How many degrees greater is the measure of one interior angle of a regular hexadecagon (a polygon with 16 sides) than the measure of one interior angle of a regular dodecagon (a polygon with 12
Sides)?

Answer: 80, 80

Step-by-step explanation:

The perimeter of a rectangle/square formula:

2L+2w=P                     >P=320

2L+2w=320                 >solve for one of variables let's pick L

2L = 320 - 2w              

L =  (320 - 2w)/2          >Simplify the dividing by 2            

L = 160-w

You also need Area formula:

A =  L(w)                            >Substitute from what we found from Perimeter

                                            formula

A = (160-w)(w)                   >Distribute w

A =  160w - w²                   >Maximum are happens at the vertex of this

                                             quadratic

A = w² - 160w

Vertex x formula in (x, y) for vertex:

a=1  b= -160     from A = w² - 160w   from standard form:  ax²+bx+c

[tex]w =- \frac{b}{2a}\\\\w= - \frac{-160}{2(1)}[/tex]

w=80

Substitute back into Perimeter to find L

L = 160-w

L = 160 - 80

L = 80

To compute 101^(4,800,000,023) mod 35 using the Chinese Remainder Theorem, we need to first decompose 35 into its prime factors: 35 = 5 × 7.

Next, we need to compute 101^(4,800,000,023) mod 5 and 101^(4,800,000,023) mod 7 separately.

To compute 101^(4,800,000,023) mod 5, we can use Fermat's Little Theorem, which states that if p is a prime number and a is a positive integer not divisible by p, then a^(p-1) ≡ 1 mod p. Since 5 is prime and 101 is not divisible by 5, we have 101^(4) ≡ 1 mod 5. Therefore, 101^(4,800,000,023) ≡ 101^(4 × 1,200,000,005 + 3) ≡ (101^4)^1,200,000,005 × 101^3 ≡ 1^1,200,000,005 × 101^3 ≡ 1 × 101^3 ≡ 1 mod 5.

To compute 101^(4,800,000,023) mod 7, we can use Euler's Totient Theorem, which states that if a and m are coprime positive integers, then a^φ(m) ≡ 1 mod m, where φ(m) is Euler's totient function. Since 7 is prime and φ(7) = 6, we have 101^6 ≡ 1 mod 7. Therefore, 101^(4,800,000,023) ≡ 101^(6 × 800,000,003 + 5) ≡ (101^6)^800,000,003 × 101^5 ≡ 1^800,000,003 × 101^5 ≡ 101^5 ≡ 4 mod 7.

Now we can use the Chinese Remainder Theorem to combine the results. Let x ≡ 1 mod 5 and x ≡ 4 mod 7. Then we can write x = 5k + 1 for some integer k. Substituting this into the second congruence, we get 5k + 1 ≡ 4 mod 7, or equivalently, k ≡ 6 mod 7. Therefore, x = 5k + 1 ≡ 5(6) + 1 ≡ 31 mod 35.

Hence, 101^(4,800,000,023) mod 35 = x = 31.

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Answer:

Step-by-step explanation:

Use the parent function, then shift as necessary

Parent function:

y= log₅ x     >put in exponential form

[tex]5^{y} =x[/tex]  

Since our variables are a bit backwards work backwards

For y = 0    x=1

For y=1   x= 5

For y= 2  x=10  and so on and so forth

Put into T table

x   |    y

1    |    0

5   |   1

10  | 2

This is your parent:   There is a stretch of 2 and a shift of 1 to right for your function

so mulitply y by 2 and move over to right by 1

x   |    y

1  +1   |    0 *2

5 +1   |     1*2

10+ 1   |      2*2

x   |    y

2   |    0

6   |   2

11  |  4

Your asymptote is x=1 because you shifted right 1

To solve the equation sin x(sin x 1) = 0, we need to find the values of x that satisfy the equation. The product of sin x and (sin x 1) equals zero when either sin x equals zero or sin x 1 equals zero. So we have two possibilities: sin x = 0 or sin x = 1.

If sin x = 0, then x can be any integer multiple of π, because sin x = 0 when x = nπ.

If sin x = 1, then x must be π/2 radians or (π/2) + 2πn radians for some integer n.

Therefore, the solutions to the equation sin x(sin x 1) = 0 are x = nπ or x = (π/2) + 2πn, where n is an integer.

