2. Graph x² + (y-2)² = 36 ​

If the third term is 31, then let's get the second term. We have to use the rule we were given and work backwards. So, we will add three and then divide by 2.

31 + 3 = 34

34 / 2 = 17

17 is the second term. Let's do the same thing we just did to find the first term: add three, divide by 2.

17 + 3 = 20

20 / 2 = 10

Answer: the first term is 10

Hope this helps!

The specified scalar or vector 5u(3v - 4w) is equal to -420i + 250j.

We can begin by distributing the scalar 5 to the vector 3v - 4w:

5u(3v - 4w) = 5u(3v) - 5u(4w)

Next, we can find the scalar multiples:

5u(3v) = 5(-4i + 2j)(3(3i - j)) = 5(-36i + 10j)

5u(4w) = 5(-4i + 2j)(4(-6j)) = 5(48i - 40j)

Now we can substitute these values back into the original expression:

5u(3v - 4w) = 5(-36i + 10j) - 5(48i - 40j)

Simplifying:

5u(3v - 4w) = -420i + 250j

Therefore, 5u(3v - 4w) = -420i + 250j.

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y^2 = 90

y = √90 = 3√10

that's it

Part A) The vertex of the graph of the function, (t, h(t)) is (2, 68).

Part B) The t-coordinate of the vertex represent is the time it takes for the ball to reach its maximum height (option b)

Part C) The h(t)-coordinate of the vertex represent is the ball's maximum height (option a).

Part A asks for the vertex of the graph of the function, which is the point where the function reaches its maximum or minimum value. To find the vertex of a quadratic function like

=> h(t) = 4 + 64t – 16t²,

we can use the formula

=>  t = -b/2a,

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c and t is the input variable (in this case, the time).

The t-coordinate of the vertex is simply the value we get when we plug this formula into our equation.

So, for

=> h(t) = 4 + 64t – 16t²,

we have a = -16, b = 64, and c = 4.

Plugging these values into the formula

=> t = -b/2a,

we get

=> t = -64/(2*(-16)) = 2.

The t-coordinate of the vertex is therefore 2.

To find the h(t)-coordinate of the vertex, we can simply plug t = 2 into the function h(t) = 4 + 64t – 16t² and evaluate it.

This gives us

=> h(2) = 4 + 64(2) – 16(2²) = 68.

Therefore, the vertex of the graph of h(t) is (2, 68).

Part B asks what the t-coordinate of the vertex represents. We know that the t-coordinate is the time at which the ball reaches its maximum height.

Therefore, the correct answer is B: the time it takes for the ball to reach its maximum height.

Part C asks what the h(t)-coordinate of the vertex represents. We just found that the h(t)-coordinate of the vertex is the maximum height of the ball.

Therefore, the correct answer is A: the ball's maximum height.

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(1) The result of the row operation R₂ ↔ R₃ is [ -5   1    0    8]

                                                                            [  8  8    -7   5 ]

                                                                            [ 2   2     6   -5]

(2) The result of the row operation  3R₁ ↔ R₁ is  [24  - 27   -21   - 18]

                                                                                [8       9      -4      -5]

                                                                                 [2       2       -7     -8]

The result of the row operation in the matrix is calculated as follows;

R₂ ↔ R₃, implies changing row 2 and row, and the result would be;

[ -5   1    0    8]

[  8  8    -7   5 ]

[ 2   2     6   -5]

The result of obtained from 3R₁;

3R₁ = [24  - 27   -21   - 18]

3R₁ ↔ R₁ is determined as; (the row interchange)

[24  - 27   -21   - 18]

[8       9      -4      -5]

[2       2       -7     -8]

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The volume of the prism is V = 4000 cm³

Given data ,

Let the volume of the prism be represented as V

Now , the value of V is

Let the height of the prism be h = 10 cm

Let the width of the prism be w = 20 cm

Let the length of the prism be l = 20 cm

So , the base area of prism = l x w

Base area = 400 cm²

Now , the volume of the prism is V = 400 x 10

V = 4000 cm³

Hence , the volume of prism is V = 4000 cm³

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a) The future value of the account after 30 years can be calculated using the formula:

FV = P * ((1 + r/n)^(n*t))

where P is the monthly deposit, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

In this case, P = $300, r = 0.02, n = 12 (monthly compounding), and t = 30. Plugging these values into the formula, we get:

FV = $300 * ((1 + 0.02/12)^(12*30)) = $150,505.60

So you will have $150,505.60 in the account after 30 years.

b) The total amount of money you will put into the account is simply the monthly deposit multiplied by the number of months in 30 years, which is 30*12 = 360 months. So the total amount of money you will put into the account is:

$300 * 360 = $108,000

c) The total interest earned can be calculated by subtracting the total amount deposited from the future value of the account. So the total interest earned is:

$150,505.60 - $108,000 = $42,505.60

Answer:

a) you will have approximately $133,381.85 in the account in 30 years.

b) a total of $108,000 into the account over 30 years.

c) a total of $25,381.85 in interest over 30 years.

