What is 48.312 rounded to the nearest tenths?

The area of the rectangle is 35.1 cm²

To find the area of a rectangle, you need to multiply its length by its width.

Given that the dimensions of the rectangle are 5.85 cm and 6 cm, respectively, the area of the rectangle is:

Area = Length x Width

Substitute the known values in the above equation, so, we have the following representation

Area = 5.85 cm x 6 cm

Evaluate the product

Area  = 35.1 cm²

Hence, the area of the rectangle to the nearest tenth is 35.1 cm²

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Complete question

The dimension of a rectangle is 5.85 cm by 6 cm. Find its area to the nearest tenth.

The expression will give 36 after evaluating.

What is an Expression?

In mathematics, an expression is a combination of numbers, variables, and mathematical operations that can be evaluated or simplified. An expression may contain one or more terms, which are separated by operators such as addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^).

Expressions can represent various mathematical concepts, such as equations, inequalities, functions, and polynomials.

Given:  m = -7 and n = 2,

We know that, the mode function converts the function into positive value.

So, |m - n| = |-7 - 2| = 9

Now,

4 |m - n| = 4 * 9 = 36

So the expression 4 |m - n| evaluates to 36 when m = -7 and n = 2.

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Answer:

9π

Step-by-step explanation:

if the circumference=6π, then you can solve for the radius. C=2πr

6π=2πr

Divide 2π from both sides to get r=3

Area=πr^2

A=π*9

Area= 9π

Answer:

Step-by-step explanation:

Let [tex]x,y[/tex] be the two numbers. Then we get

    [tex]2x+3y=-1[/tex]     [tex](a)[/tex]

       [tex]x+4y=2[/tex]       [tex](b)[/tex]

We solve equations [tex](a)[/tex] and [tex](b)[/tex] simultaneously:

    equation [tex](b)\times2[/tex] :

       [tex]2x+8y=4[/tex]        [tex](c)[/tex]  

   [tex](c)-(a)[/tex] gives:

       [tex]5y=5\rightarrow y=1[/tex]

    Sub [tex]y=1[/tex] into [tex](a)[/tex] :  

       [tex]2x+3\times1=-1\rightarrow2x=-4\rightarrow x=-2[/tex]

The solution is -2,1

The sum of the first 10 terms in the sequence is 195.The sequence given above is an arithmetic sequence.

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each successive number is found by adding a fixed number, called the common difference, to the previous number in the sequence. The common difference is the same for each pair of consecutive numbers.

An arithmetic sequence is a sequence of numbers where each successive number is obtained by adding a fixed amount, the common difference, to the preceding number. In this particular sequence, the common difference is 5.

The first term in the sequence is a₁ = -3, and the general term is an = 3an-1 + 5, where n > 2. To find any term in the sequence, we can use the formula an = a₁ + (n - 1)d, where d is the common difference. In this case, a₁ = -3 and d = 5, so the formula becomes an = -3 + (n - 1)5.

To find the 9th term in the sequence, we can substitute n = 9 into the above formula and solve for an. This gives us an = -3 + (9 - 1)5 = -3 + 40 = 37. Therefore, the 9th term in the sequence is 37.

To find the sum of the first 10 terms in the sequence, we can use the formula Sₙ = (n/2)(2a₁ + (n - 1)d). In this case, a₁ = -3 and d = 5, so the formula becomes Sₙ = (n/2)(2(-3) + (n - 1)5). Substituting n = 10 into the formula gives us Sₙ = (10/2)(2(-3) + (10 - 1)5) = (10/2)(-6 + 45) = (10/2)(39) = 195. Therefore, the sum of the first 10 terms in the sequence is 195.

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Right Question:-

a₁ = -3,

an=3an-1+5

for

n>2.

For given sequence, Find the sum of first 10 terms?

The prοbability οf getting either 2 οr οdd number is B)4/49.

Prοbability is a way οf calculating hοw likely sοmething is tο happen. It is difficult tο prοvide a cοmplete predictiοn fοr many events. Using it, we can οnly fοrecast the prοbability, οr likelihοοd, οf an event οccurring. The prοbability might be between 0 and 1, where 0 denοtes an impοssibility and 1 denοtes a certainty.

Here the Tοtal number οf οutcοmes = 7

When spinner is spun first we can get οn favοrable οutcοme , which is 2 οr οdd number.