To solve the equation sin x(sin x 1) = 0, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So we set sin x = 0 and sin x 1 = 0 and solve for x.

If sin x = 0, then x = nπ for some integer n. This is because sin x = 0 when x = nπ, where n is an integer.

If sin x 1 = 0, then sin x = 1, which means x is either π/2 radians or (π/2) + 2πn radians for some integer n.

Therefore, the solutions to the equation sin x(sin x 1) = 0 are x = nπ or x = (π/2) + 2πn, where n is an integer.

In conclusion, the solutions to the equation sin x(sin x 1) = 0 are x = nπ or x = (π/2) + 2πn, where n is an integer. This is because the product of sin x and (sin x 1) equals zero when either sin x equals zero or sin x 1 equals zero. We use the zero-product property to find the values of x that satisfy the equation.

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A value of theta in the first quadrant that satisfies the equation is approximately 0.4548 radians or 26.1 degrees.

Starting with the equation:

6sin(2θ) = 5

Divide both sides by 6:

sin(2θ) = 5/6

We know that sine is positive in the first and second quadrants. Since we are looking for a value of theta in the first quadrant, we can use the inverse sine function to solve for 2θ:

2θ = sin⁻¹(5/6)

Using a calculator, we get:

2θ ≈ 0.9095 radians

Dividing by 2, we get:

θ ≈ 0.4548 radians

To convert to degrees, we can use the conversion formula:

1 radian = 180/π degrees

So:

θ ≈ 0.4548 radians = (180/π) * 0.4548 degrees ≈ 26.1 degrees

Therefore, a value of theta in the first quadrant that satisfies the equation is approximately 0.4548 radians or 26.1 degrees.

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The statement "Chi-square is nonnegative in value; it is zero or positively valued" is true.

No, the chi-square value is always nonnegative, meaning it can only be zero or a positive value.

Chi-square is a statistical measure used in hypothesis testing and is calculated by summing the squared differences between observed and expected frequencies.

The chi-square value is a nonnegative statistical measure that is commonly used in hypothesis testing to assess the relationship between observed and expected frequencies in categorical data.

It is calculated by summing the squared differences between the observed frequencies and the expected frequencies.

The resulting value follows a chi-square distribution, which is always nonnegative.

A value of zero indicates that the observed and expected frequencies match perfectly, while positive values indicate increasing deviations from the expected frequencies.

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11. The first question an epidemiologist should ask before making judgments about any apparent patterns in this data is: "What is the source and validity of the data?"

It is crucial to assess the reliability and accuracy of the data used to create the map. Validity refers to whether the data accurately represent the true occurrence of Lyme disease cases in each county. Epidemiologists need to ensure that the data collection methods were standardized, consistent, and reliable across all counties.

They should also consider the source of the data, whether it is from surveillance systems, medical records, or other sources, and evaluate the quality and completeness of the data. Without reliable and valid data, any interpretation or conclusion drawn from the map would be compromised.

12. Population size in each county is not a concern when looking for patterns with this map because the data is presented as cases per 100,000 population.

By standardizing the data, it eliminates the influence of population size variations among different counties. The use of rates per 100,000 population allows for a fair comparison between counties with different population sizes. It provides a measure of the disease burden relative to the population size, which helps identify areas with a higher risk of Lyme disease.

Therefore, the focus should be on the rates of Lyme disease cases rather than the population size in each county.

13. The map provides information about the incidence or prevalence of Lyme disease in different counties in Virginia in 2012. It specifically presents the number of reported cases per 100,000 population, categorized into different ranges.

The map allows for a visual representation of the spatial distribution of Lyme disease cases across the state. It highlights areas with higher rates of Lyme disease and can help identify regions where the disease burden is more significant. It provides a broad overview of the relative risk and distribution of Lyme disease across the counties in Virginia during that specific time period.

14. Two additional pieces of information that would be helpful to interpret this map are:

a) Temporal trends: Knowing the temporal aspect of the data would provide insights into whether the patterns observed on the map are consistent over time or if there are variations in incidence rates between different years. This information would help identify any temporal trends, such as an increasing or decreasing trend in Lyme disease cases. It could also assist in determining if the patterns observed are stable or subject to fluctuations.

b) Risk factors and exposure data: Understanding the underlying risk factors associated with Lyme disease transmission and exposure patterns in different regions would enhance the interpretation of the map. Factors such as outdoor recreational activities, proximity to wooded areas, tick bite prevention measures, and public health interventions can influence the incidence of Lyme disease.

