Each side of a square is lengthened by 2 inches. the area of the new, larger square is 121 square inches. find the length of a side of the original square.

We are finding the volume of a solid obtained by rotating a region bounded by specific curves about the x-axis. This involves the method of disk washers and the calculation of an integral. However, the calculation is impossible with this exact set of curves due to the undefined values at x = π/2 and x = -π/2.

To answer your question, let's first understand what is happening. We are taking the region between the curves y=sec(x), y=1, x=-1, and x=1 and rotating it about the x-axis. This creates a type of solid shape called a solid of revolution. We can find the volume of such a shape using the method of cylindrical shells or disk washers.

The volume V of the solid obtained by rotating about the x-axis the region confined by the given curves is given by the formula:

V = ∫ (from a to b) π [R(x)² - r(x)²] dx

where R(x) is the distance from the x-axis to the outer curve (y=sec(x)), and r(x) is the distance from the x-axis to the inner curve (y=1).

However, calculating the integral ∫ (from -1 to 1) π [sec(x)² - 1] dx directly can be difficult because the function sec(x) is undefined at x = π/2 and x = -π/2.

A typical way around such difficulties is to use a suitable trigonometric substitution, but in this case, the function sec(x) is periodic with a period of 2π, so we can't avoid these points, both of which lie in the interval from -1 to 1. Hence, it is impossible to find the volume of the solid as stated by rotating about the x-axis the region between the curves y = sec(x), y = 1, x = -1, and x = 1.

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Defective rate can be expected to keep an eye on a Poisson distribution. Mean is equal to 800(0.02) = 16, Variance is 16, and so standard deviation is 4.
X = 800(0.04) = 32, Using normal approximation of the Poisson distribution Z1 = (32-16)/4 = 4.
P(greater than 4%) = P(Z>4) = 1 – 0.999968 = 0.000032, which implies that having such a defective rate is extremely unlikely.

If the defective rate in the random sample is 4 percent then it is very likely that the assembly line produces more than 2% defective rate now.

The probability that the defective rate exceeds 4% in the sample is approximately 0.0006, indicating a significant deviation from the expected 2%.

To solve this problem, we need to use the concept of binomial distribution and the normal approximation to the binomial distribution due to the large sample size.

Step 1: Understanding the problem

- The assembly line historically has a defective rate of 2%.

- A random sample of 800 components is drawn.

- We are interested in the probability that the defective rate is greater than 4%.

Step 2: Calculate the parameters

- Population defective rate (historical rate): [tex]\( p = 0.02 \)[/tex]

- Sample size: [tex]\( n = 800 \)[/tex]

- Sample defective rate (given): [tex]\( \hat{p} = 0.04 \)[/tex]

Step 3: Probability that defective rate is greater than 4%

- We need to find [tex]\( P(\hat{p} > 0.04) \).[/tex]

Since [tex]\( \hat{p} \)[/tex] is approximately normally distributed (by the Central Limit Theorem because [tex]\( n \)[/tex] is large), we can use the normal approximation to the binomial distribution.

Step 4: Calculate standard error of sample proportion

The standard error of the sample proportion [tex]\( \hat{p} \)[/tex] is given by:

[tex]\[ SE(\hat{p}) = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \][/tex]

Substitute the values:

[tex]\[ SE(\hat{p}) = \sqrt{\frac{0.04 \cdot 0.96}{800}} \][/tex]

[tex]\[ SE(\hat{p}) = \sqrt{\frac{0.0384}{800}} \][/tex]

[tex]\[ SE(\hat{p}) \approx 0.0062 \][/tex]

Step 5: Z-score calculation

To find the Z-score for [tex]\( \hat{p} = 0.04 \)[/tex]:

[tex]\[ Z = \frac{\hat{p} - p}{SE(\hat{p})} \][/tex]

[tex]\[ Z = \frac{0.04 - 0.02}{0.0062} \][/tex]

[tex]\[ Z \approx 3.23 \][/tex]

Step 6: Find the probability

Now, find the probability that [tex]\( \hat{p} > 0.04 \)[/tex]:

[tex]\[ P(\hat{p} > 0.04) = P(Z > 3.23) \][/tex]

Using the standard normal distribution table or a calculator:

[tex]\[ P(Z > 3.23) \approx 0.0006 \][/tex]

Conclusion:

The probability that the defective rate in the sample is greater than 4% is approximately [tex]\( 0.0006 \)[/tex], or [tex]\( 0.06\% \)[/tex].

