derivative test.

Therefore, the concavity test is a useful method for determining the nature of critical points and whether they represent a relative minimum or maximum.

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Complete question is:

What we need to examine for a method for determining whether a critical point is a relative minimum or maximum using concavity.

The median means that as many as friends have less than A. 1. 5 pets as those that have more than A. 1. 5 pets.

The median is a measure of central tendency in statistics that represents the middle value of a dataset when it is ordered from smallest to largest. The median is often used as a more robust measure of central tendency than the mean, because it is less affected by extreme values in the dataset.

From the box plot, the data set of friends with pets would be:

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, ,2 ,2, 2, 2, 2, 3, 4, 4, 7.

The median here is:

= ( 10 th position + 11 th position ) / 2

= ( 1 + 2 ) / 3

= 1. 5

This therefore means that as many friends have more than 1. 5 pets as those with less than 1. 5 pets because the median shows the number which had the same number above, and the number below.

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Answer:

1/221.

Step-by-step explanation:

Probability(first card is a King) = 4/52 = 1/13 (as there are 4 kings in the pack).

Now there are 51 cards left in the pack, 3 of which are Kings, so:

Probability(second card is a King) = 3/51 = 1/17.

These 2 events are independent so we multiply the probabilities:

Required probability =

1/13 * 1/17

= 1/221.

Answer: ∠b = 105°

Step-by-step explanation:

     We know that a straight line is equal to 180 degrees. We will create an equation and solve for ∠b.

180° = ∠b + 75°

∠b = 180° - 75°

∠b = 105°

The distance from Xavier and David to the ranch can be found using the distance formula:

distance = sqrt((change in x)^2 + (change in y)^2 + (change in z)^2)

In this case, we are given the distances DR and XR, which represent the change in x, y, and z coordinates between the tents and the ranch. We know that the tents are located at equal distances from the ranch, so the change in x, y, and z coordinates for both David and Xavier will be the same.

Let's call the distance from each tent to the ranch "d", then we have:

DR = XR = d

Substituting the given values, we get:

12.3z + 12.4 = 10.5z + 34

Solving for z, we get:

z = 6.8

Now we can find the distance from each tent to the ranch using the formula:

distance = sqrt((change in x)^2 + (change in y)^2 + (change in z)^2)

For David's tent:

distance = sqrt((12.3z)^2 + 0^2 + (12.4)^2) = sqrt((12.3*6.8)^2 + (12.4)^2) = 87.9 meters (rounded to one decimal place)

For Xavier's tent:

distance = sqrt((10.5z)^2 + 0^2 + (34)^2) = sqrt((10.5*6.8)^2 + (34)^2) = 95.6 meters (rounded to one decimal place)

Therefore, David and Xavier are 87.9 meters and 95.6 meters away from the ranch, respectively.

A: P(t) [tex]= $567,000 * (1 + 0.126)^t[/tex]

B: [tex]567,000 * (1 + 0.126)^2 = $671,448.14[/tex]

A: To model the price of the house over time, we can use the exponential formula: P(t) = P₀ * (1 + r)^t, where P₀ is the initial price, r is the rate of appreciation, and t is the time in years.

In this case, the initial price (P₀) of the house is $567,000 and the rate of appreciation (r) is 12.6% expressed as a decimal, which is 0.126. Therefore, the exponential formula to model the price of the house over time would be: P(t) = 567,000 * (1 + 0.126)^t.

B: To find the price of the house in 2022, we substitute t = 2022 - 2020 = 2 into the exponential formula.

P(2) = 567,000 * (1 + 0.126)^2

P(2) = 567,000 * (1.126)^2

P(2) ≈ 567,000 * 1.268

P(2) ≈ $719,976.00

Therefore, the price of such a house in 2022 would be approximately $719,976.00

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The missing side length is 5√3 centimeters.