Gathering data on these factors, such as survey results on behaviors and preventive measures, would help explain any variations in the reported cases and provide context for the observed patterns.

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True. the relationship of your data. Even if the data is not linear, there may still be a correlation that can be used to calculate the relationship between the variables.

Correlation refers to the strength and direction of the relationship between two variables, and it can be measured using a variety of correlation coefficients such as Pearson's correlation coefficient, Spearman's rank correlation coefficient, and Kendall's tau correlation coefficient. These coefficients can be used to quantify the strength and direction of the relationship between the variables, regardless of whether the relationship is linear or not. However, it's worth noting that correlation does not imply causation. Just because two variables are correlated does not necessarily mean that one variable causes the other variable. Additional analysis is needed to establish causality.

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our antiderivative is correct.

To find the antiderivative of the function f(x) = 7x^2sqrt(x), we can use integration by substitution. Let u = x^2, then du/dx = 2x, and dx = du/(2x).

Substituting expressions into the integral,

∫ 7x^2sqrt(x) dx = ∫ 7u^(1/2) du/(2x)

= (7/2) ∫ u^(1/2)/x du

= (7/2) ∫ u^(1/2) u^(-1/2) du (since x = u^(1/2))

= (7/2) ∫ du

= (7/2) u + C (where C is the constant of integration)

Substituting back u = x^2, we get:

= (7/2) x^2 + C

Therefore, the most general antiderivative of the function f(x) = 7x^2sqrt(x) is (7/2) x^2 + C.

To check our answer, we can differentiate (7/2) x^2 + C with respect to x:

d/dx [(7/2) x^2 + C] = 7x

Substituting x = sqrt(x^2), we get:

f(x) = 7sqrt(x^2) x = 7x^2sqrt(x)

which is the original function we started with. Hence, our antiderivative is correct.

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P(t1 < t-1 < t²) is the probability that t1 is less than t raised to the power of -1, which is less than t squared.

To calculate the probability P(t1 < t-1 < t²), you need to determine the range of values for t that satisfy this inequality. Start by isolating t:

1. t1 < t-1 → t1 + 1 < t (by adding 1 to both sides)
2. t-1 < t² → 1/t < t (by rewriting t-1 as 1/t)

Now, find the range of t values that satisfy both inequalities. Graph these inequalities on a number line, and identify the intersection of the two ranges. The probability P(t1 < t-1 < t²) will be the proportion of this intersection relative to the total possible range of values for t.

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A loan of $17,500 with a fixed annual percentage rate (APR) of 9% for a term of 5 years will result in a monthly payment of approximately $355.62.

To calculate the monthly payment, we can use the formula for the monthly payment of a fixed-rate loan, which takes into account the loan amount, the interest rate, and the loan term. The formula is:

M = [tex]P * (r * (1 + r)^n) / ((1 + r)^n - 1)[/tex]

Where:

M = Monthly payment

P = Loan amount

r = Monthly interest rate (APR divided by 12)

n = Total number of payments (loan term in months)

In this case, the loan amount (P) is $17,500, the annual percentage rate (APR) is 9%, and the loan term is 5 years (or 60 months). To calculate the monthly interest rate (r), we divide the APR by 12 (months). Therefore, r = 0.09 / 12 = 0.0075.

Plugging in the values into the formula, we get:

M = 17500 * (0.0075 * [tex](1 + 0.0075)^{60})[/tex] / ([tex](1 + 0.0075)^{60}[/tex] - 1)

M ≈ $355.62

Therefore, the monthly payment for the loan of $17,500 at a fixed APR of 9% for 5 years is approximately $355.62.

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In this circuit, we have a MOSFET amplifier with given parameters: VDD = VSS = 5V, I0 = 500µA, RL = 7kΩ, RSig = 1kΩ. The MOSFET parameters are: [tex]VT = 2V, (W/L)*kn' = 4mA/V^2[/tex], and [tex]λ = 0V^{-1[/tex].

The circuit represents a common-source amplifier configuration with an n-channel MOSFET. It operates with a supply voltage of 5V, and the input signal is connected to a 1kΩ resistor. The load resistor is 7kΩ, and the MOSFET has a threshold voltage of 2V, a transconductance parameter of 4mA/V^2, and negligible channel-length modulation.

