QUICKLY HELP ME! Trigonometry

The height of Trevor's sandcastle is 3/10 foot taller than his sister's.

Given that, Trevor's sandcastle was 1/2 of a foot tall and his sister's was 1/5 of a foot tall.

Difference in the height of sandcastles = 1/2 - 1/5

= 5/10 - 2/10

= (5-2)/10

= 3/10

So, the difference in heights = 3/10 foot

Therefore, the height of Trevor's sandcastle is 3/10 foot taller than his sister's.

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The original selling price of the kayak set, given the discounts, would be $ 520. 63

The kayak is being sold such that a discount was offered of 10 % and then an additional discount was offered for 30 %.

The first step to the original price is:

= 328 / ( 1 - 30 %)

= 328 / 0. 70

= $ 468. 57

Then, the original price, would then account for the original discount of 10 % to become:

= 468. 57 / ( 1 - 10 %)

= 468. 57 / 0. 90

= $ 520. 63

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The length of side c is approximately 4.62 units.

Calculating the value of side c can be done by applying the Law of Sines on two angles and two sides.

Initially, determining angle C is necessary:

The corresponding formula for Angle C: 180° - (Angle A + Angle B) evaluates to 90° after substituting in reference values; 60°, and 30° respectively.

Implementing the Law of Sines culminates in this expression:

a/sin(A) = b/sin(B) = c/sin(C)

Replace with known angles and evaluate as follows:

Since the measures are known;

c/sin(C) = a/sin(A).

Subsequently,

by simple algebra c= a * sin(C) / sin(A) leads to desired outcome.

Putting relevant points into the equation above gives:

c = 4 * sin(90°) / sin(60°)

By using sine values from a calculator:

the preceding expression becomes,

c = 4 * 1 / (sqrt(3)/2)

After solving for c:

c = 8 / sqrt(3)

Using square root's rationalization: multiply numerator, and denominator by √3 resulting in

c = (8 * sqrt(3)) / 3

The length of side c is approximately 4.62 units.

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Let's assume we have a triangle with angles A and B and sides a and b, where angle A is opposite to side a, and angle B is opposite to side b. Given angle A = 60°, angle B = 30°, side a = 4 units, and side b = 2 units, find the length of side c.

The function that models the population t years after 2000 is P(t) = 14400 * (1.04)^t

The population growth function is of the form:

P(t) = P₀ * (1 + r)^t

Where:

P(t) is the current population after t years

P₀ is the starting population

r is the annual growth rate in percent

Thus, P₀ = 14400 and 4% = 0.04

P(t) = P₀ * (1 + r)^t

P(t) = 14400 * (1 + 0.04)^t

P(t) = 14400 * (1.04)^t

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Complete Question

The fox population in a certain region has an annual growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 14400.

a) Find a function that models the population t years after 2000 (t=0 for 2000).

Your answer is P(t) =

Answer:

Step-by-step explanation:

In 2004, the net income of the Apex Company would be $110 million + ($30 million x 9 years) = $380 million.

The net income of the Best Corporation in 2004 would be $170 million + ($20 million x 9 years) = $350 million.

Therefore, Apex Company earned more in 2004.

To find the year when Apex surpassed Best, we need to set their net incomes equal and solve for time:

110 + 30t = 170 + 20t

10t = 60

t = 6

Therefore, Apex surpassed Best in the year 1995 + 6 = 2001.

Answer:

4 students

Step-by-step explanation:

Let's call the total number of students in the school "x". We know that a certain percentage of them had over 80% in math, and the rest (100% - that percentage) had 80% or less.

Let's call the percentage of students who had over 80% "p". Then, we can set up the following equation:

p% of x + (100% - p%) of x = x

We can simplify this to:

p/100 * x + (100 - p)/100 * x = x

Multiplying both sides by 100 to get rid of the denominators, we get:

px + (100 - p)x = 100x

Simplifying further:

px + 100x - px = 100x

100x = 465

x = 465/100 = 4.65 (rounded to two decimal places)

So the total number of students in the school is approximately 4.65. However, we can't have a fraction of a student, so let's round up to the nearest whole number and assume there are 5 students in the school.