If prοbability οf getting 2

=> Number οf favοrable οutcοme/Tοtal number οf οutcοme

=> 1/7

Nοw number οf οdd number frοm 1 tο 7 = 4

Prοbability οf getting οdd number in secοnd spun = 4/7

Then prοbability οf getting either 2 οr οdd number = [tex]\frac{1}{7}\times\frac{4}{7}[/tex] = [tex]\frac{4}{49}[/tex].

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Answer:

b) The graph will shift down 5 units from its parent graph.

Step-by-step explanation:

Your answer is in the image.

The width of the arch is 20 meters and the height of the arch is 0 meters.

The height of an arc is the sagitta. It is the line perpendicular to the chord's midpoint and to the arc itself.

The equation for the arch is provided as y = -0.04x2. The junction points (-10, 4) and (10, -4) can be used to calculate the arch's width and height.

The breadth of the arch, which is equal to the separation between the two x-intercepts, is first determined:

Width is equal to 10 minus 10 metres.

Then, we determine the arch's height, which is its highest point. We know that the largest value of y occurs at the parabola's vertex since the equation for the arch has the form y = ax2, where an is a negative constant. The vertex's x-coordinate is provided by the formula x = -b/2a, where a = -0.04 and b = 0. (since there is no linear term in the equation). The vertex's x-coordinate is as a result:

x = -b/2a = -0/2(-0.04) = 0

We change x = 0 in the equation of the arch to obtain the equivalent y-coordinate of the vertex.

y = -0.04x² = -0.04(0)² = 0

As a result, the parabola's vertex is (0, 0), and the arch's greatest height is 0 metres.

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Answer:

Step-by-step explanation:

1) To solve the equation √(6x) = x, we can first square both sides of the equation to eliminate the square root:

(√(6x))^2 = x^2

6x = x^2

Next, we can rearrange the equation into standard quadratic form by subtracting 6x from both sides:

x^2 - 6x = 0

Now, we can factor out an x from the left-hand side of the equation:

x(x - 6) = 0

Setting each factor equal to zero, we find two solutions:

x = 0 or x - 6 = 0

Therefore, the solutions to the equation √(6x) = x are x = 0 and x = 6.

2)To solve √(5x-6) = x, we can square both sides of the equation:

(√(5x-6))^2 = x^2

5x-6 = x^2

Rearranging this quadratic equation to the standard form ax^2 + bx + c = 0, we get:

x^2 - 5x + 6 = 0

This can be factored into:

(x - 2)(x - 3) = 0

Therefore, the solutions to the equation √(5x-6) = x are x = 2 and x = 3.

We should check if these solutions satisfy the original equation:

When x=2: √(5x-6) = √(5(2)-6) = √4 = 2, which satisfies the equation.

When x=3: √(5x-6) = √(5(3)-6) = √9 = 3, which also satisfies the equation.

Therefore, the solutions are x = 2 and x = 3.

3) To solve the equation √(2x-1) = x-3, we can square both sides of the equation to eliminate the square root:

(√(2x-1))^2 = (x-3)^2

Simplifying the left-hand side gives:

2x-1 = (x-3)^2

Expanding the right-hand side gives:

2x-1 = x^2 - 6x + 9

Rearranging the equation gives:

x^2 - 8x + 10 = 0

We can solve this quadratic equation :

x^2 - 8x + 10 = 0

Now we can solve this quadratic equation by using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -8, and c = 10. Substituting these values into the formula:

x = (-(-8) ± √((-8)^2 - 4(1)(10))) / 2(1)

Simplifying:

x = (8 ± √(64 - 40)) / 2

x = (8 ± √24) / 2

x = 4 ± √6

Therefore, the solutions to the equation √(2x - 1) = x - 3 are x = 4 + √6 and x = 4 - √6.

4) has no real solutions.

when you have to remove the square root you have to power the both sides of the equation.