The common-source amplifier configuration uses the MOSFET in the triode region for signal amplification. With a bias current (I0) of 500µA flowing through the MOSFET, a voltage drop develops across RSig, generating an input signal voltage. The MOSFET operates in the saturation region, given VT = 2V. The transconductance parameter ((W/L)*kn') determines the amplification capability of the MOSFET, with a higher value resulting in higher gain. The load resistor RL sets the output impedance of the amplifier. In this case, RL = 7kΩ. The MOSFET's λ parameter, representing channel-length modulation, is negligible (λ = 0V^-1), indicating minimal dependence of the drain current on the drain-to-source voltage. Overall, this circuit configuration allows for amplification of the input signal and provides an amplified output signal at the drain of the MOSFET.

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To prove that triangle PQR is a right triangle, we need to show that it satisfies the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, we need to check if 12.5^2 + 30^2 = 32.5^2 holds true.

In triangle PQR, let's label the sides as follows: PQ = 12.5 cm, QR = 30 cm, and RP = 32.5 cm.

To determine if triangle PQR is a right triangle, we need to apply the Pythagorean theorem. According to the theorem, the sum of the squares of the two shorter sides should be equal to the square of the longest side, which is the hypotenuse.

Calculating the squares of the side lengths:

PQ^2 = (12.5 cm)^2 = 156.25 cm^2

QR^2 = (30 cm)^2 = 900 cm^2

RP^2 = (32.5 cm)^2 = 1056.25 cm^2

Now, we check if PQ^2 + QR^2 = RP^2:

156.25 cm^2 + 900 cm^2 = 1056.25 cm^2

Since the equation is true, i.e., both sides are equal, we can conclude that triangle PQR satisfies the Pythagorean theorem and is, therefore, a right triangle.

Therefore, triangle PQR is a right triangle based on the given side lengths.

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The fact that the median value of retirement accounts per household is $0 indicates that a significant percentage of households have no retirement accounts.

This means that there is a wide wealth gap in the country and many households are not saving for their retirement, or they are using other forms of savings such as real estate or investments.

While the mean value of retirement accounts is $94,500, this does not give a complete picture of the distribution of retirement account balances. The mean is highly influenced by extreme values or outliers, such as households with very high balances. Therefore, it is important to consider both the mean and median when analyzing the distribution of retirement account balances.

The median value of $0 suggests that there is a large number of households with no retirement accounts, which could be due to several reasons. For instance, some households may not have access to employer-sponsored retirement plans, or they may not have enough disposable income to contribute to individual retirement accounts. Additionally, some households may not prioritize saving for retirement or may choose to rely on other sources of income in retirement, such as Social Security.

The fact that a significant percentage of households do not have retirement accounts can have serious implications for their financial well-being in retirement. Without adequate savings, households may be forced to rely on Social Security or other forms of government assistance, which may not be sufficient to cover all their expenses. This underscores the importance of encouraging households to save for retirement, as well as providing access to retirement savings plans and education on financial planning.

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Based on the information provided, we have a pair of parallel lines intersected by a transversal. The angles formed by the transversal and the parallel lines are related to each other in specific ways.

In this case, we are given that angle ZA is equal to 75°. Since the figure has parallel lines, we can determine that angle ZA is corresponding to angle OA (denoted as angle ΟΑ), meaning they have the same measure. Therefore, angle OA is also 75°.

To summarize:

ZA = 75°

OA = 75°

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Answer:

The surface area of a cylinder can be calculated using the formula:

SA = 2πr^2 + 2πrh

where r is the radius of the base of the cylinder, h is the height of the cylinder,

Substituting r = 3 and h = 1 into the formula, we get:

SA = 2π(3)^2 + 2π(3)(1)

SA = 2π(9) + 2π(3)

SA = 18π + 6π

SA = 24π

Therefore, the surface area of the cylinder is 24π square units.

g∘f(x) or g(f(x)) is equal to 27x + 43.

To find g∘f(x) or g(f(x)), we need to substitute the function f(x) into the function g(x).

Given:

f(x) = 3x + 4

g(x) = 9x + 7

To find g∘f(x), we substitute f(x) into g(x) as follows:

g(f(x)) = g(3x + 4)

Now, we substitute 3x + 4 for x in the function g(x):

g(f(x)) = 9(3x + 4) + 7

Expanding and simplifying:

g(f(x)) = 27x + 36 + 7

g(f(x)) = 27x + 43

Therefore, g∘f(x) or g(f(x)) is equal to 27x + 43.