We can use the Pythagorean theorem to find the missing side length. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. In equation form, this looks like:

a² + b² = c²

where a and b are the lengths of the legs, and c is the length of the hypotenuse.

To use this formula to solve for the missing side length, we can plug in the values we know:

5² + b² = 10²

We can simplify this equation by squaring 5 and 10:

25 + b² = 100

Next, we can isolate the variable (b) on one side of the equation by subtracting 25 from both sides:

b² = 75

Finally, we can solve for b by taking the square root of both sides:

b = √(75)

This simplifies to:

b = 5*√(3)

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Complete Question:

By using the Pythagoras theorem, Find the value of the Other side when the value of hypotenuse is 10 cm and the value of the side is 5 cm.

1. The coordinates are: START: (0,0) END: (6,-4), 2. The perimeter of rectangle YARD is 20 units,3. The lengths of YX and YZ are 4 units and 6 units, respectively, 4. The area of Mary's room is 24 square units,

1-To get from one corner of his yard to another, Jason travels 4 units down and then 6 units right. Starting from the origin, his starting point is (0,0). From there, he moves 4 units down to the point (0,-4), and then 6 units right to reach his endpoint, which is at (6,-4).

2-The rectangle has two sides of length 4 and two sides of length 6. The perimeter is the sum of the lengths of all sides, so it is equal to 2(4) + 2(6) = 8 + 12 = 20 units.

3-The coordinates of points Y, X, and Z are not given, so we cannot calculate the lengths directly. However, we know that the sides of a rectangle are perpendicular, so we can use the Pythagorean theorem to find the lengths. Let Y be the origin (0,0), and let X be the point (0, -4). Then YX has length 4 units. Similarly, let Z be the point (6, 0), so YZ has length 6 units.

4.To find the area of a rectangle, we can multiply the lengths of its sides. From problem 3, we know that the lengths of the sides are 4 and 6 units, so the area is 4 x 6 = 24 square units.

5. The sides AB and AC form a right angle.

To determine which sides of the triangle form a right angle, we need to find the slope of each side. The slope of AB is (3-8)/(6-3) = -5/3, and the slope of AC is (3-3)/(6-3) = 0. Since the product of the slopes of two perpendicular lines is -1, we can see that AB is perpendicular to AC. Therefore, sides AB and AC form a right angle.

6. Tyler walks 9 blocks in all.

To find the distance Tyler walks, we need to calculate the length of sides AB and AC. Using the distance formula, we can find that the length of AB is sqrt[(6-3)² + (3-8)²] =√[(34) units, and the length of AC is 3 units. Therefore, Tyler walks 3 + √[34 units along the two sides that form the right angle. This is approximately 9.4 blocks, so he walks 9 blocks in all.

7. The distance between two points in a coordinate plane can be found using the distance formula:

d = √[(x₂-x₁)² + (y₂-y₁)²]

where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points, and d is the distance between them. The formula is derived from the Pythagorean theorem, which relates the sides of a right triangle.

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The rate can be expressed as "3 players per sport"

A rate is a ratio that compares two quantities with different units. In this case, we have 30 players and 10 sports. To express this as a rate, we want to compare the number of players to the number of sports. We can write this as:

30 players / 10 sports

To simplify this ratio, we can divide both the numerator (30 players) and denominator (10 sports) by the same factor to get an equivalent ratio. In this case, we can divide both by 10:

(30 players / 10) / (10 sports / 10)

This simplifies to:

3 players / 1 sport

So the rate can be expressed as "3 players per sport" or "3:1" (read as "three to one"). This means that for every one sport, there are three players.

Alternatively, we can express the rate as a fraction or decimal by dividing the number of players by the number of sports:

30 players / 10 sports = 3 players/sport = 3/1 = 3 or 3.0 (as a decimal)

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Answer:

  C.  Box B

Step-by-step explanation:

You want to know which of these two boxes has a volume greater than 9.5 ft³:

The volume of each box is found by multiplying its dimensions:

  Box A: (3 ft)(2 ft)(1.5 ft) = 9 ft³

  Box B: (2.5 ft)(2 ft)(2 ft) = 10 ft³

Only box B is large enough, choice C.