Now we can use the information given to find the number of students who had over 80%:

"of the students had over 80% in math"

implies that

p% of x = number of students who had over 80%

So if we plug in the values we have:

p% of 5 = number of students who had over 80%

Simplifying:

0.01p * 5 = number of students who had over 80%

0.05p = number of students who had over 80%

We don't know the value of p, but we can solve for the number of students who had over 80% for different values of p. For example:

If p = 90, then:

0.05(90) = 4.5

So 4.5 students had over 80%. Since we can't have half a student, we can assume that 4 students had over 80%.

Alternatively, we can solve for p using the information given:

"of the students had over 80% in math"

implies that

p% of x = number of students who had over 80%

So:

p% of x = number of students who had over 80%

p% of 5 = number of students who had over 80%

0.01p * 5 = number of students who had over 80%

0.05p = number of students who had over 80%

We know that the number of students who had over 80% is some integer value between 0 and 5, inclusive. We can test different values of p within this range to see if they give us an integer solution:

If p = 90, then:

0.05(90) = 4.5

This is not an integer solution, so p = 90 is not the correct answer.

If p = 80, then:

0.05(80) = 4

This is an integer solution, so p = 80 is the correct answer. Therefore, 4 students had over 80%.

Answer:

reverse or flip

In a case whereby  stack of two hundred eighty cards is placed next to a ruler, and the height of stack is measured to be 7/ 8 inches the thickness  of one card is 1/320 inches

Based on  the provided information, two hundred eighty cards is placed next to a ruler ten we can set up the expression as

( 7/ 8) / 280

7/ 8 * 1/280

1/320

Therefore, based on the given information, this implies that Each card is 1/320 inches

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(a)The arrow will reach a maximum height of 130 feet.

(b) After around 5 seconds, the arrow will strike the ground.

(c) At 1 second and 4 seconds after launch, the arrow will be 114 feet high.

a) The maximum height of the arrow occurs at the vertex of the parabolic pathway. The x-coordinate of the vertex is given by -b/2a, where a=-16 and b=80. So, t= -b/2a = -80/(2x(-16)) = 2.5 seconds. To find the maximum height, plug in t=2.5 into the equation: h(2.5) = -16(2.5)² + 80(2.5) + 50 = 130 feet.

Therefore, the maximum height of the arrow is 130 feet.

b) To find the time it takes for the arrow to reach the ground, find the value of t when h(t)=0 (since the arrow hits the ground when h=0). We can use the quadratic formula to solve for t:

t = (-V₀ ± √(V₀² - 4ah₀)) / 2a

where a=-16, V₀=80, and h₀=50.

t = (-80 ± √(80² - 4x(-16)50)) / 2(-16) = 5 seconds or -1.5625 seconds

Since time can't be negative, the arrow will hit the ground after about 5 seconds.

c) To find the time it takes for the arrow to be 114 feet high, solve for t when h(t) = 114.

-16t² + 80t + 50 = 114

-16t² + 80t - 64 = 0

Dividing both sides by -16 gives:

t² - 5t + 4 = 0

Factoring gives:

(t-4)(t-1) = 0

So t=4 seconds or t=1 second.

Therefore, the arrow will be 114 feet high at 1 second and 4 seconds after launch.

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Answer:

The domain in interval notation is (-infinity, infinity), or all real numbers.

Answer:

Step-by-step explanation:

First, we need to find the z-scores for q1 and q3.

Q1:

Using the formula for z-score, we get:

z = (x - μ) / σ

where x is the IQ score we want to find the z-score for, μ is the mean IQ of the population, and σ is the standard deviation of the population.

For the first quartile (q1), we want to find the z-score such that 25% of the population has an IQ score below that value. From the standard normal distribution table, we find that the z-score corresponding to a cumulative area of 0.25 is -0.674.

