Forces with magnitudes of 85 pounds and 50 pounds act on a single point at angles of 45° and 60°, respectively, with the positive x-axis. Find the direction and magnitude of the resultant of these
forces.

The number of ways to choose a president, vice president and secretary from a set of 24 individuals is given as follows:

12,144 ways.

The number of possible permutations of x elements from a set of n elements is given by:

[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]

The permutation formula is used when the order in which the elements are chosen is important, which is the case for this problem. The order is important as there are different roles, that is, president, vice president and secretary.

For this problem, 3 people are chosen from a set of 24, hence the number of ways is given as follows:

P(24,3) = 24!/21! = 12144.

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The length of arc length s is 1152π² or 11358.26.

We have,

The arc length of a circle can be calculated using the formula:

Arc Length = 2πrθ/360

Where:

π is a mathematical constant approximately equal to 3.14159.

r is the radius of the circle.

θ is the central angle subtended by the arc, measured in degrees.

The arc length of a circle can be written as:

s = angles/360 x 2πr _______(1)

Now,

r = 4 cm

Angle = (2/5)π

Now,

Substitute in (1).
s = angles/360 x 2πr

s = (2/5)π/360 x 2π x 4

s = 2π x 72 x 2π x 4

s = 4π² x 288

s = 1152π²

or

s = 11358.26

Thus,

The length of arc length s is 1152π² or 11358.26.

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Answer:The correct answer is C. 3 liters.

Step-by-step explanation:

To determine the total amount of liquid in the pot, we need to add the volume of the broth and the volume of the water.

The broth has a volume of 500 milliliters, and the water has a volume of 1,500 milliliters. Adding these together, we get:

500 mL + 1,500 mL = 2,000 mL

Since there are 1,000 milliliters in a liter, we can convert the volume to liters:

2,000 mL ÷ 1,000 = 2 liters

Therefore, the total amount of liquid in the pot is 2 liters.

(a) The distribution of the proportion of people in t high school diploma = 0.0158.

(b)  the probability that at least 88% of the sample of 1600 young Americans will have earned their high school diploma is extremely small.

Given that,

(a) Based on the central limit theorem, the normal distribution may be used to approximate the fraction of persons in the sample who have a high school diploma.

The mean proportion of individuals in the population who have earned their high school diploma can be estimated as

⇒ 879/1600 = 0.5494.

The standard deviation can be estimated as the square root of (0.5494*(1-0.5494)/1600)

=0.0158

b) To find the probability that at least 88% of the sample of 1600 young Americans will have earned their high school diploma,

We need to use the normal distribution with a mean of 0.5494 and a standard deviation of 0.0158.

We can standardize the value of 88% to the corresponding z-score:

z = (0.88 - 0.5494) / 0.0158

  = 20.99

Using a standard normal distribution table or calculator, we find that the probability of a z-score this large or larger is essentially zero,

So the probability that at least 88% of the sample will have earned their high school diploma is extremely small.

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To find the unit rate, we need to determine how much distance Margo covers in one unit of time. We can do this by dividing the distance by the time.

Distance = 1/4 mile

Time = 1/12 hour

Unit rate = Distance ÷ Time

Unit rate = (1/4 mile) ÷ (1/12 hour)

We can simplify this division by multiplying both the numerator and denominator by the least common multiple of 4 and 12, which is 12.

Unit rate = (1/4 mile) ÷ (1/12 hour) x (12/12)

Unit rate = (3/4 mile) ÷ 1 hour

Unit rate = 3/4 mile per hour

Therefore, Margo's unit rate is 3/4 mile per hour. This means that she can cover a distance of 3/4 mile in one hour of walking.

Answer:

3mph

Step-by-step explanation:

1/12 of an hour will be 5 min. In 5 min she can walk 1/4 mile then in one hour she can walk 1/4 x 12. This means her rate will be 3 miles per hour.

60/12 = 5

12 x 1/4 = 12/4 = 3

A. The expressions that are squared of linear expressions are: c) (1/2x + 4)²; f) x² + 36.        B. The equivalent expressions are: a) (x+3)² = 16; f) x² + 6x + 9 = 16.

