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Rebecca is x years old.Mary is 8 years older than rebecca. Jill is three times older than mary. The sum of their ages is 67.
A) form an expression in terms of x

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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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.

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:

Point A (-3,3)

Point B about (3,-2.75)

Point C (-4,-4)

Point D (1,0)

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

163 money t for time 2 for 2hours

The triangles which are similar or congruent discussed below.

If the triangles are similar then the corresponding side ratios are equal.

1. 6/9 = 2/3

9/13.5= 2/3

8/12= 2/3

Thus, the triangle are similar.

2. The second case is neither similar or congruent.

3. The pair has one right angle common and one side measure 6 unit.

So, the triangle are neither similar or congruent.

4. The triangles have two angles equal.

so, the triangle are similar.

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Step-by-step explanation:

The answer is:

- Control of significant inventory balances

This is because just-in-time processing is a system that emphasizes on producing goods or services at the exact time they are needed, without accumulating inventory. Therefore, it does not prioritize the control of significant inventory balances. The other options are benefits of just-in-time processing.

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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Answer:

The new prism has 9 times the surface area of the original prism

Step-by-step explanation:

There are three ways in which you can answer this question.

The hard way:

The surface area of a rectangular prism is given by
A = 2(LW+ LH+ WH)
where L= length, W= width and H = height)

If we were to triple the sides we would get the new side measures as
3L, 3W, 3H

New surface area becomes:
A' = 2 (3L · 3W + 3L · 3H · 3W· 3H)

A' = 2(9LW + 9 LH + 9 WH)

Factoring out 9 from the brackets we get

A' = 2 · 9 (LW+ LH+ WH)

A'/A = 2 · 9 (LW+ LH+ WH) /2(LW+ LH+ WH)

The common term 2(LW+ LH+ WH) cancels out from numerator and denominator leaving 9 as the answer

A smarter and easy way of doing this

A cube is nothing but a rectangular prism with all sides equal. Let a be the length of a side of the cube

A cube has 6 sides. The surface area of each side = a x a = a²

So total surface area A = 6a²

If each side is tripled, each side becomes 3a.
New surface area A' = 6 (3a)² = 6 (9a²)

A'/A = 6 (9a²)/6(a²) = 9

An even easier way

Again we take a cube. But instead of using a variable, let's assign the side of the cube a length of 1 unit

Surface area A = 6 · 1² = 6

After tripling each side becomes 3 units long
New surface area A' = 6 · 3² = 6.9 = 54

A'/A = 54/6 = 9

Choose whichever method you feel comfortable with

Answer: Mean = 1.15       Median = 1

Step-by-step explanation: add all numbers up and then divide it by 4 which will give you the mean.

Median: 0.9 and 1.1 are the 2 middle numbers in which you will add together then divide by 2 to get the exact middle value which is 1 exactly.

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:

please insert question

Step-by-step explanation:

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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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%.

the final result of the double integral is `(4/3)*a. we have to Integrate the inner integral with respect to z.

what is inner integral ?

An inner integral is a mathematical term that refers to the integral function that is evaluated first in a double integral.

In the given question,

To solve this double integral, we will use the following steps:

Integrate the inner integral with respect to z.

Evaluate the result of the inner integral at  upper and lower limits of z.

Substitute the result of the inner integral into the outer integral and integrate with respect to y.

Evaluate the result of the outer integral at the upper and lower limits of y.

Simplify the expression.

Now, let's apply these steps to solve the given double integral:

Integrate the inner integral with respect to z:

∫(x²*z + y²*z + z³) dz = x²/2*z² + y²/2*z² + z^4/4 + C

where C is the constant of integration.

Evaluate the result of the inner integral at the upper and lower limits of z:

(x²/2*(a²-x²-y²)¹⁵ + y²/2*(a²-x²-y²)¹⁵ + (a²-x²-y²)²/4)

- (x²/2*(-a²+x²+y²)¹⁵ + y²/2*(-a²+x²+y²)¹⁵ + (-a²+x²+y²)²/4)

Substitute the result of the inner integral into the outer integral and integrate with respect to y:

markdown

∫[(x²/2*(a²-x²-y²)¹⁵ + y²/2*(a²-x²-y²)¹⁵ + (a²-x²-y²)²/4)

- (x²/2*(-a²+x²+y²)¹⁵ + y²/2*(-a²+x²+y²)¹⁵ + (-a²+x²+y²)²/4)] dy

Evaluate the result of the outer integral at the upper and lower limits of y:

= ∫[(x²/2*(a²-x²-y²)¹⁵ + y²/2*(a²-x²-y²)¹⁵ + (a²-x²-y²)²/4)- (x²/2*(-a²+x²+y²)¹⁵ + y²/2*(-a²+x²+y²)¹⁵ + (-a²+x²+y²)²/4)] dy

from y = -sqrt(a²-x²) to y = sqrt(a²-x²)

= (2/3)*x²*(a²-x²)¹⁵ + (2/3)*(a²-x²)²⁵

- (2/3)*x²*(-a²+x²)¹⁵ + (2/3)*(-a²+x²)²⁵

Simplify the expression:

= (4/3)*a³ - (4/3)*a*x²

Therefore, the final result of the double integral is `(4/3)*a

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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

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.

Answer:

reverse or flip

If the recipe requires 3(1/2) pounds of flour for every (3/4) cups of sugar, then for 1 cup of sugar , Joseph should use 4.67 pounds of flour.

A "Proportion" is defined as a statement that two fractions are equal. It expresses the relationship between two quantities that are in the same ratio.