So we have:

-0.674 = (x - 110) / 16

Solving for x, we get:

x = 99.8

Therefore, q1 is approximately 99.8.

Q3:

Similarly, for the third quartile (q3), we want to find the z-score such that 75% of the population has an IQ score below that value. From the standard normal distribution table, we find that the z-score corresponding to a cumulative area of 0.75 is 0.674.

So we have:

0.674 = (x - 110) / 16

Solving for x, we get:

x = 120.8

Therefore, q3 is approximately 120.8.

Answer: 4m-24n+3

Step-by-step explanation:

Expand and collect like terms!

[tex](17m-12n-1) + (4-13m-12n)\\= 17m - 12n - 1 + 4 - 13m - 12n\\=4m-24n+3[/tex]

Hope this helps <3

The value of line CD is 20

From the diagram shown, we have that;

Line BE = 3x - 11

Line CD = 9x - 43

Note that the length of line BE is half the length of line CD

this is represented as;

2BE = CD

Now, substitute the values

2(3x - 11) = 9x - 43

expand the bracket

6x - 22 = 9x - 43

collect the lie terms

6x - 9x = -43 + 22

-3x = -21

Make 'x' the subject

x = 7

Then, CD = 9(7) - 43 = 20

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Answer:

780

Step-by-step explanation:

In this example we will call Original Price = y

72%=0.72

218.40=0.72*y

1-0.72=0.28

218.4÷0.28=780

Answer: D

Step-by-step explanation:

When using deduction and starting from a given set of rules and conditions, you can determine what must be true. Therefore, the correct answer is:

D. What must be true

The first three terms of the expression (2 · x + 1 / x)¹² are 4096 · x¹², 24576 · x¹⁰ and 67584 · x⁸, respectively.

In this problem we find the case of a expression of the form (a + b)ⁿ, where any term of the expression can be found by binomial theorem:

[tex](a + b)^{n} = \sum\limits_{k = 0}^{n} \frac {n!}{k! \cdot (n - k)!}\cdot a^{n - k}\cdot b^{k}[/tex]

Where:

If we know that a = 2 · x, b = 1 / x and 12, then the first three terms of the power of the binomial are, respectively:

n = 0

[tex]C_{0} = \frac{12!}{0! \cdot (12 - 0)!}\cdot (2\cdot x)^{12 - 0} \cdot (\frac {1}{x})^{0}[/tex]

C₀ = 4096 · x¹²

n = 1

[tex]C_{1} = \frac{12!}{1! \cdot (12 - 1)!}\cdot (2\cdot x)^{12 - 1} \cdot (\frac {1}{x})^{1}[/tex]

C₁ = 12 · (2048 · x¹¹) · (1 / x)

C₁ = 24576 · x¹⁰

n = 2

[tex]C_{2} = \frac{12!}{2! \cdot (12 - 2)!}\cdot (2\cdot x)^{12 - 2} \cdot (\frac {1}{x})^{2}[/tex]

C₂ = 66 · (1024 · x¹⁰) · ( 1 / x²)

C₂ = 67584 · x⁸

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Answer:

D

Step-by-step explanation:

All the numbers that have fraction are irrational therefore the numbers are real

Answer:

9. 40 = 2πr

r = 20/π inches = 6.4 inches

d = 40/π inches = 12.7 inches

10. 256 = 2πr

r = 128/π feet = 40.7 feet

d = 256/π feet = 81.5 feet

The odds against the paper having the letter C written on it is 10 : 11

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

MATHEMATICS

In the above word, we have

Letter C = 1

letters = 11

So, the probability of C is

Probability = 1/11

When converted to odds we have

Odds = Letter - Letter C : Letters

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

Odds = 11 - 1 : 11

Evaluate

Odds = 10 : 11

Hence, the odds is 10 : 11

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Answer:

Step-by-step explanation:

a. Variables needed to solve the problem:

Sample size: n = 1,026

Proportion of the sample that responded "Make some major changes": p = 0.39

Confidence level: 95%

b. To determine if a normal distribution can be used to approximate the data, we need to check if the sample size is large enough to meet the requirements for a normal approximation. The sample size should be at least 10 times larger than the number of successes (np) and 10 times larger than the number of failures (n(1-p)). In this case, we have:

np = 1026 x 0.39 = 399.14

n(1-p) = 1026 x 0.61 = 626.86

Both np and n(1-p) are greater than 10, so we can assume that a normal distribution can be used to approximate the data.

c. The standard deviation of the proportion can be calculated using the following formula:

standard deviation = sqrt(p(1-p) / n)

standard deviation = sqrt(0.39 x 0.61 / 1026) = 0.024

d. The point estimate of the proportion of all United States adults who would respond "Make some major changes" is simply the sample proportion, which is p = 0.39. The margin of error can be calculated using the following formula:

margin of error = z* * standard deviation

where z* is the z-score associated with the 95% confidence level. Using a standard normal distribution table or a calculator, we find that the z-score for a 95% confidence level is approximately 1.96. Therefore:

margin of error = 1.96 * 0.024 = 0.047

e. The confidence interval can be calculated using the following formula:

confidence interval = point estimate ± margin of error

confidence interval = 0.39 ± 0.047

confidence interval = (0.343, 0.437)

Therefore, we are 95% confident that the proportion of all United States adults who would respond "Make some major changes" is between 0.343 and 0.437.

Answer:

B

Step-by-step explanation:

163 money t for time 2 for 2hours

a) The value of the mean is μ = 22.5

   The value of the standard deviation is σ = 3.5

b) The Value of 15 girls or fewer is significantly low.

   The value of 30 girls or more is significantly high.

c) The result 36 is significantly high because 36 is greater than 30 girls. A         result of 36 girls is not necessarily definitive proof of the method's effectiveness.

The standard deviation is a measure of the amount of variability or dispersion in a set of data values. It is a statistical measure that tells you how much, on average, the values in a dataset deviate from the mean or average value.

a) Since the probability of having a girl for each couple is 0.5, the number of girls each couple will have can be modeled as a binomial distribution with parameters n=1 and p=0.5.

Let X be the random variable denoting the number of girls in 45 couples. Then, X follows a binomial distribution with parameters n=45 and p=0.5.

The mean of a binomial distribution is given by μ = np, so in this case, the mean number of girls in a group of 45 couples is:

μ = np = 45 x 0.5 = 22.5

Therefore, we expect to see around 22-23 girls in a group of 45 couples.

The standard deviation of a binomial distribution is given by σ = √(np(1-p)), so in this case, the standard deviation of the number of girls in a group of 45 couples is:

σ = √(np(1-p)) = √(45 x 0.5 x 0.5) = 3.535

Therefore, we can expect the number of girls in a group of 45 couples to have a standard deviation of around 3.5.

b) In this case, we can assume that the number of girls in a group of 45 couples follows a normal distribution due to the Central Limit Theorem.

Using the standard deviation we found in the previous answer (σ = 3.535), we can calculate the values that separate the results that are significantly high and significantly low.

Significantly high:

Mean + 2σ = 22.5 + 2(3.535) = 29.57

Significantly low:

Mean - 2σ = 22.5 - 2(3.535) = 15.43

c) To determine if the result of 36 girls is significantly high, we need to compare it to the values we calculated in the previous answer.

Mean + 2σ = 22.5 + 2(3.535) = 29.57

Since 36 is greater than 29.57, we can conclude that the result of 36 girls is significantly high.

This suggests that the method of gender selection may be having an effect on the probability of having a girl. However, we cannot conclusively say this without conducting further analysis or testing.

It is also important to note that the result of 36 girls is not necessarily definitive proof of the method's effectiveness.

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The next 20 octal numbers after 7 are: 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33

In the decimal system, we count from 0 to 9 before moving to the next place value. In the octal system, we count from 0 to 7 before moving to the next place value. Therefore, the next 20 octal numbers after 7 are:

10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33

To understand how this pattern works, consider the decimal number 32. To convert it to octal, we can repeatedly divide it by 8 and record the remainders:

32 ÷ 8 = 4 with remainder 0

4 ÷ 8 = 0 with remainder 4

So the octal representation of 32 is 40. Similarly, we can convert any decimal number to octal by repeatedly dividing by 8 and recording the remainders.

