Determine the number of real solutions:
1. y=-3x^2+x+12

Answer:

the correct answer is the fifth one 1-3x

Answer:

tan (A) = 1/2. tan (B) = 2

Step-by-step explanation:

tan θ = opposite/adjacent

for tan(A), the opposite is 5. adjacent is 10. hypotenuse is 5√5.

tan (A) = 5/10 = 1/2.

for tan (B), the opposite is 10, adjacent is 5. hypotenuse is 5√5.

tan(B) = 10/5 = 2

Check the picture below.

[tex]60+\stackrel{ \measuredangle ABC }{(3x+15)}+(20x-10)~~ = ~~180\implies 65+23x=180 \\\\\\ 23x=115\implies x=\cfrac{115}{23}\implies x=5 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \measuredangle EBF }{(3x+15)}\implies 3(5)+15\implies \text{\LARGE 30}^o[/tex]

Answer:

There are 6 children in the group and 4 adults

Step-by-step explanation:

Because the adult ticket cost $8, so there are 4 adults and a total of $32 for adults.

Then I took 62-32=$30

$30 (total for children price) / $5 (the children's tickets) = 6 children

The fully simplified product of (b minus 2 c)(negative 3 b + c) can be found by using the distributive property and combining like terms. The product is -3b^2 + 7bc - 2c^2, which has 3 terms.

Therefore, the statement "The simplified product has 2 terms" is false and the statement "The simplified product has 4 terms" is also false. The degree of each term is the sum of the exponents of the variables,

so the degree of the product is the highest degree among the terms. In this case, the highest degree is 2, which means that the statement "The simplified product has a degree of 2" is true and the statement "The simplified product has a degree of 4" is false.

Lastly, we can count the number of negative terms in the product to determine if the statement "The simplified product, in standard form, has exactly 2 negative terms" is true. We see that there are two negative terms, -3b^2 and -2c^2, so this statement is true.

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1/5 = 200 pennies

5/5 = 1000 pennies

28.

Arc BC = 107 degrees

---Since the given angle is on the center of the circle, the arc has the same measure.

Angle BAC = 53.5 degrees

---Since this is an inscribed angle, the measure of the angle is half of the corresponding arc.

29.

Arc BC = 92 degrees

---Since we are given an inscribed angle, the corresponding arc is double the measure of the angle.

Hope this helps!

The nearest Whole number, the coordinator should use a sample size of 753.

To find the sample size needed when the population proportion is known, we can use the formula:

n = (z^2 * p * q) / E^2

where:

n= sample size

z = z-score for the desired level of confidence (we'll use 1.96 for 95% confidence)

p = population proportion

q = 1 - p (the proportion of the population that does not have the characteristic)

E = margin of error (we'll use 0.03 for 3% margin of error)

Substituting the values given in the problem:

n = (1.96^2 * 0.3 * 0.7) / 0.03^2

n ≈ 752.9

Rounding up to the nearest whole number, the coordinator should use a sample size of 753.

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part a.

The average velocity on the interval [0,2] is -16 ft/sec

part b.

The instantaneous velocity when t=2  is found as -48 ft/sec

part c.

Time taken to reach the ground is 3 seconds.

part d.

velocity of the ball when it hits the ground is found as  80 ft/sec downward.

The average velocity on the interval [0,2] is determined by :

average velocity = change in distance / change in time

change in distance :

s(2) - s(0) = (-16(2)^2 + 16(2) + 96) - (-16(0)^2 + 16(0) + 96) = -32

where the change in time:

2 - 0 = 2

average velocity = -32/2

average velocity = -16 ft/sec

b. The instantaneous velocity :

s'(t) = -32t + 16

and  t=2:

s'(2) = -32(2) + 16 =

s'(2)=  -48 ft/sec

the velocity of the ball when it hits the ground is at time = 3

s'(t) = -32t + 16

s'(3) = -32(3) + 16 =

s'(3)  =  -80 ft/sec

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To solve the inequality 2x < 12, we need to isolate x on one side of the inequality sign. We can do this by dividing both sides by 2:

2x < 12

x < 6

Therefore, the solution to the inequality is x < 6. This means that all values of x that are less than 6 satisfy the inequality. We can represent this graphically on a number line:

<=======()------6-------------->

The open circle () indicates that x cannot be equal to 6, but can be any value less than 6. The arrow indicates that the inequality is true for all values of x to the left of the open circle.