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The β-coordinate vector of x is [c1, c2] = [(3x1 - x2 - 5x3)/20, (x2 - 2x1)/10 + (3x1 - x2 - 5x3)/40]. This is the vector representation of x in the basis β.

To find the β-coordinate vector of x, we need to express x as a linear combination of b1 and b2. Let the β-coordinate vector of x be [c1, c2]. Then we have:

x = c1*b1 + c2*b2

Substituting the given values for b1 and b2, we get:

[x1, x2, x3] = c1*[2, -2, 4] + c2*[6, 1, -3]

This gives us a system of equations:

2c1 + 6c2 = x1
-2c1 + c2 = x2
4c1 - 3c2 = x3

We can solve this system using Gaussian elimination or other methods to get the values of c1 and c2. The solution is:

c1 = (3x1 - x2 - 5x3)/20
c2 = (x2 - 2x1)/10 + c1/2

Therefore, the β-coordinate vector of x is [c1, c2] = [(3x1 - x2 - 5x3)/20, (x2 - 2x1)/10 + (3x1 - x2 - 5x3)/40]. This is the vector representation of x in the basis β.

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The expected price of Tricki Corp, when it begins trading ex-rights, is $90.

We have,

When a stock goes ex-rights, the right to buy additional shares at a discounted price is no longer available to new investors.

Therefore, the value of the right is subtracted from the current stock price.

In this case,

To purchase one share of Tricki Corp, an investor would need to buy 10 rights at a cost of $10 each, for a total cost of $100.

With the subscription price of $90, the total cost of one share is $190.

Before going ex-rights, the stock price is $100.

After going ex-rights, the value of the right is $190 - $100 = $90.

The expected price of Tricki Corp, when it begins trading ex-rights.

= $100 - $90

= $10.

The new stock price will be $100 - $10 = $90.

Thus,

The expected price of Tricki Corp, when it begins trading ex-rights, is $90.

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1/12 is a fraction that is smaller than 1/6, and the product of 1/6 and 1/2 relates to 1/6 by being a fraction that is smaller than it.

The product of 1/6 and 1/2 is 1/12, which is not directly related to 1/6200.

The divide 1 by 1/6200, the result would be 6200, which is 12 multiplied by 516.67.

This shows that 1/6200 is equivalent to 1/12 of 516.67, which is a way to indirectly relate it to the product of 1/6 and 1/2.
The product of 1/6 and 1/2 relates to 1/6 because when you multiply these two fractions, you get a smaller fraction as a result. In this case, (1/6) x (1/2) = 1/12.

Which is smaller than both original fractions.

This demonstrates that when multiplying two fractions, the product is typically smaller than the original fractions.

The product of 1/6 and 1/2 which is (1/6) x (1/2) = 1/12 is smaller than 1/6.

This is because multiplying 1/6 by a fraction less than 1 (such as 1/2) results in a product that is smaller than the original fraction.

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11,572.28 = Pe^1.35Now we need to solve for P. Divide both sides by e^1.35:11,572.28/e^1.35 = PApproximating to the nearest cent:4,000.00 = PTherefore, the deposit was $4,000.00.

To solve this problem, we will use the formula for continuous compounding which is given as A = Pert. A = the amount after t years, P = principal amount, e = the constant, r = annual interest rate (as a decimal), t = number of years.Assuming that the amount deposited 15 years ago is P, we can substitute the values we know into the formula:A = Pert11,572.28 = Pe^(0.09*15)Simplifying:11,572.28 = Pe^1.35Now we need to solve for P. Divide both sides by e^1.35:11,572.28/e^1.35 = PApproximating to the nearest cent:4,000.00 = PTherefore, the deposit was $4,000.00.

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The volume of Sam's model is 136.5 cubic inches.

The volume of the prism is 6.5 * 7 * 9 = 409.5 cubic inches.

The volume of the rectangular pyramid is given by 1/3*Base area*height.

In this case, the base area of the pyramid is the same as the base of the prism which is 6.5*7 = 45.5 square inches.

The height of the pyramid is the same as the height of the prism which is 9 inches.

Substituting these values in the formula above we get:

1/3*45.5*9 = 136.5 cubic inches.

Therefore, the volume of Sam's model is 136.5 cubic inches.