Step-by-step explanation:

From the interior angle theorem:

The Interior Angle Theorem states that if two secants or chords intersect inside a circle, then the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs.

Considering the given circles:

m∠AED = ¹/₂(45 + 109)

m∠AED = 77°

m∠AEB = 180 - 77

m∠AEB = 103°

m∠LK = (2 * 62) - 74)

m∠LK = 50°

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The horizontal distance from the base of the skyscraper out to the ship will be 1140 feet.

Given that:-

The angle is = 45

The height of the skyscraper is 1140 feet.

The horizontal distance will be calculated by applying the angle property in the right angle triangle.

tan45  =  (  P  /  B  )

B  =  P  /  tan45

B  =  P                                 Since ( tan45 =1 )

B  =  1140 feet.

Therefore the horizontal distance from the base of the skyscraper out to the ship will be 1140 feet.

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From the observation deck of a skyscraper, Brandon measures a 45^{\circ} ∘ angle of depression to a ship in the harbor below. If the observation deck is 1140 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest tenth of a foot if necessary.

Carson can line up his textbooks on his bookshelf in 24 different ways

What is Permutation?

Permutations are a way to count the number of ways that a set of objects can be arranged in a particular order. A permutation is an ordered arrangement of objects.

What is Combination?

The combination is a way to count the number of ways that a set of objects can be selected without regard to order. A combination is an unordered selection of objects.

According to the given information:

Carson has four textbooks that he wants to line up on his bookshelf. The number of different ways that he can do this is given by the permutation formula:

n! / (n - r)!

where n is the total number of objects (in this case, 4 textbooks), and r is the number of objects that he wants to arrange in a particular order (in this case, all 4 textbooks).

On substituting the values in the formula,

4! / (4 - 4)! = 4! / 0! = 4 x 3 x 2 x 1 / 1 = 24

Therefore, Carson can line up his textbooks on his bookshelf in 24 different ways.

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The correct answer is: D) distance traveled since leaving Bay Town; x ≥ 0

The equation y = -x + 175 represents the distance left to travel to Oak Glen.

In this equation, x is the distance traveled since leaving Bay Town, and y is the distance left to reach Oak Glen.

The domain represents the possible values of x, which is the distance traveled since leaving Bay Town.

Since distance traveled cannot be negative, the domain is x ≥ 0. Therefore, the correct answer is:

D) distance traveled since leaving Bay Town; x ≥ 0

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The question in English is:

The distance between Bay Town and Oak Glen is 175 miles. If the equation y = -x +175 represents the distance to go to Oak Glen, what does the domain represent? What is the domain?

To find the slope in this problem, we can use the formula for calculating slope, which is change in distance divided by change in time. In this case, the change in distance is 350 miles - 200 miles = 150 miles, and the change in time is 4:00 pm - 1:00 pm = 3 hours.

So, the slope (rate of driving) is 150 miles / 3 hours = 50 miles per hour.

Therefore, Sue will drive an additional 50 miles by 5:00 pm, making her total distance driven by 5:00 pm 200 miles + 150 miles + 50 miles = 400 miles. So, Sue will drive 400 miles by 5:00 pm if she continues at the same rate. Note that in this problem, we are assuming that Sue maintains a constant speed throughout her drive. If her speed changes, the solution may be different.

Answer:

To determine the longest rod that can be carried to the loading dock, we want to find the shortest distance from point T to line segment CD. We can use the Pythagorean theorem for this.

First, we need to find the equation of the line containing segment CD. We can find the slope of the line CD as:

m = (y2 - y1) / (x2 - x1) = (36 - 48) / (48 - 0) = -12/48 = -1/4

where (x1, y1) = (0, 48) and (x2, y2) = (48, 36).

Using point-slope form, we get the equation of the line CD as:

y - 48 = (-1/4)(x - 0)

y = (-1/4)x + 48

Now, we can find the perpendicular distance from point T to the line CD as follows:

d = |(-1/4)(30) + 72 - 48| / sqrt((-1/4)^2 + 1)

d = 42 / sqrt(17) ≈ 10.21

Therefore, the longest rod that can be carried to the loading dock is approximately 10.2 inches long (rounded to the nearest tenth of an inch).