1)6x= x^2

 0 = x^2 - 6x

0= x(x-6)

x=0 or x-6=0

            x=6

2) [tex]\sqrt{5x-6}[/tex] = x

   5x-6=x^2

   x^2 -5x+6=0

   (x-3)(x-2)=0

    x-3=0 or x-2= 0

    x=3 or x=2

3)

[tex]\sqrt{2x-1}[/tex] = x-3

2x-1 = (x-3)^2

2x-1= x^2 - 6x +9

x^2 -8x +10=0

this equation can't solve by factoring method easily. so we have to use the completing square method.

x^2-8x+10=0

x^2-8x= -10

x^2 - 8x+(-4)^2 = -10+ ( -4)^2

(x-4)^2 = 6

[tex]x= 4 +\sqrt{6}[/tex] or  [tex]x= 4 -\sqrt{6}[/tex]

4) [tex]x= 2 + \sqrt{2x-11}[/tex]

x^2= 4+4 [tex]\sqrt{2x-11}[/tex] + 2x-11

x^2= 7+ 2x +4[tex]\sqrt{2x-11}[/tex]

x^2-2x-7= 4[tex]\sqrt{2x-11}[/tex]

(x^2-2x-7)^2= 16 (2x-11)

x^4 +4x^2 +49-4x^3 + 28x - 14x^2 = 32x-176

x^4 - 4x^3 - 10x^2 - 4x + 225= 0

x(x^3 -4x^2 - 10x-4)+225=0

this is a power 4 th equation this equation can solve by normal steps.

The solutions to the absolute value inequalities 1 to 20 are shown below

|x + 8| < 16

So, we have

-16 < x + 8 < 16

-24 < x < 8

|r + 1| ≤ 2

So, we have

-2 ≤ r + 1 ≤ 2

-3 ≤ r ≤ 1

|2c - 1| ≤ 7

So, we have

-7 ≤ 2c - 1 ≤ 7

-6 ≤ 2c ≤ 8

-3 ≤ r ≤ 4

Using the above as a guide, the solution to the other inequalities are

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Answer: 7 marbles

Step-by-step explanation:

Lin starts with 27 and creates 3 equal groups.

When separating, you divide.

27 divided by 3 is 9.

Each group has 9 marbles.

This means that by giving one group to Jill, she receives 9 marbles.

Jill gives away 2 of her 9 marbles, thus leaving her with 7 marbles.

9-2=7.

Answer:

The probability of winning the lottery, denoted by P(A), can be calculated as follows:

P(A) = number of winning outcomes / total number of possible outcomes

Since there is only one winning four-digit number and there are 2,352 possible four-digit numbers, we have:

P(A) = 1/2352

Using a calculator, we get:

P(A) ≈ 0.00042

Therefore, the value of P(A) is approximately 0.00042 or 4.2 × 10^(-4) (rounded to five decimal places).

Step-by-step explanation:

Answer:

-10

Step-by-step explanation:

[tex]f(x) = 3x + 2[/tex]

Replace x in the function with the given value x = -4:

[tex]f( - 4) = 3 \times ( - 4) + 2 = - 12 + 2 = - 10[/tex]

Answer:

To write 9/11 in the form √a, we need to simplify the fraction so that it has a square root in the numerator or the denominator. We can do this by multiplying the numerator and denominator by √11:

9/11 × (√11/√11) = (9√11)/(11√11) = (√(9 × 11))/(√(11 × 11)) = √99

Therefore, 9/11 can be written in the form √99, where a = 99.

Option D : Based on the given data, the best prediction for the number of students who prefer ice cream at the college, which is approximately 1,440 students.

To make prediction of the number of students who prefer ice cream, we need to use the proportion of students who prefer ice cream from the sample to the entire population of 4,000 students.

Based on the given data, 81 out of 225 students prefer ice cream. To estimate the number of students who prefer ice cream in the entire college, we can use the ratio of students who prefer ice cream in the sample to the total number of students in the college.

The total number of students in the college is 4,000, so we can set up a proportion:

81/225 = x/4000

Solving for x, we get:

x = (81/225) * 4000 = 1,440

Therefore, the best prediction for the number of students who prefer ice cream in the entire college is 1,440.

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A college cafeteria is looking for a new dessert to offer its 4,000 students. The table shows the preference of 225 students.

Ice Cream Candy Cake Pie Cookies

81 9 72 36 27

Which statement is the best prediction about the scoops of ice cream the college will need?

A. The college will have about 480 students who prefer ice cream.

B. The college will have about 640 students who prefer ice cream.

C. The college will have about 1,280 students who prefer ice cream.

D. The college will have about 1,440 students who prefer ice cream.

The measure of the angle A as required to be determined in the task content is; 16.7°.

It follows from the task content that the measure of the angle A is to be determined.

By using the trigonometric ratio;

tan (A) = BC / AC

Hence, by substitution; it follows that;

tan (A) = 6 / 20

Hence, by simplification; we have that;

tan A = 3/10

tan A = 0.3

Therefore, to determine A;

A = tan-¹ (0.3)

A = 16.7°.

