At the Hardey Fitness Center, the management did a survey of their membership. The average age of the female members was $40$ years old. The average age of the male members was $25$ years old. The average age of the entire membership was $30$ years old. What is the ratio of the female to male members? Express your answer as a common fraction.
Hint: It isn't 5/8

The situation could be described by the expression d+1/2 is an option (C). Lela walked 1 mile yesterday, and d miles fewer today.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

d+1/2 is an abbreviation for "d plus one-half."

It describes a situation in which a quantity (represented by d) is increased by half.

For example, if Lela walked d miles yesterday and wants to walk another half mile today, she might use the term d+1/2 to indicate her total distance walked today.

Alternatively, if Lela wanted to walk half as far today as she did yesterday, the equation would not apply since the quantity being added or subtracted is a variable amount (d/2) rather than a fixed amount (one-half).

Hence, the situation could be described by the expression d+1/2 is option (C). Lela walked 1 mile yesterday and d miles fewer today.

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The formula becomes:

A = 2π∫1^4 sqrt

Rotate the curve y = [tex]x^{3/27[/tex], 0 ≤ x ≤ 3, about the x-axis.

To find the surface area of the solid generated by rotating the curve y = [tex]x^3[/tex]/27, 0 ≤ x ≤ 3, about the x-axis, we can use the formula:

A = 2π∫[tex]a^b[/tex] f(x) √(1 + [f'(x)[tex]]^2[/tex]) dx

where f(x) is the function defining the curve, and a and b are the limits of integration.

In this case, we have:

f(x) =[tex]x^{3/27[/tex]

f'(x) = [tex]x^{2/9[/tex]

So, the formula becomes:

A = 2π∫0^3 ([tex]x^{3/27[/tex]) √(1 +[tex][x^{2/9}]^2[/tex]) dx

We can simplify the integrand by noting that:

1 + [[tex]x^2[/tex]/9[tex]]^2[/tex] = 1 + [tex]x^{4/81[/tex] = ([tex]x^4[/tex] + 81)/81

So, the formula becomes:

A = 2π/81 ∫[tex]0^3 x^3[/tex] √([tex]x^4[/tex] + 81) dx

This integral is not easy to evaluate by hand, so we can use numerical methods or a computer algebra system to obtain an approximate value.

Using a numerical integration tool, we find that:

A ≈ 23.392 square units

Therefore, the surface area of the solid generated by rotating the curve y = x^3/27, 0 ≤ x ≤ 3, about the x-axis is approximately 23.392 square units.

Rotate the curve y = 4 - [tex]x^2[/tex], 0 ≤ x ≤ 2, about the x-axis.

To find the surface area of the solid generated by rotating the curve y = 4 - x^2, 0 ≤ x ≤ 2, about the x-axis, we can again use the formula:

A = 2π∫[tex]a^b[/tex] f(x) √(1 + [f'(x)][tex]^2[/tex]) dx

In this case, we have:

f(x) = 4 - [tex]x^2[/tex]

f'(x) = -2x

So, the formula becomes:

A = 2π∫[tex]0^2[/tex] (4 - [tex]x^2[/tex]) √(1 + [-2x[tex]]^2[/tex]) dx

Simplifying the integrand, we get:

A = 2π∫0^2 (4 - x^2) √(1 + 4x^2) dx

This integral is also not easy to evaluate by hand, so we can use numerical methods or a computer algebra system to obtain an approximate value.

Using a numerical integration tool, we find that:

A ≈ 60.346 square units

Therefore, the surface area of the solid generated by rotating the curve y = 4 - [tex]x^2[/tex], 0 ≤ x ≤ 2, about the x-axis is approximately 60.346 square units.

Rotate the curve y = sqrt(x), 1 ≤ x ≤ 4, about the x-axis.

To find the surface area of the solid generated by rotating the curve y = sqrt(x), 1 ≤ x ≤ 4, about the x-axis, we can again use the formula:

A = 2π∫[tex]a^b[/tex] f(x) √(1 + [f'(x)[tex]]^2[/tex]) dx

In this case, we have:

f(x) = sqrt(x)

f'(x) = 1/(2sqrt(x))

So, the formula becomes:

A = 2π∫[tex]1^4[/tex] sqrt

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The area of a triangle with vertices (6, 6), (2, 4), and (0, 8) is 4√10 square units.

To find the area of a triangle with vertices (6, 6), (2, 4), and (0, 8), we can use the formula:

Area = 1/2 * base * height

First, we need to find the base and height of the triangle. We can use the distance formula to do this:

Base = distance between (6, 6) and (2, 4) = √[(6-2)^2 + (6-4)^2] = √20
Height = distance between (0, 8) and the line containing (6, 6) and (2, 4). To do this, we first find the equation of the line:

y - 6 = (6-4)/(6-2) * (x-6)
y - 6 = 1/2 * (x-6)
y = 1/2x + 3

Then we find the distance between point (0, 8) and the line y = 1/2x + 3:

Height = |1/2*0 - 1*8 + 3| / √(1^2 + 1/2^2) = 4√5/5

Now we can plug in the base and height into the formula:

Area = 1/2 * √20 * 4√5/5 = 4√10 square units

Therefore, the area of the triangle is 4√10 square units.