Answer:90% probability

Step-by-step explanation:

To calculate the experimental probability of the cup landing on its side, we can use the formula:

experimental probability = number of times the cup landed on its side / total number of trials

For the first set of trials, the experimental probability is:

experimental probability = 19/20 = 0.95

For the second set of trials, the experimental probability is:

experimental probability = 81/90 = 0.9

So the experimental probability of the cup landing on its side for the first set of trials is higher than for the second set of trials. However, since the number of trials is relatively small, it is possible that this difference is due to chance and does not reflect a real difference in the probability of the cup landing on its side. To determine this with more confidence, we would need to conduct more trials and perform statistical tests to determine whether the difference is statistically significant.

Answer:

123 feet/41 yards

Step-by-step explanation:

When solving problems like this, we need to understand and note down what we already know to make our lives easier.

- He walks 13 different dogs

- the park is a perfect square

- he takes each dog for one full lap

- That day, he walked 6396ft

- He takes each dog the same distance

Now, let's find the unit value (how much ft does he walk if he takes JUST ONE dog around Greenpoint Park?)

Just divide: 6396 / 13 = 492 ft

Meaning, that when Harrold takes JUST ONE dog around the park, which is a perfect square, he walks 492 feet.

One full lap around the square shaped park is just 4 sides of the square.

So, he walks 492 ft every 4 sides.

However, the question specifically asked for one side.

Just divide again to get your answer:

492 / 4 = 123 ft

Therfore, the length of one side of the park is 123 feet, which is 41 yards.

The probability is given as follows:

P(34 < X < 46) = 0.5468 = 54.68%.

The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 40, \sigma = 8[/tex]

The probability is the p-value of Z when X = 46 subtracted by the p-value of Z when X = 34, hence:

Z = (46 - 40)/8

Z = 0.75

Z = 0.75 has a p-value of 0.7734.

Z = (34 - 40)/8

Z = -0.75

Z = -0.75 has a p-value of 0.2266.

Hence:

0.7734 - 0.2266 = 0.5468 = 54.68%.

The probability is:

P(34 < X < 46).

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Answer:

  x = ±1, ±√65

Step-by-step explanation:

You want the solution to (x² -1)³ = 64(x² -1)².

Subtracting the right side, we have ...

  (x² -1)³ -64(x² -1) = 0

  (x² -1)²(x² -1 -64) = 0

The solutions make the factors zero:

  x² -1 = 0   ⇒   x = ±1

  x² -65 = 0   ⇒   x = ±√65

Solutions are x = ±1 and x = ±√65.

__

Additional comment

The solutions ±1 are each multiplicity 2.

The cubic polynomial is given as y = 0.25(x³ - 12x + 16). Then the value of y when x = 6 will be 40.

Therefore the option  C is correct.

A polynomial expression is described as an algebraic expression with variables and coefficients.

If the zeroes of the polynomial are negative 4, 2, and 2.

Then the factors will be (x + 4), (x - 2), and (x - 2).

Then the cubic polynomial will be

we can write the polynomial equation as:

y = C(x³ - 12x + 16)

Then the polynomial is maximum at (-2, 8) then the value of C will be 0.25.

y = 0.25 (x³ - 12x + 16)

y = 0.25 (6³ - 12 × 6 + 16)

y = 0.25 (216 - 72 + 16)

y = 0.25 (160)

y = 40

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The given trigonometric expression sin⁻¹(-3√(2)/2) in radians is approximately -2.2143 radians.

As we know that sin⁻¹(x) = -cos⁻¹(x) + π/2, which is the angle in the fourth quadrant whose cosine is 3sqrt(2)/2.

We have:

cos²θ + sin²θ = 1

Since sine is negative and cosine is positive, we know that:

sinθ = -sqrt(1 - cos²θ)

Substituting cosθ = 3√(2)/2, we get:

sinθ = -√(1 - (3√(2)/2)²) = -√(1 - 9/8) = -√(1/8) = -√(2)/2

Therefore, sin^(-1)(-3sqrt(2)/2) = -cos^(-1)(3sqrt(2)/2) + π/2.

Since cosine is positive in the fourth quadrant, we have:

cos⁻¹(3√(2)/2) = π/4

Substituting this value, we get:

sin⁻¹(-3√(2)/2) = -π/4 + π/2 = π/4

Therefore, sin⁻¹(-3√(2)/2) in radians is approximately -2.2143 radians.

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Answer:

it would be 9x

Step-by-step explanation:

because subtract 3 from 12 and leave the x

Answer:

  a)  |x -8| = 6

  b)  |x -15| = 0

Step-by-step explanation:

You want the values of 'b' and 'c' for the cases where the solutions to the equation |x -b| = c are ...

