Are all rectangles similar

Ina case whereby Andrew believes that the probability that he will win the tennis match is 2/9.the probability that he will lose the tennis match is [tex]\frac{7}{9}[/tex]

Probability is the likelihood that something will occur. When we're not sure how something will turn out, we can discuss the likelihood of different outcomes, or their probabilities. Statistics is the study of events that follow a probability distribution.

p(he will win the tennis match )= 2/9.

P(he will lose the tennis match) =1 - 2/9

[tex]\frac{9}{9} -\frac{2}{9}[/tex]

=[tex]\frac{7}{9}[/tex]

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Answer:

The value of the constant of proportionality is [tex]k=5[/tex]

Step-by-step explanation:

we know that

A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]

The value of the constant k is equal to the value of the slope

In this problem we have

[tex]y=5x[/tex] ------> is a linear direct variation

The slope is [tex]m=5[/tex]

therefore

The value of the constant of proportionality is [tex]k=5[/tex]

The constant of proportionality on the equation is 5.

To obtain the constant of proportionality ;

The direct relationship between y and x is :

y α x

y = kx - - - (1)

Where, y and x are variables and k is the constant of proportionality.

Comparing equation (1) with the equation given :

y = kx

y = 5x

We can observe that, k in the equation (1) equals to 5 in the given equation.

Hence, the constant of proportionality is 5

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Answer:

She rode [tex]1\frac{1}{2}[/tex] miles from her friends house to school.

Step-by-step explanation:

Geneva rode her bike a total =  [tex]2\frac{1}{2}[/tex] miles

She rode from her house to the park = [tex]\frac{4}{5}[/tex] miles

She rode from the park to her friend's house = [tex]\frac{1}{5}[/tex] miles

total distance from her house to her friend's house = [tex]\frac{4}{5}+\frac{1}{5}[/tex] = 1 mile

Finally she rode the rest of the way to her school.

distance between her friend's house to school = [tex]2\frac{1}{2}-1[/tex] = [tex]\frac{3}{2}[/tex] miles

[tex]1\frac{1}{2}[/tex] miles she rode from her friends house to school

The time required for the principal amount to double is 13 years 10 months and 10 days approx.

Compound interest simply refers to the fact that an investment, loan, or bank account's interest accrues exponentially over time as opposed to linearly over time. The term "compound" is crucial here.

CI Formula. C.I. = Principal (1 + Rate)^time − Principal.

Given, A sum of $5000 is invested at an interest rate of 5% per year.

hence, principle = 5000

Amount = 10000

rate = 5%

let's assume the time is x

Let's solve for time

A(t)= 10000 = 5000* e⁵ˣ/¹⁰⁰

2 = (e)^x/20

taking logs on both sides

ln 2 = x / 20

x = 20 ln2

time required = 20 ln2 = 20 * 0.69 = 13.86

Therefore, To make the amount double at the interest rate of 5% we need time approx 13 years 10 months, and 10 days.

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To find the probability of the sample mean price being between $2.691 and $2.732, calculate the Z-scores for each bound and use a Z-table or statistical software. This utilizes the central limit theorem and the concept of Z-scores in statistics.

The question is about finding the probability that the mean price for a sample of 32 gas stations was between $2.691 and $2.732, assuming the population mean was $2.715 and standard deviation was $0.049. To solve this, we use the Z-score formula for each sample mean, Z = (X - μ) / (σ/√n), where μ is the population mean, σ is the population standard deviation, and n is the sample size. Then, we find the area between these Z-scores using a Z-table or statistical software to get the probability.

To calculate:

Refer to a Z-table or software with these Z-scores to find the probability of the sample mean price being between $2.691 and $2.732. This method exemplifies how statisticians use Z-scores and the central limit theorem to estimate probabilities relating to sample means.

The set of all real numbers less than or equal to -27 in interval notation is (-∞, -27].

The set of all real numbers that are less than or equal to -27 can be represented in interval notation as (-∞, -27]. In interval notation, a square bracket indicates that the endpoint is included in the set. In this case, -27 is included because the question states 'less than or equal to.' The symbol '-∞' represents negative infinity, indicating that there is no lower bound on the set.

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The equation represents a direct variation is y = x/3. So option (b) is correct.

A direct variation is a relationship between two variables where one variable is a constant multiple of the other. To identify a direct variation, we look for equations in the form y = kx, where k is a constant.

Let's analyze each equation:

A) y = x - 1

This equation doesn't represent a direct variation because it includes a constant term (-1) in addition to the variable x.

B) y = x/3

This equation represents a direct variation because it's in the form y = kx, where k = 1/3.

C) y = x + 5

Similar to option A, this equation doesn't represent a direct variation because it includes a constant term (+5) in addition to the variable x.

D) y = 2x - 3

Similar to option A, this equation doesn't represent a direct variation because it includes a constant term (-3) in addition to the variable x.