In summary, the solution to the inequality 2x < 12 is x < 6.

I'm 15 BTW

The standard deviation of this sample of shopping time is equal to 50.

In order to calculate the standard deviation of this sample of shopping time, we would have to first determine the mean of the data set.

Mathematically, the mean for these data sets would be calculated by using this formula:

Mean = [F(x)]/n

For the total sum of data, we have:

F(x) = 42 + 27 + 22 + 37 + 32

F(x) = 132.1.

Mean = 160/5

Mean = 32

Now, we can calculate the standard deviation by using this formula:

Standard deviation, S = √(1/n × ∑(xi - u₁)²)

Standard deviation, S = √(1/5 × ∑(42 - 32)² + 1/5 × ∑(27 - 32)² + 1/6 × ∑(22 - 32)² + 1/5 × ∑(37 - 32)² + 1/5 × ∑(32 - 32)²

Standard deviation, S = 250/5

Standard deviation, S = 50.

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The vertices of the triangle XYZ after a dilation with a scale factor of [tex]4[/tex] are [tex]X'(-8, 4)[/tex], [tex]Y'(12, -4)[/tex], and [tex]Z'(4, 16)[/tex].

To find the vertices of the image after a dilation with a scale factor of [tex]4[/tex], we need to multiply the coordinates of each vertex by the scale factor.

Given the triangle with vertices [tex]X(-2,1), Y(3,-1), and \ Z(1,4)[/tex], we can apply the dilation to each vertex.

Let's start with vertex [tex]X(-2,1)[/tex]:

The new x-coordinate,[tex]\(x'\)[/tex], can be obtained by multiplying the original x-coordinate by the scale factor: [tex]\(x' = 4 \times (-2) = -8\)[/tex].

Similarly, the new y-coordinate, [tex]\(y'\)[/tex], is obtained by multiplying the original y-coordinate by the scale factor: [tex]\(y' = 4 \times 1 = 4\)[/tex].

Thus, the image of vertex X is [tex]X'(-8, 4)[/tex].

Now let's apply the same process to vertices [tex]Y(3,-1) \ and \ Z(1,4):[/tex]

For vertex Y:

[tex]\(x' = 4 \times 3 = 12\)\\\y' = 4 \times (-1) = -4\)[/tex]

The image of vertex Y is Y'(12, -4).

For vertex Z:

[tex]\(x' = 4 \times 1 = 4\)\\\(y' = 4 \times 4 = 16\)[/tex]

The image of vertex Z is [tex]Z'(4, 16)[/tex].

Therefore, the vertices of the triangle XYZ after a dilation with a scale factor of [tex]4[/tex] are [tex]X'(-8, 4)[/tex], [tex]Y'(12, -4)[/tex], and [tex]Z'(4, 16)[/tex].

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Answer:

47
Step-by-step explanation:

(26) The length FE in the intersecting chords is 6.

(27) The length of arc AE is 64⁰.

The length FE is calculated by applying intersecting chord theorem as follows;

The intersecting chord theorem, also known as the power of a point theorem, states that:

If two chords of a circle intersect inside the circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

FE ≅ AB = 6

The length of arc AE is calculated as follows;

arc AE + arc AB + arc ED = 180 (sum of angles of a semi circle)

arc AE + 58 + 58 = 180

arc AE + 116 = 180

arc AE = 64

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(a) The Goran's monthly payment is  605.9.

(b) His total amount to repay the loan is 50,895.6.

(c) The amount of interest he will pay is 10,395.6.

The Goran's monthly payment, is calculated using the following formula;

P = r(PV) / (1 - (1 + r)⁻ⁿ)

where;

r = 6.7% / 12 = 0.00558

PV = 40,500

n = 7 years x 12 months/year = 84 months

P = 0.00558(40,500) / (1 - (1 + 0.00558)⁻⁸⁴)

P  = 605.9

Total amount = 84 x 605.9 = 50,895.6

The total interest paid on the loan is the difference between the total amount repaid and the amount borrowed.