Ultimately the measure of angle A as required to be determined is; 16.7°.

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Let's use variables to represent the width and length of the rectangle:

Let w be the width of the rectangle (in cm)

Then, the length of the rectangle can be expressed as 5w - 18 (since it is 18 cm less than five times its width).

The area of the rectangle can be calculated as:

A = w * (5w - 18)

We are given that the area of the rectangle is 35 square cm. Substituting this value into the equation above, we get:

35 = w * (5w - 18)

Expanding the right-hand side, we get:

35 = 5w^2 - 18w

Rearranging the terms and setting the equation equal to zero, we get:

5w^2 - 18w - 35 = 0

To solve for w, we can use the quadratic formula:

w = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 5, b = -18, and c = -35.

Plugging in the values, we get:

w = (-(-18) ± sqrt((-18)^2 - 4(5)(-35))) / 2(5)

w = (18 ± sqrt(784)) / 10

w = (18 ± 28) / 10

So w = 2.6 or w = -1.4

Since the width cannot be negative, we reject the solution w = -1.4.

Therefore, the width of the rectangle is approximately 2.6 cm.

To find the length, we can use the expression we derived earlier:

length = 5w - 18

length = 5(2.6) - 18

length = 13 - 18

length = -5

Since the length cannot be negative, we know that there is an error in our calculation. We made an assumption that the width is less than the length, which is not necessarily true. We should check our work and try again using the positive root instead of the negative one.

Using the positive root for the width, we get:

w = (18 + 28) / 10

So w = 4.6

Therefore, the width of the rectangle is 4.6 cm.

To find the length, we can use the expression we derived earlier:

length = 5w - 18

length = 5(4.6) - 18

length = 23 - 18

length = 5

So the length of the rectangle is 5 cm.

Therefore, the dimensions of the rectangle are width = 4.6 cm and length = 5 cm.

Answer:

Step-by-step explanation:

A bandana is in the shape of a triangle. The base of the bandana is 30 in. wide and the height is 12 in.

Triangle Area = 1/2 b x h

1/2 30 x 12 =

15 x 12 = 180 in^2

The wοrk dοne in pushing the car up the incline is 39,672 fοοt-pοunds.

The dοt prοduct is a mathematical οperatiοn that takes twο vectοrs and prοduces a scalar. It is the prοduct οf the magnitudes οf the twο vectοrs and the cοsine οf the angle between them.

Tο calculate the wοrk dοne, we need tο find the fοrce applied tο the car and the distance it was mοved.

First, let's calculate the fοrce required tο push the car up the incline. We can use the weight οf the car and the angle οf the incline tο find the fοrce required.

The weight οf the car is given as 1850 pοunds. The fοrce required tο push the car up the incline is equal tο the cοmpοnent οf the weight that acts parallel tο the incline. This is given by:

fοrce = weight * sin(angle)

where the angle is 7 degrees.

Plugging in the values, we get:

fοrce = 1850 * sin(7) = 220.4 pοunds

Nοw, we need tο find the distance that the car was pushed up the incline. This is given as 180 feet.

Using the dοt prοduct, the wοrk dοne can be calculated as:

wοrk = fοrce * distance * cοs(angle)

where the angle is the angle between the fοrce and the displacement vectοrs. In this case, the fοrce and displacement are in the same directiοn, sο the angle is 0 degrees and cοs(0) = 1.

Plugging in the values, we get:

wοrk = 220.4 * 180 * cοs(0) = 39,672 fοοt-pοunds

Therefοre, the wοrk dοne in pushing the car up the incline is 39,672 fοοt-pοunds.

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Answer:

$200.00

Step-by-step explanation:

We are asking what is 75% of what number is 150.  If we take off 1/4 or 25% we are leaving on 3/4 or 75%

.75 x = 150  Divide both sides by .75

[tex]\frac{.75x}{.75}[/tex] = [tex]\frac{150}{.75}[/tex]

x = 200

Check:

200 x .25 = 50  To find the discount.

200 - 50 = 150   Subtract the discount from the original price to find the sale's price.

Helping in the name of Jesus.