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To determine the change in angular momentum (ΔL) of a cylinder from t = 0 to t = t0, a student can use the equation:

ΔL = I * Δω

where ΔL is the change in angular momentum, I is the moment of inertia of the cylinder, and Δω is the change in angular velocity.

To calculate Δω, the student needs to know the initial and final angular velocities, ω0 and ωt0, respectively. The change in angular velocity can be calculated as:

Δω = ωt0 - ω0

Once Δω is determined, the student can use the moment of inertia (I) of the cylinder to calculate ΔL using the equation mentioned earlier.

The moment of inertia (I) depends on the mass distribution and shape of the cylinder. For a solid cylinder rotating about its central axis, the moment of inertia is given by:

I = (1/2) * m * r^2

where m is the mass of the cylinder and r is the radius of the cylinder.

By substituting the known values for Δω and I into the equation ΔL = I * Δω, the student can calculate the change in angular momentum (ΔL) of the cylinder from t = 0 to t = t0.

It's important to note that this method assumes that no external torques act on the cylinder during the time interval. If there are external torques involved, the equation for ΔL would need to include those torques as well.

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(a) The number of bacteria at t = 0 hours is 1000.

b) The growth rate of the bacteria is 0.01.

c)  The graph will be an exponential growth.

d) The population after 4 hours is 1221.40 bacteria.

e) The number of bacteria will reach 1700 after about 23.5 hours.

(f)  The number of bacteria will double after about 69.3 hours.

(a) To determine the number of bacteria at t = 0 hours, we substitute t = 0 into the given model:

N(0) = [tex]1000e^{(0.01)(0)[/tex] = 1000e⁰ = 1000

So, the number of bacteria at t = 0 hours is 1000.

(b) The growth rate of the bacteria is the coefficient of t in the exponent, which is 0.01.

(c) The graph will be an exponential growth curve that starts at (0, 1000) and approaches infinity as t approaches infinity.

(d) To find the population after 4 hours, we substitute t = 4 into the given model:

N(4) = 1000[tex]e^{(0.01)(4)[/tex] ≈ 1221.40

So, the population after 4 hours is 1221.40 bacteria.

(e) To find when the number of bacteria will reach 1700, we set N(t) = 1700 and solve for t:

1700 = 1000[tex]e^{(0.01t)[/tex]

1.7 = [tex]e^{(0.01t)[/tex]

ln(1.7) = 0.01t

t ≈ 23.5

So, the number of bacteria will reach 1700 after about 23.5 hours.

(f) To find when the number of bacteria will double, we set N(t) = 2000 and solve for t:

2000 = [tex]e^{(0.01t)[/tex]

2 = [tex]e^{(0.01t)[/tex]

ln(2) = 0.01t

t ≈ 69.3

So, the number of bacteria will double after about 69.3 hours.

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Answer:

d. 0

Step-by-step explanation:

To solve the given trigonometric equation, let's use the trigonometric identity: sin²θ + cos²θ = 1. We can rewrite the equation provided as follows:

cos a + cos² B + cos² y = 3

Using the identity, we can rewrite it as:

1 - sin² a + 1 - sin² B + 1 - sin² y = 3

Simplifying further, we have:

3 - (sin² a + sin² B + sin² y) = 3

Rearranging the equation, we get:

sin² a + sin² B + sin² y = 3 - 3

sin² a + sin² B + sin² y = 0

Therefore, the value of sin² a + sin² B + sin² y is 0 (option d).

The town of Lantana still needs to raise $1,496 for the new playground.

To find out how much more money the town of Lantana needs to raise for a new playground, you need to add up the amount of money each school has raised and subtract that total from the total cost of the playground.So:

Total amount raised = $5,538 + $2,834 + $4,132

Total amount raised = $12,504

To find how much more is needed, you subtract the total amount raised from the total amount needed:

Total amount needed - Total amount raised = $14,000 - $12,504

= $1,496

So the town of Lantana still needs to raise $1,496 for the new playground.

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The relationship of angle 1 and angle 2 is same side interior angles.

From the information given, we have that;

The quadrilateral ABCD is a parallelogram.

Now, we need to know the properties of a parallelogram. These properties includes;

We can see from the diagram shown that;

<1 and <2 are same side interior angles and are thus supplementary.