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The required balance after 10 years with continuous compounding at a 7% interest rate would be approximately $1007.

To calculate the balance after 10 years with a continuous compounding interest rate of 7%, we can use the formula for continuous compound interest:

[tex]F = P * e^{rt}[/tex]

Where:

F is the future balance or the final amount

P is the principal amount (initial deposit)

e is the base of the natural logarithm (approximately 2.71828)

r is the interest rate (expressed as a decimal)

t is the time in years

In this case, the initial deposit (principal) is $500, the interest rate is 7% (0.07 as a decimal), and the time is 10 years. Plugging these values into the formula, we get:

[tex]F = 500 * e^{0.07 * 10}[/tex]

Using a calculator, we can evaluate e^(0.07 * 10) ≈ 1.96728. Multiplying this by 500 gives us:

[tex]F=500 * 1.96728[/tex]
F = $1007

Therefore, the balance after 10 years with continuous compounding at a 7% interest rate would be approximately $1007.

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Answer:

Point A (-3,3)

Point B about (3,-2.75)

Point C (-4,-4)

Point D (1,0)

Step-by-step explanation:

The two points on the line are (5, -9/5) and (-5/3, 3).

The other point on the line is (0, -12/7).

The equation [tex]\frac{m}{5} +\frac{n}{3} =0[/tex] can be rewritten as:

3m + 5n = 0

When m=5:

3m + 5n = 0

3(5) + 5n = 0

n = -9/5

Simplify the above equation,

When n=3:

3m + 5n = 0

3m + 5(3) = 0

m = -5/3

the two points on the line are (5, -9/5) and (-5/3, 3)

The equation [tex]\frac{m}{10} -\frac{7n}{6} =4[/tex] can be simplified as follows:

m/10 - 7n/6 = 4

m/10 = 7n/6 + 4

m = 70n/6 + 40

m = 35n/3 + 20

When n=0:

m = 35n/3 + 20

m = 20

So, the point on the line is (20, 0).

When m=0:

35n/3 + 20 = 0

35n/3 = -20

n = -12/7

So, the other point on the line is (0, -12/7).

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The graph is positive and increases as the value of x increases.

The behavior of the graph of y = √(4x) is determined by substituting some value of x into the function and check the corresponding value of y.

When x = 0, the value  of y is calculated as;

y = √(4(0))

y = 0

When x = 1, the value of y is calculated as;

y = √(4(1))

y = 2

When x = 4, the value of y is calculated as;

y = √(4(4))

y = 4

When x = 9,  the value of y is  calculated as;

y = √(4(9))

y = 6

From the data above, the value of y increases as x increases, although not at equal increment.

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The solution is :  the perimeter and area of a square if the length of its diagonal is 16 mm is, 45.3 mm and 512mm².

Here, we have,

Use the basic 45-45-90 triangle with side length 1 as the building block here.  If the length of one side is 1, then the perimeter is 1 + 1 + 1 + 1, or 4, and the length of the diagonal is √2.

We are told that the length of the diagonal of the given square is 16 m.

Determine the length of one side of this square, using an equation of proportions:

16        x

------ = -------

√2       1

                                            16

Then (√2)x = 16, and x = -----------

                                            √2

The perimeter of the given square (with diagonal 16 mm)  is 4 times the side length found above, or:

     16                      16

4 ---------- = (2)(2) -----------  =  (2)(√2)(16) = 32√2 (all measurements in mm)

      √2                    √2

This perimeter, rounded to the nearest tenth, is 45.3 mm.

so, area of the  square is:

(16/√2)² = 512mm².

Hence, The solution is :  the perimeter and area of a square if the length of its diagonal is 16 mm is, 45.3 mm and 512mm².

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