= 50,895.6 - 40,500

= 10,395.6

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The proof is shown in the solution.

Given is an expression (2k + x)", we need to expand and prove,

Expand the binomials and cancel the 2k's:

n(n−1) / 2 (2k) = n(n−1)(n−2) / 6

Solving,

3(2k) = n−2

n = 6k+2

Hence proved.

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The probability of exactly 5 occurrences is (Rounding to three Decimal places), we get P(X ≤ 3) ≈ 0.251.

a) The probability of exactly 5 occurrences is given by the Poisson probability mass function:

P(X = 5) = (e^(-λ) * λ^5) / 5! = (e^(-5.1) * 5.1^5) / 120 ≈ 0.1755

Rounding to four decimal places, we get P(X = 5) ≈ 0.1755.

b) The probability of more than 6 occurrences can be calculated as the complement of the probability of 6 or fewer occurrences:

P(X > 6) = 1 - P(X ≤ 6) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6))

Using the Poisson probability mass function and the given value of λ, we can calculate each of the probabilities:

P(X = 0) ≈ 0.006

P(X = 1) ≈ 0.031

P(X = 2) ≈ 0.079

P(X = 3) ≈ 0.135

P(X = 4) ≈ 0.174

P(X = 5) ≈ 0.1755

P(X = 6) ≈ 0.1493

Substituting these values into the formula, we get:

P(X > 6) ≈ 1 - (0.006 + 0.031 + 0.079 + 0.135 + 0.174 + 0.1755 + 0.1493) ≈ 0.249

Rounding to three decimal places, we get P(X > 6) ≈ 0.249.

c) The probability of 3 or fewer occurrences is given by the cumulative distribution function:

P(X ≤ 3) = ∑ P(X = k), for k = 0, 1, 2, 3.

Using the Poisson probability mass function and the given value of λ, we can calculate each of the probabilities:

P(X = 0) ≈ 0.006

P(X = 1) ≈ 0.031

P(X = 2) ≈ 0.079

P(X = 3) ≈ 0.135

Adding these probabilities, we get: P(X ≤ 3) ≈ 0.251

Rounding to three decimal places, we get P(X ≤ 3) ≈ 0.251.

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The other zero of the function given that one zero is 4 include: B. 8.

The correct graph of the quadratic function represented in the table is: graph A.

In Mathematics and Geometry, the standard form of a quadratic function is represented by the following equation;

ax² + bx + c = 0

Next, we would create a system of equation by using the data points provided in the table above;

a(2)² + b(2) + c = 12

4a + 2b + c = 12     .....equation 1.

a(6)² + b(6) + c = -4

36a + 6b + c = -4     .....equation 2.

a(7)² + b(7) + c = -3

49a + 7b + c = -3     .....equation 3.

By solving the system of equations simultaneously, the values of a, b, and c are as follows;

a = 1

b = -12

c = 32

Therefore, the required quadratic function is given by;

ax² + bx + c = 0

x² - 12x + 32 = 0

x² - 8x - 4x + 32 = 0

x(x - 8) - 4(x - 8) = 0

(x - 4)(x - 8) = 0

x = 4 or x = 8

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Step-by-step explanation:

To determine which point is a solution of the linear inequality 92 + 3y < 6, we can substitute the values of each point into the inequality and check if the inequality holds true.

Let's evaluate the inequality for each point:

1. Point (1, -1):

92 + 3(-1) < 6

92 - 3 < 6

89 < 6

The inequality is not true for this point.

2. Point (2, -2):

92 + 3(-2) < 6

92 - 6 < 6

86 < 6

The inequality is not true for this point.

3. Point (1, 1):

92 + 3(1) < 6

92 + 3 < 6

95 < 6

The inequality is not true for this point.

4. Point (1, 3):

92 + 3(3) < 6

92 + 9 < 6

101 < 6

The inequality is not true for this point.

None of the given points satisfy the inequality. Therefore, none of the points (1, -1), (2, -2), (1, 1), or (1, 3) are solutions to the linear inequality 92 + 3y < 6.

Answer:

Step-by-step explanation:

There are 5 marbles

Answer: 35, 48, 63

Step-by-step explanation:

This sequence is known as an arithmetic progression sequence.