The solutions to |x -b| = c are the solutions to ...

  x -b = c   ⇒   x = b +c

  x -b = -c   ⇒   x = b -c

Given the two solutions P and Q, the values of 'b' and 'c' can be found from ...

  P = b +c

  Q = b -c

Adding these two equations gives ...

  P +Q = 2b   ⇒   b = (P +Q)/2

Subtracting the second equation from the first gives ...

  P -Q = 2c   ⇒   c = (P -Q)/2

  b = (2 +14)/2 = 8

  c = (14 -2)/2 = 6

The equation is ...

  |x -8| = 6

  b = (15 +15)/2 = 15

  c = (15 -15)/2 = 0

The equation is ...

  |x -15| = 0

The scenario that can be represented by the inequalities 1.25 < x < 1.5 is: A. A container of milk costs at least $1.25 but less than $1.50

The inequality 1.25 < x < 1.5 represents a range of values between 1.25 and 1.5, where x falls within that range.

Option A, which states that a container of milk costs at least $1.25 but less than $1.50, fits this range.

Option B, which states that a student spends at least 1 hour 15 minutes, but no more than 1 hour 30 minutes on homework, does not fit this range, as it represents a range of time values and not a range of numerical values.

Option C, which states that a tip added to a restaurant bill is less than or equal to 25% or less than or equal to 50%, does not fit this range either, as it represents two separate ranges of values.

Option D, which states that the point value of a test item is more than 1.25 points and less than 1.5 points, fits this range as well.

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Answer:

15

Step-by-step explanation:

Let's call the number we try to find "[tex]x[/tex]"

In the problem, we know that "26 is less than twice a number" is the same as "4". So we can write this as an equation:

[tex]2x - 26 = 4[/tex]

Now we can solve for x by isolating it on one side of the equation.

First, we isolate "2x" by flipping "-26" to the other side, which will now be "26".

[tex]2x = 4 + 26[/tex]

Then we add

[tex]2x = 30[/tex]

finally, we isolate the x by dividing each side with 2

2x ÷ 2 = 30 ÷ 2
x = 15

Answer:

O (x + 1)2 + (y + 2)² = 9

Step-by-step explanation:

This is because the distance between points A and B is 3 units, and the distance between points B and C is 4 units, making the triangle a 3-4-5 right triangle. Point B is located at (-1, 1), which is 3 units away from point A (-1, -2) and 4 units away from point C (-5, 1). Therefore, the circle with center (-1, -2) and radius 3 would pass through point B, and its equation would be:

(x + 1)2 + (y + 2)² = 3²

(x + 1)2 + (y + 2)² = 9

Answer:

Answer is D.

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The smaller angle made by the two hands of the clock at 3 o'clock is 90 degrees.

A figure known as an angle is created by two rays or line segments that meet at a place known as the vertex of the angle. The sides or legs of the angle are other names for the rays or line segments.

To determine the smaller angle made by the two hands of a clock when it is 3 o'clock, we can use the following formula:

angle = |(11/2) * m - 30h|

where:

m is the number of minutes past the hour (in this case, since it is 3 o'clock, m = 0)

h is the hour (in this case, h = 3)

By using this formula, we get:

angle = |(11/2) * 0 - 30(3)| = |0 - 90| = 90

Therefore, the smaller angle made by the two hands of the clock at 3 o'clock is 90 degrees.

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The value of variable y from the system of vertical angles is equal to 125.

In this problem we need to determine by algebra properties the value of a variable y from a system of two pairs of vertical angles. The system is represented by the following expression:

x + 20 = 3 · x - 50

2 · x = 70

x = 35

Then, by definition of supplementary angles:

(x + 20) + y = 180

55 + y = 180

y = 125

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For every 3 days Rochelle helped, she did not help for 1 day.

For every 4 days Rochelle helped, she did not help for 1 day.

For every 7 days Rochelle helped, she did not help for 6 days.

For every 28 days Rochelle helped, she did not help for 21 days.

Let's use the terms provided to complete the sentences comparing Rochelle's time in the garden.
Since Rochelle helped in the garden for 7 days during the last 4 weeks, we can calculate the total number of days in 4 weeks and then find the number of days she did not help.
There are 7 days in a week, so in 4 weeks, there are 4 x 7 = 28 days.
Rochelle helped for 7 days, so she did not help for 28 - 7 = 21 days.
Now, let's complete the sentences:
For every 1 day(s) Rochelle helped, she did not help for 3 day(s).
For every 3 day(s) Rochelle did not help, she helped 1 day(s).

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