To solve the problem, we set up a proportion to relate the amount of flour to the amount of sugar:

We know that, recipe requires 3(1/2) pounds of flour for every (3/4) cups of sugar,

Which means,

⇒ (3.5 pounds of flour)/(0.75 cups of sugar) = (x pounds of flour)/(1 cup of sugar),

where x is = amount of flour needed for 1 cup of sugar.

We "cross-multiply" and simplify;

⇒ 3.5 × 1 = 0.75 × "x" pounds of flour,

⇒ 3.5/0.75 = x,

⇒ x ≈ 4.67 pounds of flour,

Therefore, Joseph should use 4.67 pounds of flour for 1 cup of sugar.

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The limit of the given sum as n approaches infinity is 0.

We have the limit as n approaches infinity of the sum from i = 1 to n of 1/(n+i). We can rewrite this sum using the hint provided as:

[tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex](1/ n + i) = [tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex] (1/n) * (1 + i/n)

To find the limit, we need to take the limit of the Riemann sum as n approaches infinity. This is equivalent to taking the limit of the area of n rectangles under the curve y=1/x as n approaches infinity.

As n becomes very large, the width of each rectangle becomes very small, and the height of each rectangle approaches 1/n. Therefore, the area of each rectangle approaches zero.

We can then express the limit as an integral:

[tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex](1/ n + i) =[tex]\lim_{n \to \infty} \sum_{\substack{i=1}} ^{n}[/tex] (1/n) * (1 + i/n) =[tex]\lim_{n \to \infty} \int ^1 _{1+1/n}[/tex] 1/x dx

Evaluating this integral gives:

[tex]\lim_{n \to \infty} \int ^1 _{1+1/n}[/tex] 1/x dx = [tex]\lim_{n \to \infty}[/tex] ln(1+1/n) = ln(1+0) = 0

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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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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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The z score of 0.166 square units of the standard normal distribution is to leave z is -0.999.

It is clear that 0.166 square units of the standard normal distribution is to the left of z. This means that the area under the standard normal distribution curve to the left of z is 0.166.

Using a standard normal distribution table or calculator, we can find the z-score corresponding to this area. For example, using a calculator, we can use the inverse normal cumulative distribution function (also called the probit function) to find the z-score:

invNorm(0.166) ≈ -0.999

Therefore, the z-score corresponding to 0.166 square units of the standard normal distribution is to the left of z is approximately -0.999.

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I can assist you in determining the r-coordinate of vertex K* following the reflection if you provide me the coordinates of vertex K in the original quadrilateral.

It is possible to multiply, divide, add, or subtract in mathematics. The following is how an expression is put together: Number, expression, and mathematical operator The components of a mathematical expression (such as addition, subtraction, multiplication or division, etc.) include numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression, often known as an algebraic expression, is any mathematical statement that contains variables, numbers, and an arithmetic operation between them. For instance, the word m in the given equation is separated from the terms 4m and 5 by the arithmetic symbol +, as does the variable m in the expression 4m + 5.

I can assist you in determining the r-coordinate of vertex K* following the reflection if you provide me the coordinates of vertex K in the original quadrilateral.

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As per the given equation the correct time, t, that it took for the ball to hit the ground is t = 3 seconds.

An equation is a mathematical statement that expresses the equality between two expressions. Equations typically include variables, which are symbols that represent unknown or varying quantities, and constants, which are known values.

An equation can be written in various forms, depending on the type of equation and the context in which it is used. Some common forms of equations include linear equations, quadratic equations, and polynomial equations.

For example, the equation x + 5 = 10 is a linear equation that has one variable, x. It can be solved by subtracting 5 from both sides of the equation to get x = 5.

A quadratic equation, such as x² + 2x + 1 = 0, has a variable raised to the second power. It can be solved using the quadratic formula or by factoring.

To solve the equation 0=−16t²+32t+48, Greg factored out a common factor of -16 from all three terms to get:

0 = -16(t² - 2t - 3)

Then, he factored the quadratic expression inside the parentheses as:

0 = -16(t - 3)(t + 1)

This gives us two solutions for t: t = 3 and t = -1.

However, we can discard the solution t = -1, since time cannot be negative in this context. Therefore, the correct time, t, that it took for the ball to hit the ground is t = 3 seconds.

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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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Answer: Yes, the triangles are congruent.

Step-by-step explanation:

According to the Side-Angle-Side (SAS) congruence theorem, if two triangles have two pairs of corresponding sides that are congruent and the included angle between those sides is also congruent, then the triangles are congruent.

In this case, we know that four parts (angles and sides) of one triangle are congruent to the corresponding four parts of another triangle. This means that two pairs of corresponding sides are congruent (since corresponding sides are equal) and the included angles between those sides are also congruent (since corresponding angles are equal). Therefore, the triangles satisfy the conditions of the SAS congruence theorem and are congruent.

It's important to note that this applies only to two triangles with exactly the same four congruent parts. If there is even one part that is not congruent between the two triangles, they cannot be proven to be congruent just by these means alone.

Answer:

If you know that 4 parts (angles and sides) of one triangle are congruent to the corresponding 4 parts of another triangle, then the triangles are congruent by the Side-Angle-Side (SAS) Congruence Postulate. This postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

The SAS postulate is one of five ways to prove that two triangles are congruent. The other four ways are Angle-Side-Angle (ASA), Side-Side-Side (SSS), Hypotenuse-Leg (HL), and Reflexive Property of Congruence.

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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