Part A: For the expressions that are squared of linear expressions:

a) 9x² - 36: This expression is not a squared linear expression because it contains a constant term (-36).

b) p² - 6p + q: This expression is not a squared linear expression because it contains a quadratic term (-6p) and a constant term (q).

c) (1/2x + 4)²: This expression is a squared linear expression because it represents the square of a linear expression, (1/2x + 4).

d) (2d + 8)(2d-8): This expression is not a squared linear expression because it represents the product of two linear expressions, (2d + 8) and (2d - 8).

e) x² + bx + 36: This expression is not a squared linear expression because it contains a quadratic term (x²) and a constant term (36).

f) x² + 36: This expression is a squared linear expression because it represents the square of the linear expression (x) and a constant term (36).

Part B: For the equations that are equivalent to x² + 6x = 16:

a) (x+3)² = 16: This equation is equivalent because it represents the square of the linear expression (x+3) equal to 16.

b) (x + 3)² = 0: This equation is not equivalent because it represents the square of the linear expression (x+3) equal to zero, not 16.

c) x² + 6x + 9 = 0: This equation is not equivalent because it represents a quadratic equation with a constant term (9), not x² + 6x = 16.

d) (x+3²) = 15: This equation is not equivalent because it represents the square of the linear expression (x+3²) equal to 15, not 16.

e) x² + bx + 9 = 25: This equation is not equivalent because it represents a quadratic equation with a constant term (9) and a different right-hand side (25), not x² + 6x = 16.

f) x² + 6x + 9 = 16: This equation is equivalent because it represents the square of the linear expression (x+3) equal to 16.

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The correct option is A, the relative frequency is 0.03

To find the relative frequency for a given outcome, we need to take the quotient between the number of times that we got that outcome, and the total number of times that the experiment is done.

Here the outcome  is "wants fruit smooties"

And for the total "number of times that the experiment is done" we need to count the number of students

We know:

Total number = 500

Number of 8th grades who want a fruit smoothies = 17

Relative frquency = 17/500 = 0.03

The correct option is the first one.

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Answer:

The simplified expression for g(a+5) - g(5) is 2a^2 + 20a.

Step-by-step explanation:

To evaluate and simplify the expression g(a+5) - g(5), we need to substitute the function g(t) = 2t^2 into the given expression.

Let's start by evaluating g(a+5):

g(a+5) = 2(a+5)^2

Expanding the expression:

g(a+5) = 2(a^2 + 10a + 25)

g(a+5) = 2a^2 + 20a + 50

Next, let's evaluate g(5):

g(5) = 2(5)^2

g(5) = 2(25)

g(5) = 50

Now we can substitute these values back into the expression g(a+5) - g(5):

g(a+5) - g(5) = (2a^2 + 20a + 50) - 50

Simplifying:

g(a+5) - g(5) = 2a^2 + 20a

Therefore, the simplified expression for g(a+5) - g(5) is 2a^2 + 20a.

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The expression that is equivalent to [tex]\\\\\frac{60m^-2n^6\\}{5m^-4 n^-2}[/tex] for all values of m and n where the 5m-4n-2 expression is defined is [tex]12m^{2} n^{8}[/tex]

In mathematics, an expression or mathematical expression  can be described as the finite combination of symbols which is been analyzed and  well-formed according by following some set of rules which could be varies base on the kind of the symbol as ll as the operation  that are involved in the expression and it s been done  depending on the context so that another expression can be gotten.

This is given as   [tex]\\\\\frac{60m^-2n^6\\}{5m^-4 n^-2}[/tex]

Then we have it defined by;  [tex]12m^{2} n^{8}[/tex]

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The domain value that corresponds to a range value of 4 is,

⇒ x = 1/2

Given that;

Function is,

y = 4x + 2

Since, the equation equal to the range value:

4 = 4x + 2

Then, we can solve for "x":

4 - 2 = 4x

2 = 4x

x = 1/2

Now that we have the value of "x", we can find the corresponding value of "y" by substituting it into the given equation:

y = 4x + 2

y = 4(1/2) + 2

y = 4 + 2

y = 6

Therefore, the domain value that corresponds to a range value of 4 is,

⇒  x = 1/2

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The maximum volume of the box is 864 cubic inches.