Answer:

$0.14

Step-by-step explanation:

[tex]8,308 - 7,998 = 310 \: miles[/tex]

We don't have information on how many gallons of gas Ms. Ward purchased, so we can't calculate the exact cost per mile. However, we can use the average fuel efficiency of her car to estimate the cost. Let's assume her car gets an average of 25 miles per gallon, which is a common fuel efficiency for a compact car.

To calculate how many gallons of gas she used, we can divide the number of miles driven by the fuel efficiency:

[tex]\frac{310 \: miles}{25 \: miles _{gallon} } = 12.4 \: gallons[/tex]

Assuming that the cost of gas was $3.50 per gallon, we can multiply the number of gallons used by the cost per gallon to find the total cost of gas:

[tex]12.4 \: gallons \times $3.50_{gallon} = $43.40[/tex]

To find the cost per mile, we can divide the total cost of gas by the number of miles driven:

[tex]\frac{$43.40}{310 \: miles} = $0.14_{mile} [/tex]

So Ms. Ward's cost per mile for this time period was approximately $0.14.

Answer:

Step-by-step explanation:

Miles travelled = 8,308 - 7,998 = 310 miles

Total cost = $26.79

Cost per mile[tex]=\frac{26.79}{310}[/tex] =0.0864

Ms. Ward's cost per mile is $0.09 (rounded to nearest cent).

A written history ​means the writing of history, such as the writing of history based on the critical examination of sources which is also known as historiography.

Also known as recorded history describes the historical events that have been recorded in a written form or other documented communication which are subsequently evaluated by historians using the historical method.

Historiography also includes the theory and history of historical writing. Modern historians strive to reconstruct a record of human activities and gain a deeper understanding of them.

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Width of the table = 3.5 feet , Length of the table = 5.5 feet and Perimeter of the table = 18 feet

What is Rectangle ?

A rectangle is a 2-dimensional shape with four straight sides, where opposite sides are parallel and equal in length. It has four right angles and its diagonals bisect each other. It is a type of quadrilateral, and its area is calculated by multiplying the length and width of the rectangle. It is commonly found in many geometric shapes and everyday objects such as doors, windows, and computer screens.

Let's convert all measurements to the same unit for easier calculation. Since the table width is given in inches, we can convert it to feet by dividing by 12 (since 1 foot = 12 inches).

Table width: 42 inches = 42÷12 = 3.5 feet

Let's denote the length of the table as 'l' feet. The perimeter of a rectangle is given by the formula:

Perimeter = 2 * (Length + Width)

Given that the perimeter is 18 feet, we can substitute the values into the formula:

18 = 2 * (l + 3.5)

Now, we can solve for 'l':

18 = 2l + 7 (distributing 2 to both terms in the parentheses)

2l = 18 - 7 (subtracting 7 from both sides)

2l = 11 (simplifying)

l = 11÷2 (dividing both sides by 2)

Therefore, Width of the table = 3.5 feet , Length of the table = 5.5 feet

and Perimeter of the table = 18 feet

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The length οf the diagοnal is 20 feet.

A rectangle is a type οf quadrilateral with parallel sides that are equal tο οne anοther and fοur vertices that are all 90 degrees apart. It is alsο knοwn as an equiangular quadrilateral fοr this reasοn. The term "parallelοgram" can alsο be used tο describe a rectangle because the οppοsing sides are equal and parallel.

Given that 16 feet and 12 feet make up a rectangle.

The diagοnal and twο cοnsecutive sides οf a rectangle make a right-angled triangle.

Tο find the diagοnal apply Pythagοrean theοrem.

The widely accepted geοmetric principle knοwn as the Pythagοrean Theοrem states that the square οn the hypοtenuse οf a right triangle equals the sum οf the squares οn its legs.

Draw a rectangle:

Cοnsider △ABC:

AB² + BC² = AC²

12² + 16² =AC²

AC² = 144+ 256

AC² = 400

Take square rοοt οn bοth sides:

AC = 20

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Using the pattern given, we found that the colour of the lightbulb in the 543rd position of a string of lightbulbs is green.

What is meant by the pattern?

A recurring arrangement of numbers, shapes, colours and other elements is known as a pattern. The Pattern can be connected to any kind of occasion or thing. When a group of numbers are arranged in a particular way, the arrangement is referred to as a pattern. Patterns can also occasionally be referred to as a series. The number of patterns can be limitless or finite. There are many distinct kinds of number patterns, including geometric, Fibonacci, and algebraic or arithmetic patterns. In mathematics, number patterns are quite prevalent.