      3 - 0 = 3

      8 - 3 = 5

      15 - 8 = 7

      24 - 15 = 9

Using this pattern, we will add 11 to 24 for the next term in the sequence. Then, we will continue this pattern for the next two terms.

➜ The pattern is resulting odd numbers from subtracting two terms in a row; 3, 5, 7, 9, [11], [13], [15].

      24 + 11 = 35

      35 + 13 = 48

      48 + 15 = 63

The maximum, minimum, and median values include the following: D. maximum = 8, minimum = –8, median = 4.

In Mathematics and Statistics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

Based on the information provided about the data set, the five-number summary for the given data set include the following:

Minimum (Min) = -8.

First quartile (Q₁) = -4.

Median (Med) = 4.

Third quartile (Q₃) = 6.

Maximum (Max) = 8.

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a)  Note that the the probability that on a given day, no machines break down is 0.3487

b) the probability that all printing   requirements on a particular day will be met is 0.2323

a) To find the probability that no machines break down, we can use the binomial distribution with

n = 10 (number of machines) and

p = 0.1 (probability of a machine breaking down).

The probability of no breakdowns  is

P(X = 0), where X is the number of breakdowns.

Using the binomial distribution formula: P(X = k) = (n choose k) * [tex]p^{k}[/tex] * (1 - p)[tex]^{n-k}[/tex]

P(X = 0) = (10 choose 0) * 0.1⁰ * 0.9¹⁰ = 0.3487

So it is right to stay that  the probability that on a given day, no machines break down is 0.3487

(b)

P(8 or more machines are functioning) = P(X = 8) + P(X = 9) + P(X = 10)

P(X = 8) = (10 choose 8) * 0.1⁸ *   0.9² = 0.1937

P(X = 9) = (10 choose 9) * 0.1⁹ * 0.9¹ = 0.0386

P(X = 10) = (10 choose 10) * 0.1¹⁰ * 0.9⁰ = 0.0000001

P(8 or more machines are functioning) = 0.1937 + 0.0386 + 0.0000001 = 0.2323

Hence, the probability that all printing requirements on a particular day will be met is 0.2323 or approximately 23.23%.

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Sam started with £5.

Now, we need to find out how much money Sam has left after spending £4.50 on chocolate and £3.00 on cold coffee. To do this, we can add the two amounts that he spent and subtract the total from his original amount of money:

£4.50 + £3.00 = £7.50

Original amount - £7.50 = Two-fifths of the original amount

Now, we know that Sam has two-fifths of his original amount of money left. We can use this information to set up an equation:

2/5 x Original amount = Amount of money left

We can solve for the original amount by multiplying both sides by 5/2:

Original amount = (5/2) x Amount of money left

Plugging in the amount of money Sam has left, we get:

Original amount = (5/2) x Amount of money left

Original amount = (5/2) x (Original amount - £7.50)

Simplifying the right side of the equation, we get:

Original amount = (5/2) x Original amount - (5/2) x £7.50

Original amount - (5/2) x Original amount = (5/2) x £7.50 (3/2) x Original amount = (15/2)

Original amount = £5

Therefore, Sam started with £5.

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The probability of getting 14 successes out of 19 trials with a probability of success of 0.8 is approximately 0.0572.

Using the binomial probability formula, we have:

P(X=14) = (n choose x) * pˣ . (1-p)⁽ⁿ⁻ˣ⁾

Plugging in the values, we get:

P(X=14) = (19 choose 14) x 0.8¹⁴ x 0.2⁵

P(X=14) ≈ 0.0572

Therefore, the probability of getting 14 successes out of 19 trials with a probability of success of 0.8 is approximately 0.0572.

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The expression y = sin(x²) cot(2x) when differentiated is 2xcot(2x)cos(x²) - 2csc²(2x)sin(x²)

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

y=sin x² cot 2x

Express properly

So, we have the following representation

y = sin(x²) cot(2x)

When each term of the expression are differentiated using the first principle and the product rule, we have

sin(x²) ⇒ 2xcot(2x)cos(x²)

cot(2x) ⇒ -2csc²(2x)sin(x²)

So, the solution is 2xcot(2x)cos(x²) - 2csc²(2x)sin(x²)

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