Let side length of the square cut from each corner is x inches.

After cutting the squares from each corner, the remaining dimensions of the sheet metal will be:

Length: 24 - 2x inches

Width: 24 - 2x inches

Height: x inches

So, the volume of the box

V = (24 - 2x) (24 - 2x)  x

V = x(24 - 2x)²

To find the maximum volume, we can take the derivative of V with respect to x, set it to zero, and solve for x:

dV/dx = (24 - 2x)² - 2x(24 - 2x) = 0

4x² - 96x + 576 = 0

Solving the quadratic equation,

x = 6 and x = 12.

We can substitute x = 6 into the volume equation to find the maximum volume:

V = 6(24 - 2(6))²

V = 6(12)²

V = 6 x 144

V = 864 cubic inches

Therefore, the maximum volume of the box is 864 cubic inches.

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Answer:

Step-by-step explanation:

Answer:   B y = ±[tex]\sqrt{x+36}[/tex]

Step-by-step explanation:

To find the inverse of any equation.  Switch the x and the y then solve for y.

y = x² - 36                      >switch variables

x = y² - 36                      >add 36 to both sides

x+36=y²                         >take square root of both sides

y = ±[tex]\sqrt{x+36}[/tex]  

B

Answer:

the answer is 54,000

Step-by-step explanation:

(540 × 45) + (540 × 55)

=24300 + 27700

=54000

Equation [1] : a(n) = 3 + (n - 1)5 represents the step where he made the mistake.

We have,

A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

We have a student who used the explicit formula a[n] = 5+3(n-1) for the sequence 3,8,13,18,23,...to find the 12th term.

The given sequence is -

3 , 8 , 13 , 18 , 23, ..

Here -

d = 8 - 3 = 13 - 8 = 5

a = 3

a(n) = a + (n - 1)d      {for arithmetic sequence}

a(n) = 3 + (n - 1)5     ...Eq[1]

a(n) = 3 + 5n - 5

a(n) = 5n - 2

Therefore, Equation [1] : a(n) = 3 + (n - 1)5 represents the step where he made the mistake.

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complete question:

A student uses the explicit formula an= 5+3(n-1) for the sequence 3,8,13,18,23,...to find the 12th term. Explain the error the student made.

If he wants an average of 84, he needs to get at least 93 points.

Remember that the average value between 3 values A, B, and C is:

(A + B + C)/3

Here we know that the first two scores are 76 and 83 points, let's say that the third score is x, if we want to have an average of 84 or more, then we need to solve:

(76 + 83 + x)/3 = 84

159 + x = 252

x = 252 - 159

x = 93

So he needs to get at least 93 points in the next exam.

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Circle B is dilation of 6 unit left and 4 unit down to the center of circle A.

We have to given that;

Circle A is transformed into Circle B using a sequence of two transformations.

And, The first transformation is a dilation centered at (6,0).

Here, Center of circle A is,

⇒ A = (6, 0)

And, Center for circle B is,

⇒ B = (0, - 4)

Hence, We get;

⇒ B = (6 - 6, 0 - 4)

⇒ B = (0, - 4)

Hence, Circle B is dilation of 6 unit left and 4 unit down to the center of circle A.

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a) The two expressions do not represent the same value. b) the surface area of a rectangular prism consists of the sum of the areas of all its faces. By doubling the areas of each face

a. To determine if Sofia and Ella's expressions represent the same value for the surface area of a rectangular prism, we can simplify their expressions and compare them.