Given,

The number of lights on a string of lights = 1000

The order in which the lights repeat is red, yellow, green and blue.

After every four lights, the order repeats.

We can get the pattern as follows.

The blue lights are in positions 4,8,12,........,4n.

We have to find the multiple of 4 close to 543.

Now we can check if 542 is a multiple of 4.

542/4 = 135.5

So it is not a multiple of 4.

Now check if 544 is a multiple of 4.

544/4 = 136

So the bulb in the 544th position is blue.

Then the bulb in the 543rd position should be green.

Therefore using the pattern given, we found that the colour of the lightbulb in the 543rd position of a string of lightbulbs is green.

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Answer:

60 inches

Step-by-step explanation:

5 ft x 12 in/ft = 60 in

The surface area of the box shown is 186 square inches (186 in²).

Surface area is the measure of the area on a two-dimensional surface. It is the sum of the areas of all the shapes that make up a two-dimensional object. It can be used to calculate the area of a sphere, a cylinder, a rectangular prism, and more.

The box shown is a three-dimensional rectangular prism. The surface area (S.A.) of a rectangular prism is calculated by adding the area of each of its six faces. The area of each face is calculated by multiplying the length of the face by its width.

For the box shown, the length is 3 inches (3 in), the width is 12 inches (12 in), and the height is 10 inches (10 in). To find the surface area, we need to calculate the area of each face and add them together. The surface area of this box is calculated as follows:

Front and back faces: 3 in x 10 in = 30 in²

Left and right faces: 12 in x 10 in = 120 in²

Top and bottom faces: 3 in x 12 in = 36 in²

Total surface area: 30 in² + 120 in² + 36 in² = 186 in²

Therefore, the surface area of the box shown is 186 square inches (186 in²).

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Therefore, the solutions for θ in the interval 0° ≤ θ ≤ 360° are approximately 131.81° and 228.19°.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. It is used to study and analyze various phenomena that involve periodic functions, such as waves, oscillations, and sound. Trigonometry also has practical applications in fields such as physics, engineering, navigation, and surveying. It involves the use of trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent to calculate the sides and angles of triangles and other geometric figures.

Here,

Let's solve for θ:

First, let's substitute u = cos(θ), so we have:

3u² - 5u - 4 = 0

Now we can use the quadratic formula:

u = [ -(-5) ± √((-5)² - 4(3)(-4))] / (2*3)

u = [ 5 ± √(49) ] / 6

u1 = (5 + 7) / 6 = 2

u2 = (5 - 7) / 6 = -2/3

Since the cosine function has a range of -1 ≤ cos(θ) ≤ 1, we can discard the solution u1 = 2.

Now we can solve for θ:

cos(θ) = u2 = -2/3

θ = cos⁻¹(-2/3) ≈ 131.81°  

θ = 360° - cos⁻¹(-2/3) ≈ 228.19°

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Answer: -1280

Step-by-step explanation:

in a geometric sequence it either multiplies or divides negatively or positively, in this case -5 to -20 is multiplied 4 positively and so is -20 to -80

then -80x4= -320

-320x4= -1280

Answer:

Rounding to the nearest whole degree, the rescue boat should travel at an angle of approximately 41 degrees off of due east to take the shortest route to the ship.

Step-by-step explanation:

We can use trigonometry to solve this problem. Let's draw a diagram to represent the situation:

            A (rescue boat)

              | \

              |   \

              |     \ C (disabled ship)

   24 mi |       \

              |        \

              |          \

              |θ          \

              B-----------D (Port Lincoln)

               48 mi

In the diagram, point B represents Port Lincoln, point C represents the disabled ship, and point A represents the rescue boat. Point D is the intersection of the eastward path and the northward path taken by the ship.

We are given that BD = 24 miles and CD = 48 miles. We want to find the angle θ, which is the angle between the line segments AB and AD.

To find θ, we can use the law of cosines:

cos(θ) = (BD² + CD² - AD²) / (2 x BD x CD)

Substituting the given values, we get:

cos(θ) = (24² + 48² - AD²) / (2 x 24 x 48)

Simplifying, we get:

cos(θ) = 0.75

To solve for θ, we can take the inverse cosine of both sides:

θ = cos⁻¹(0.75)

Using a calculator, we get:

θ ≈ 41.41°

Rounding to the nearest whole degree, the rescue boat should travel at an angle of approximately 41 degrees off of due east to take the shortest route to the ship.