Therefore, the correct answer is B) y = x/3, as it represents a direct variation.

Answer:

Brenda      Claire      Jim     Micah        Peter

 4.3             5.5       6 1/6     15/4           4 3/8

 4.3             5.5       6.17      3.75        4.375   (all values in decimal)

A. order the weights of the donations from greatest to least.

6.17 , 5.5 , 4.375 , 4.3 , 3.75

B. Which of the friends will receive a free movie coupon? Which will not?

As given - Each person who donates at least 4.25 pounds of aluminum receives a free movie coupon.

So, Brenda, Claire, Jim and Peter will receive coupons and Micah will not receive.

C.What if? Would the person with the smallest donation win a movie coupon if he or she had collected 1/2 pound more of aluminum?

Micah got 3.75 pounds and if he got 0.5 pounds more, then he would have;

[tex]3.75+0.5=4.25[/tex] pounds

Then he would have got the coupon as a minimum of 4.25 pounds is required.

Answer-

The company repaid the money in 3 years.

Solution-

A company borrowed $25,000 at 3.5% and was charged $2,625 in interest.

Considering the interest as simple interest,

[tex]\text{interset}=\dfrac{\text{Principal}\cdot \text{Rate of interest}\cdot \text{Time period}}{100}[/tex]

Here,

Interest = $2625

Principal = $25000

Rate of interest = 3.5% annually

Putting the values,

[tex]\Rightarrow 2625=\dfrac{25000\times 3.5\times t}{100}[/tex]

[tex]\Rightarrow t=\dfrac{2625\times 100}{25000\times 3.5}[/tex]

[tex]\Rightarrow t=3\ years[/tex]

Therefore, the company repaid the money in 3 years.

Answer:

Hence, option: D is true.

D) 3×(2)^t = 1,536; t = 9

Step-by-step explanation:

Jane is saving her money in order to purchase a new racing bike. She initially saves $3 and plans to double the amount she saves each month.

he bike Jane wants is $1,536 at the local bike shop.

The equation that represents this situation, and after how many months, t, will Jane have enough money to purchase the bike is:

D) 3×(2)^t = 1,536; t = 9

since in the first month the cost is:

[tex]C_1=3\times 2[/tex]

in next month the cost is:

[tex]C_2=3\times 2\times 2\\\\C_2=3\times 2^2[/tex]

in t=3 months the expression is:

[tex]C_3=3\times 2^2\times 2\\\\C_3=3\times 2^3[/tex]

Hence in general we say:

[tex]C_t=3\times 2^t[/tex]

Hence,

[tex]3\times 2^t=1536\\\\2^t=\dfrac{1536}{3}\\\\2^t=512\\\\2^t=2^9\\\\t=9[/tex]

Hence, option: D is true.

D) 3×(2)^t = 1,536; t = 9

The domain of the function y = ln (-x + 3/2) is all real numbers less than 3/2, represented in interval notation as (-∞, 3/2). This is because the argument of the logarithm must be positive, leading to an inequality that restricts x to be less than 3/2.

The domain of a function refers to the complete set of possible values of the independent variable. For the function y = ln (-x + 3/2), we must consider where the argument of the natural logarithm, -x + 3/2, is greater than zero. This is because the natural logarithm function is defined only for positive arguments.

To find the domain, set the argument of the logarithm function greater than zero: -x + 3/2 > 0. Solving this inequality, we find -x > -3/2, hence x < 3/2. This means that the domain of the function is all real numbers less than 3/2, which can be written in interval notation as (-∞, 3/2).

The domain of a function is the set of all possible input values. In the function y = ln(-x + 3/2), the natural logarithm function is defined only for positive numbers, so to find the domain, we need to identify the values of x that make the argument of the logarithm greater than zero.

Solve for the argument inside the logarithm to be greater than zero: -x + 3/2 > 0

Simplify the inequality to find the valid domain: x < 3/2

Therefore, the domain of the function y = ln(-x + 3/2) is x < 3/2.

The coefficient of the squared term in the parabola's equation, with a vertex at (3,5) and passing through the point (-1,6), is 1/16.

The student is asking about the coefficient of the squared term in the parabola's equation given that the vertex is at (3,5) and another point on the parabola is (-1,6). Since we know the vertex, we can write the vertex form of a parabolic equation as:

y = a(x - h)
^2 + k

Plugging the vertex into the equation, we get:

5 = a(3 - 3)
^2 + 5

This simplifies down to:

5 = a(0) + 5

So, we cannot determine 'a' from the vertex alone. However, we can use the other point to find 'a'. Plugging the coordinates (-1,6) into the vertex form, we get:

6 = a(-1 - 3)^2 + 5

6 = a(-4)^2 + 5

6 = 16a + 5

1 = 16a

a = 1/16

Therefore, the coefficient 'a' is 1/16.