Interpretation:

Since the probability is very low, it suggests that a defective rate of 4% in the sample is highly unlikely to occur if the true defective rate on the assembly line is 2%. This could indicate a potential issue or change in the process affecting the defective rate, warranting further investigation or quality control measures.

The density of a material is equal to its mass divided by its volume.

The equation that represents the relationship between density (D), mass (M), and volume (V) is: D = M/V.

This equation shows that the density of a material is equal to its mass divided by its volume.

For example, if you have an object with a mass of 10 grams and a volume of 2 cubic centimeters, you can calculate its density by dividing the mass by the volume: D = 10g / 2cm³ = 5g/cm³.

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The density (D) of a material is directly proportional to the mass (M) and inversely proportional to its volume (V). The mathematical equation for this relationship is D = k * (M / V).

The density (D) of a material is directly proportional to the mass (M) and inversely proportional to its volume (V). To translate this into a mathematical equation, we can write:

D = k * (M / V)

Where D represents the density, M represents the mass, V represents the volume, and k represents the proportionality constant.

This equation demonstrates that as the mass of the object increases, the density also increases, while as the volume increases, the density decreases.

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To calculate the expected return, multiply each amount that can be won by its corresponding probability, and sum these values. The expected return of the game is $1 1/6, which corresponds to answer choice (b).

The student is asking how to calculate the expected return in a game with different probabilities of winning different amounts. To find the expected return, you multiply each outcome by its probability and then sum these products. The possible wins are $1, $5, and $0, with probabilities of 1/3, 1/6, and 1/2, respectively.

To calculate the expected return:

Add up these expected values to get the total expected return:

$1/3 + $5/6 + $0 = $2/6 + $5/6 = $7/6

The expected return is $7/6, which simplifies to $1 1/6. Therefore, the correct answer is (b).

The holes of the graph are located at (5, 3) and (-1, 11)

How to determine the hole of the graph

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

The graph

The holes of the graph are at (5, 3) and (-1, 11)

This is so because the function is not defined at this value

Also, we can see that

The function has it vertical asymptotes are x = 2 and x = -2

Answer:

2

Step-by-step explanation:

Took the test ;)

Answer:

The rate of change is 25. It means the cost increased by $25 per day.

Step-by-step explanation:

The given table is

Time (days)       Cost ($)

       3                   75

       4                  100

       5                  125

       6                  150

It means the graph of this constant function passing through the points (3,75) and (4,100).

If a line passing through two points, then the slope of the line is

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{100-75}{4-3}[/tex]

[tex]m=\frac{25}{1}[/tex]

[tex]m=25[/tex]

Therefore the rate of change is 25. It means the cost increased by $25 per day.

Final answer:

The probability that a randomly selected blue marble came from the first box is 7/31, which is approximately 0.2258 when rounded to four decimal places.

Explanation:

The probability that the blue marble came from the first box can be found using Bayes' theorem and the concept of conditional probability. First, we need to determine the probability of drawing a blue marble from either box (P(Blue)). Then, we calculate the probability of drawing a blue marble from the first box (P(Blue|First box)). Finally, we apply Bayes' theorem to find the probability that the blue marble came from the first box (P(First box|Blue)).

Here are the relevant probabilities:

P(First box) = 1/2 (since there are only two boxes)

P(Second box) = 1/2

P(Blue|First box) = 3/5 (3 blue out of 5 total marbles)

P(Blue|Second box) = 2/7 (2 blue out of 7 total marbles)

Using these probabilities, we calculate P(Blue):

P(Blue) = P(Blue|First box) * P(First box) + P(Blue|Second box) * P(Second box) = (3/5) * (1/2) + (2/7) * (1/2) = 3/10 + 1/7 = 21/70 + 10/70 = 31/70

Now, we apply Bayes' theorem to get P(First box|Blue):

P(First box|Blue) = [P(Blue|First box) * P(First box)] / P(Blue) = [(3/5) * (1/2)] / (31/70) = (3/10) / (31/70) = (3/10) * (70/31) = 21/310 = 7/31 or approximately 0.2258 (rounded to four decimal places)

Therefore, the probability that the marble came from the first box is 7/31.