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Dina can reach a maximum kinetic energy of 2450 Joules when she skis to the bottom of the slope.

Potential energy (PE) = m x g x h

where m = mass, g = acceleration due to gravity, and h = height

Here, Dina's mass (m) = 50 kg, height (h) = 5 m, and acceleration due to gravity (g) = 9.8 m/s².

So, PE = 50 x 9.8 x 5

PE = 2450 Joules

When Dina skis down the slope, all of her potential energy will be converted into kinetic energy (KE) at the bottom of the slope, neglecting any losses due to friction and air resistance.

Thus, the maximum kinetic energy (KE) Dina can reach at the bottom of the slope is also 2450 Joules.

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The maximum kinetic energy she can reach when she skis to the bottom of the slope is 2452 joules

To find the maximum kinetic energy that Dina can reach when she skis to the bottom of the slope, we need to use the principle of conservation of energy, which states that the total energy of a system remains constant.

At the top of the slope, Dina has potential energy due to her position relative to the ground. This potential energy is given by:

PE = m × g × h

where m is Dina's mass (50 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slope (5 m).

PE = 50 kg × 9.8 m/s^2 × 5 m = 2450 J

When Dina skis to the bottom of the slope, all of her potential energy is converted into kinetic energy, which is given by:

KE = 1/2 × m × v^2

where v is her velocity at the bottom of the slope.

To find the maximum velocity, we can use the fact that the total energy of the system remains constant:

PE = KE

2450 J = 1/2 × 50 kg × v^2

v^2 = 98 m^2/s^2

v = sqrt(98) = 9.90 m/s

Finally, we can substitute this velocity into the kinetic energy equation to find the maximum kinetic energy that Dina can reach:

KE = 1/2 × 50 kg × (9.90 m/s)^2 = 2452 J

Therefore, Dina can reach a maximum kinetic energy of 2452 Joules when she skis to the bottom of the slope.

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The price of one apple is $0.70, obtained by solving the system of equations 4x + 9y = 12.70 and 8x + 11y = 17.70 using elimination.

Let's use a system of equations caculation the problem.

Let x be the price of one apple and y be the price of one banana.

From the first sentence, we know that:

4x + 9y = 12.70

From the second sentence, we know that:

8x + 11y = 17.70

Now we can solve for x by using either substitution or elimination.

Let's use elimination.

We can multiply the first equation by 11 and the second equation by -9, then add them together:

44x + 99y = 139.70

-72x - 99y = -159.30

-28x = -19.60

Dividing both sides by -28, we get:

x = 0.70

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To find the temperature at 6pm, we need to start with the temperature at 9am and then add or subtract the changes in temperature that occurred during the day.

We know that the temperature at 9am was -1.9°F. Between 9am and noon, the temperature rose 11.3°F, so at noon the temperature was:

-1.9 + 11.3 = 9.4°F

Between noon and 3pm, the temperature dropped 7.9°F, so at 3pm the temperature was:

9.4 - 7.9 = 1.5°F

Between 3pm and 6pm, the temperature dropped 12.7°F, so at 6pm the temperature was:

1.5 - 12.7 = -11.2°F

Therefore, the temperature at 6pm was -11.2°F.

The length that will give a plot with the maximum area allowed is 6 meters.

To find the length that will give a plot with the maximum area, we need to use the formula for the area of a rectangle, which is A = lw, where A is the area, l is the length, and w is the width. In this case, we are given that the area is 45 square meters, and the width is 7.5 meters.

Substituting these values into the formula, we get:

45 = l(7.5)

To solve for l, we divide both sides by 7.5:

l = 45/7.5

Simplifying, we get:

l = 6

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