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For the sample of weight of corn chips dispensed in dispensing machine, probability that the sample mean weight of these 100 bags falls between 10.50 and 10.80 ounces is equals to 0.4332.

We have a sample of weight of corn chips dispensed by the dispensing machine.

Dispensed weight of bag by the dispensing machine = 10 ounces

The sample of weight of bags follows the normal distribution with, sample mean, [tex] \bar x[/tex] = 10.5 ounces

standard deviations = 2 ounces

Randomly selected from this dispensing machine. Sample size, n = 100

We have to determine probability that the sample mean weight of these 100 bags falls between 10.50 and 10.80 ounces,

[tex]P( 10.50 < \bar x < 10.80),[/tex]

Using Z-score formula for sample mean in normal distribution, [tex]Z = \frac{ \bar x - \mu}{ \frac{\sigma}{\sqrt{n}} }[/tex]

where μ--> population mean

σ -->standard deviations

n --> Sample size

Now, the required probability is [tex]P( 10.50 < \bar x < 10.80)[/tex]

= [tex]P(\frac{ 10.50 - \mu }{\frac{\sigma}{\sqrt{n}}} < \frac{ \bar x - \mu}{ \frac{\sigma}{\sqrt{n}} } < \frac{ 10.80 - \mu }{\frac{\sigma}{\sqrt{n}}} )[/tex]

= [tex]P(\frac{ 10.50 - 10 }{\frac{2}{\sqrt{100}}} < z< \frac{ 10.80 - 10 }{\frac{2}{\sqrt{100}}} )[/tex]

= [tex]P(\frac{ 0.50 }{\frac{2}{10} }< z< \frac{ 0.80 }{\frac{2}{10}})[/tex]

= [tex]P(2.5 < z< 4)[/tex]

= 0.4332

Hence, required value is 0.4332.

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The limit of this expression as x approaches 0 is 1. To prove this, we can use L'Hospital's Rule.

Take the natural log of both sides and use the chain rule to simplify:

lim x→0 cot(3x)sin(9x) = lim x→0 ln(cot(3x)sin(9x))

Apply L'Hospital's Rule:

lim x→0 ln(cot(3x)sin(9x)) = lim x→0 [3cos(3x)cot(3x) - 9sin(9x)sin(9x)]/[3sin(3x)cot(3x) + 9cos(9x)sin(9x)]

Apply L'Hospital's Rule again:

lim x→0 [3cos(3x)cot(3x) - 9sin(9x)sin(9x)]/[3sin(3x)cot(3x) + 9cos(9x)sin(9x)] = lim x→0 [3(−sin(3x))cot(3x) - 9(cos(9x))sin(9x)]/[3(−cos(3x))cot(3x) + 9(−sin(9x))sin(9x)]

Simplify each side of the equation:

lim x→0 [3(−sin(3x))cot(3x) - 9(cos(9x))sin(9x)]/[3(−cos(3x))cot(3x) + 9(−sin(9x))sin(9x)] = lim x→0 −3/9

= -1/3

Since the limit of both sides of the equation is the same, the original limit must also be -1/3.

However, since cot(0) and sin(0) both equal 0, the limit of the original expression is 1.

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The limit of the expression lim(x→0) cot(3x) sin(9x) is 1.

We can use the properties of trigonometric functions to simplify the expression without needing to apply L'Hôpital's rule.

Recall that cot(x) = cos(x) / sin(x). Applying this to the expression:

lim(x→0) (cos(3x) / sin(3x)) sin(9x)

The sin(3x) term in the numerator and denominator cancels out:

lim(x→0) cos(3x) sin(9x) / sin(3x)

Next, we can simplify the expression further by applying the identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B) to sin(9x):

lim(x→0) cos(3x) (sin(3x)cos(6x) + cos(3x)sin(6x)) / sin(3x)

Now, we can cancel out the sin(3x) term in the numerator and denominator:

lim(x→0) cos(3x) (cos(6x) + cos(3x)sin(6x)) / 1

As x approaches 0, all trigonometric functions in the expression approach their respective limits. Therefore, we can evaluate the limit directly:

lim(x→0) cos(3x) (cos(6x) + cos(3x)sin(6x)) / 1 = cos(0) (cos(0) + cos(0)sin(0)) / 1 = 1(1 + 1(0)) = 1(1 + 0) = 1

Hence, the limit of the expression lim(x→0) cot(3x) sin(9x) is 1.

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