Sofia's expression: (3 cm x 4 cm) + (3 cm x 5 cm) + (4 cm x 5 cm) + (4 cm x 5 cm)

            = 12 cm² + 15 cm² + 20 cm² + 20 cm²

            = 67 cm²

Ella's expression: 2(3 cm x 4 cm) + 2(3 cm x 5 cm) + 2(4 cm x 5 cm)

          = 2(12 cm²) + 2(15 cm²) + 2(20 cm²)

          = 24 cm² + 30 cm² + 40 cm²

          = 94 cm²

The two expressions do not represent the same value. Sofia's expression calculates the surface area by adding the areas of each face once, while Ella's expression calculates the surface area by doubling the areas of each face and then summing them up.

b. Ella's expression, 2(3 cm x 4 cm) + 2(3 cm x 5 cm) + 2(4 cm x 5 cm), shows more clearly the fact that the surface area of a rectangular prism consists of the sum of the areas of all its faces. By doubling the areas of each face

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Answer:

Perimeter of a rectangle

Step-by-step explanation:

You have to combine all of the sides which results in 2 lengths and 2 widths being combined (2l+2w)

The required interquartile range of the given data set is 7.

To find the interquartile range (IQR), we first need to order the data set from least to greatest:

4, 6, 6, 11, 12, 13, 13, 13, 14

Next, we calculate the first quartile (Q1) and the third quartile (Q3).

Q1 is the median of the lower half of the data set. Since there are 9 numbers, the lower half consists of the first four numbers: 4, 6, 6, 11. The median of these numbers is the average of the middle two, which is (6 + 6) / 2 = 6.

Q3 is the median of the upper half of the data set. The upper half consists of the last four numbers: 12, 13, 13, 14. The median of these numbers is the average of the middle two, which is (13 + 13) / 2 = 13.

Now that we have Q1 = 6 and Q3 = 13, we can calculate the interquartile range (IQR) as the difference between Q3 and Q1:

IQR = Q3 - Q1 = 13 - 6 = 7

Therefore, the interquartile range of the given data set is 7.

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Answer:

The expected value for this game is -$0.75, indicating that, on average, players would expect to lose $0.75 per game.

Step-by-step explanation:

Expected Value = (Probability of Winning * Payout for Win) - Cost of Playing

In this case:

Probability of Winning = 0.05

Payout for Win = $5

Cost of Playing = $1

Expected Value = (0.05 * $5) - $1

Expected Value = $0.25 - $1

Expected Value = -$0.75

Answer:

k = - 4

Step-by-step explanation:

given a quadratic equation in standard form

ax² + bx + c = 0 ( a ≠ 0 ) , then the product of the roots is [tex]\frac{c}{a}[/tex]

3x² + 5x + 3k = 0 ← is in standard form

with a = 3 and c = 3k , then

[tex]\frac{c}{a}[/tex] = - 4 , that is

[tex]\frac{3k}{3}[/tex] = - 4 ( multiply both sides by 3 to clear the fraction )

3k = - 12 ( divide both sides by 3 )

k = - 4

The value of "a" is approximately -3.784 when the function f(x) is divided by (x - a) and leaves a remainder of -5.

To find the value of "a" when the function f(x) = x^3 + 16x^2 + 60x + 40 is divided by (x - a) and leaves a remainder of -5, we can use the Remainder Theorem.

According to the Remainder Theorem, if a polynomial f(x) is divided by (x - a), the remainder is equal to f(a).

In this case, the remainder is -5, so we have f(a) = -5.

Substituting a into the function, we get:

f(a) = a^3 + 16a^2 + 60a + 40 = -5

Now, we need to solve this equation to find the value of "a."

a^3 + 16a^2 + 60a + 40 = -5

Rearranging the equation:

a^3 + 16a^2 + 60a + 45 = 0

To find the exact value of "a," we can use numerical methods such as factoring, synthetic division, or using a graphing calculator. Unfortunately, the solution to this equation is not straightforward and requires numerical approximations.

Using numerical methods or a graphing calculator, we find that the value of "a" is approximately -3.784.

Therefore, the value of "a" is approximately -3.784 when the function f(x) is divided by (x - a) and leaves a remainder of -5.

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The binomials represents the length of each of it sides = a + 5.

The given expression is;

a² + 10a + 25

It represents the area of square.