Answer:

There are 22 cats and 8 dogs.

Step-by-step explanation:

Let the cats be represented by = c

Let the dogs be represented by = d

Given, that the pet store has 30 pets.

First equation forms :

[tex]c+d=30[/tex]            ........(1)

Also given, the cats cost $50 each and dogs cost $100 each and the total for all is $1900. Now second equation forms:

[tex]50c+100d=1900[/tex]     .........(2)

From equation (1) we get [tex]c=30-d[/tex]

Putting this value of c in equation 2:

[tex]50(30-d)+100d=1900[/tex]

[tex]1500-50d+100d=1900[/tex]

=> [tex]50d=400[/tex]

=> [tex]d=8[/tex]

Now,[tex]c+d=30[/tex]

So, [tex]c=30-8[/tex]

[tex]c=22[/tex]

Hence, there are 22 cats and 8 dogs.

Answer:

there isn't a number to multiply

Step-by-step explanation:

The solutions to the quadratic equation x² = 7x + 4 are found by using the quadratic formula, which results in two solutions: [tex](\frac{7 + \sqrt{65}}{2 },\frac{7 - \sqrt{65}}{2 })[/tex]

The solutions to the quadratic equation x² = 7x + 4 can be found by first rewriting the equation in standard form as x² - 7x - 4 = 0. To solve this quadratic equation, we can factor it, complete the square, or use the quadratic formula. In this case, let's factor the equation if possible. Unfortunately, this quadratic does not factor neatly. Therefore, we apply the quadratic formula which is [tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex] , where a = 1, b = -7, and c = -4.

After substitution, we get [tex]x = \frac{7 \pm \sqrt{49 + 16}}{2}[/tex]. This simplifies to x = [tex]\frac{7 \pm \sqrt{65}}{2}[/tex], resulting in two solutions: [tex]x = \frac{7 + \sqrt{65}}{2 }[/tex]and [tex]x = \frac{7 - \sqrt{65}}{2 }[/tex]. Therefore, the solution set is [tex](\frac{7 + \sqrt{65}}{2 },\frac{7 - \sqrt{65}}{2 })[/tex]

The simplified expression is (x - 5) / (2x).

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.com

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

The expression given is 3 (x - 5) / (6x)

We can simplify this expression by following the order of operations, which is a set of rules that tells us which operations to perform first in a mathematical expression.

The order of operations.

Perform any calculations inside parentheses first.

Exponents (ie: powers and square roots, etc.)

Multiplication and Division (from left to right)

Addition and Subtraction (from left to right)

Using the order of operations, we can simplify the expression as follows:

We start by simplifying the expression inside the parentheses.

x - 5 represents five less than x.

Next, we multiply the result of step 1 by 3.

= 3 (x - 5)

= 3x - 15

Finally, we divide the result of step 2 by the product of 6 and x.

= (3x - 15) / (6x)

= (3(x - 5)) / (6x)

= (x - 5) / (2x)

Therefore,

The simplified expression is (x - 5) / (2x).

Learn more about expressions here:

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By using the continuous compound interest the balance is growing $4,673.56 after 6 years.

Interest that compounded continuously to the principal amount. This interest rate provides exponential growth to period of time.

Formula of continuous compound interest rate;

[tex]P(t) = P_0e^{rt}[/tex] , where P₀ is the principal amount, r is the interest rate and t is the time period.

Given that the principal amount is $20000 and and interest rate 3.5% in a year.

And here we use formula of continuous compound interest rate;

[tex]P(t) = P_0e^{rt}[/tex]

Here, we have the value P₀ = $20000 , r = 3.5 % / 100 = 0.035% in a year and t = 6 years

Substitute these above values in the formula;

p(t) = $20000 × [tex]e^{0.035}[/tex] ×[tex]e^{6}[/tex]

P{t} = $24673.56

P{t} = $24673.56 nearest one cent

The final balance is $24673.56.

Therefore, the total continuous compound interest is $4,673.56.

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