Since area of square = (side)²

Therefore,

(side)² = a² + 10a + 25

           =  a² + 5a + 5a + 25

           =  a(a+5) + 5(a+5)

           =   (a+5)(a+5)

           =   (a+5)²

⇒(side)² = (a+5)²

Taking square root both sides

Hence

side of square = a + 5

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The sample data suggest that the population proportion is not significantly greater than 0.59 at the significant level of 0.01.

In this case, the test statistic is calculated as follows:

z = (106/150 - 0.59) / sqrt(0.59(1-0.59)/150)

= 1.55

The p-value is calculated as the area under the standard normal curve to the right of the test statistic. In this case, the p-value is calculated as follows:

p-value = p(Z > 1.55)

= 0.123

Since the p-value is greater than the significance level of 0.01, we fail to reject the null hypothesis. Therefore, we cannot conclude that the population proportion is significantly greater than 0.59 at the significant level of 0.01.

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We can solve for x by equating the two ratios:

a/b = 6/8 = 3/4

We can conclude that x is equal to 3/4.

To find the value of x in the given scenario, where triangles AABC and ADEF are similar, we can use the concept of corresponding sides in similar triangles.

From the given information, we know that the lengths of sides AB and DE are in proportion with each other, as the triangles are similar. Let's denote the length of AB as a and the length of DE as b. Similarly, let's denote the length of BC as c and the length of EF as d.

Since the corresponding sides are in proportion, we can set up the following equation:

AB/DE = BC/EF

Substituting the given values, we have:

a/b = 6/8

To find the value of x, we need to determine the ratio of the corresponding side lengths. Dividing both sides of the equation by 6, we get:

a/6 = b/8

Cross-multiplying, we have:

8a = 6b

Now, we can solve for x by equating the two ratios:

a/b = 6/8 = 3/4

We can conclude that x is equal to 3/4.

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The function that has a greater output value for x = 10 is table B

From the question, we have the following parameters that can be used in our computation:

The table of values

The table A is a linear function with

A(x) = 1 + 0.3x

The table B is an exponential function with the equation

B(x) = 1.3ˣ

When x = 10, we have

A(10) = 1 + 0.3 * 10 = 4

B(10) = 1.3¹⁰ = 13.79

13.79 is greater than 4

Hence, the function that has a greater output value for x = 10 is table B

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The area of the sector in the circle shown above is given as follows:

32.3 in².

The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:

A = πr²

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle, hence it is given as follows:

r = 10 in.

Then the area of the entire circle is given as follows:

A = π x 10²

A = 314 in².

The entire circumference of the circle is of 360º, while the angle is of 37º, hence the area of the sector is given as follows:

37/360 x 314 = 32.3 in².

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Answer: the woodworker should choose the dimensions of the box to be approximately 3.825 inches by 3.825 inches by 1.603 inches, with a total surface area of approximately 33.512 square inches.

Step-by-step explanation:

Let the side length of the square base be x, and the height of the prism be h. Then the volume of the prism is:

V = x^2h

We're given that V = 61.3 cubic inches, so:

x^2h = 61.3

We want to minimize the surface area of the box, which consists of the area of the two square bases (2x^2) plus the area of the four rectangular faces (4xh). So the total surface area is:

A = 2x^2 + 4xh

We can solve the first equation for h:

h = 61.3/x^2

Substituting this into the equation for A, we get:

A = 2x^2 + 4x(61.3/x^2)

A = 2x^2 + 245.2/x

To minimize A, we take the derivative and set it equal to zero:

dA/dx = 4x - 245.2/x^2 = 0

4x = 245.2/x^2

x^3 = 61.3

x = (61.3)^(1/3)

x ≈ 3.825

So the length of each side of the square base should be approximately 3.825 inches. We can use the equation for h to find the height:

h = 61.3/x^2

h ≈ 1.603

So the height of the prism should be approximately 1.603 inches.

Therefore, the woodworker should choose the dimensions of the box to be approximately 3.825 inches by 3.825 inches by 1.603 inches, with a total surface area of approximately 33.512 square inches.