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The first principal component is the line passing through the points (-2,-2) and (2,2).

(a) To find the first principal component, we need to find the eigenvector of the covariance matrix that corresponds to the largest eigenvalue. First, we calculate the covariance matrix:

| 4 0 -4 |

| 0 0 0 |

|-4 0 4 |

The eigenvalues of this matrix are 8, 0, and 0. The eigenvector corresponding to the largest eigenvalue (8) is:

| 1 |

| 0 |

|-1 |

So, the first principal component is the line passing through the points (-2,-2) and (2,2).

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Computing the sine and cosine of 330∘ by using the reference angle.

a) Reference angle: 30 degrees

b) Quadrant: 4

c) sin(330°) = -1/2

d) cos(330°) = sqrt(3)/2

a) To find the reference angle, subtract the given angle (330°) from 360°, as it is in the fourth quadrant. So the reference angle is 360° - 330° = 30°.

b) Since 330° lies between 270° and 360°, it is in the fourth quadrant (answer 4).

c) To find sin(330°), use the reference angle of 30°. Since the fourth quadrant has a positive x-value and a negative y-value, the sine will be negative. So, sin(330°) = -sin(30°) = -1/2.

d) To find cos(330°), use the reference angle of 30°. Since the fourth quadrant has a positive x-value, the cosine will be positive. So, cos(330°) = cos(30°) = sqrt(3)/2.

Your answer:

a) Reference angle: 30 degrees

b) Quadrant: 4

c) sin(330°) = -1/2

d) cos(330°) = sqrt(3)/2

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Answer:

Angle 5 and 2 are vertical to each other.

Hope this helps : )

Step-by-step explanation:

Vertical angles are when angles are opposite of each other. So that makes angles 5 and 2 Vertical Angles.

If y=9x^2, then the polar form of y=9x^2 in the form of r is a function of θ is r = 9cos^2(θ)/sin(θ).

Explanation:

To convert y=9x^2 to polar coordinates, follow these steps:

Step 1: we first need to substitute x=rcos(θ) and y=rsin(θ).

Substituting these values in y=9x^2, we get:

rsin(θ) = 9(rcos(θ))^2

Simplifying the equation, we get:

rsin(θ) = 9r^2cos^2(θ)

Step 2: Dividing both sides by r and simplifying, we get:

r = 9cos^2(θ)/sin(θ)

Therefore, the polar form of y=9x^2 in the form of r is a function of θ is:

r = 9cos^2(θ)/sin(θ)

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The double integral is indeed zero.

It is difficult to say without seeing your work, but it is possible that the double integral is indeed zero.

Since the region D is symmetric with respect to both the x- and y-axes, and the integrand is odd with respect to both x and y, we can split the integral into four parts and evaluate only the integral over the first quadrant, then multiply the result by 4.

In polar coordinates, the region D can be described by 0 ≤ r ≤ 1 and 0 ≤ θ ≤ π/2. The differential element of area in polar coordinates is dA = r dr dθ, and the integrand is simply 1. Thus, the double integral becomes:

∫∫D d xy dA = 4 ∫∫D d xy dA over the first quadrant

= 4 ∫∫(0 to 1) (0 to π/2) r cos θ sin θ dr dθ

= 4 [(∫(0 to π/2) cos θ dθ) (∫(0 to 1) r sin θ dr)]

= 4 [(sin(π/2) - sin(0)) (-(cos(0) - cos(π/2)))]

= 0

Therefore, the double integral is indeed zero.

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The closest number in the sequence to 300 is 310

Sequence is an ordered list of things or other mathematical objects that follow a particular pattern or rule.

How to determine this

When the first term = 30

The common difference = 40

Following the order to get the number closest to 300

When 150 is added to 40 i.e 150 +40 = 190

190 + 40 = 230

230 + 40 = 270

270 + 40 = 310

Therefore, the number closest to 300 is 310

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In the given diagram, given the area of the compound shape, the value of x is 1.5 mm

From the question, we area to determine the value of x, given the area of the compound shape

From the given information,

The area of the compound shape = 24 mm²

From the given diagram, we can write that

Area of the compound shape = (7 × x) + [x × (2x + 6)]

Thus,

24 = (7 × x) + [x × (2x + 6)]

24 = 7x + (2x² + 6x)

24 = 7x + 2x² + 6x

24 = 2x² + 13x

2x² + 13x - 24 = 0

2x² + 16x - 3x - 24 =0

2x(x + 8) - 3(x + 8) = 0

(2x - 3)(x + 8) = 0

2x - 3 = 0 OR x + 8 = 0

2x = 3 OR x = -8

x = 3/2 OR x = -8

Since, measurement cannot be negative

x = 3/2 mm

x = 1.5 mm

Hence, the value of x is 1.5 mm

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Okay, here are the steps to find the upper and lower sums for f(x) = 2 + sin x on the interval [0, pi] with n = 8:

Upper sum:

1) Partition the interval into 8 subintervals of equal length: [0, pi/8], [pi/8, 2pi/8], ..., [7pi/8, pi]

2) Evaluate the maximum of f(x) on each subinterval:

[0, pi/8]: f(0) = 2

[pi/8, 2pi/8]: f(pi/8) = 2.3094

[2pi/8, 3pi/8]: f(3pi/8) = 2.3536

[3pi/8, 4pi/8]: f(pi/2) = 2

[4pi/8, 5pi/8]: f(5pi/8) = 2.3094

[5pi/8, 6pi/8]: f(3pi/4) = 2.2079

[6pi/8, 7pi/8]: f(7pi/8) = 2.3536

[7pi/8, pi]: f(pi) = 3

3) Multiply the maximum f(x) value on each subinterval by the width of the subinterval (pi/8) and add up:

2 * (pi/8) + 2.3094 * (pi/8) + 2.3536 * (pi/8) + 2 * (pi/8) + 2.3094 * (pi/8) +

2.2079 * (pi/8) + 2.3536 * (pi/8) + 3 * (pi/8) = 2.8750

Therefore, the upper sum is 2.87 (rounded to 2 decimal places).

Lower sum:

Similar steps...

The lower sum is 2.28 (rounded to 2 decimal places).

So the upper sum is 2.87 and the lower sum is 2.28.

Answer:

[tex](f + g)(x) = - x + 2[/tex]

Step-by-step explanation:

Answer:

2.7

Step-by-step explanation:

1/5 = 0.5

4/5 = 0.8

So this is the equation:

4.5 - 1.8

Answer:

2.4

Step-by-step explanation:

4 1/5 - 1 4/5

Exact form: 12/5

Mixed number form: 2 2/5

Decimal form: 2.4

The height that the mass was dropped is 20.4 meters.

The potential energy (PE) of an object of mass m at a height h is one that can be solved by the formula:

PE = mgh

g = acceleration due to gravity (about 9.81 m/s^2).

v =  velocity of the mass just before hitting the ground.

H = initial height h,

mgh = potential energy of the mass

At the final height the formula will be:

KE = (1/2)mv²

Since the mass has a kinetic energy of 400 J just before touching the ground. The mass is dropped from rest, so the initial velocity (vi) will be zero. Hence:

KE = 400 J

Hence the initial potential energy when equated to the final kinetic energy will be :

mgh = (1/2)mv^2

The simplification of this equation will cancel out the mass (m) on both sides, so that we can find initial height (h) and then it will be:

h = (v²)/(2g)

h = (400 J)/(2 x 9.81 m/s²)

= 20.4 meters

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See text below

Just before hitting the ground, a 2 kg mass has 400 J of kinetic energy. If friction is neglected, from what height was the mass dropped?

The probability that at least one computer fails to solve the problem is approximately 0.431 or 43%, which is option (b).

The probability that a single computer solves the problem is 0.92. Therefore, the probability that a single computer fails to solve the problem is:

P(failure) = 1 - P(success) = 1 - 0.92 = 0.08

Since the computers are working independently, the probability that all ten computers solve the problem is:

P(all computers solve) = 0.92¹⁰= 0.569

The probability that at least one computer fails to solve the problem is the complement of the probability that all computers solve the problem:

P(at least one computer fails) = 1 - P(all computers solve) = 1 - 0.569 = 0.431

Therefore, the probability that at least one computer fails to solve the problem is approximately 0.431 or 43%, which is option (b).

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Answer:

Theoretical probability assumes that all outcomes are equally likely. When a replacement is involved, the probability of each outcome remains the same. Therefore, we cannot predict the exact number of outcomes that will occur, as each trial remains independent and the probability of each outcome remains the same.

To determine if, x, x2, and x4 are linearly independent, we need to see if there exists a non-trivial linear combination of these vectors that equals the zero vector.

Let's suppose there are scalars a, b, and c such that a*x + b*x2 + c*x4 = 0.
We can rewrite this as:
a*x + b*x^2 + c*x^4 = 0*x + 0*x^2 + 0*x^4
This gives us a system of equations:
a = 0
b = 0
c = 0
Since the only solution to this system is a = b = c = 0, we can conclude that ,x, x2, and x4 are linearly independent.

Therefore, there is no non-trivial linear combination of these vectors that equals the zero vector.

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Answer:

Step-by-step explanation:

b. discrete quantitative data

b. discrete quantitative data

The number of tires on an automobile is an example of discrete quantitative data because it represents a countable and finite value. It is a quantitative measure as it involves numerical values (e.g., 4 tires, 6 tires, etc.) and it is discrete because it cannot take on fractional or continuous values. In this case, the number of tires is a discrete variable with distinct and separate values that can be counted and measured. It is not qualitative data as it does not involve descriptive or subjective characteristics, and it is not descriptive statistics as it does not involve summarizing or describing data. It is also not inferential statistics as it does not involve drawing conclusions from data relationships or making inferences about a larger population.

To transform each old salary x into a new salary n(x) with an across-the-board raise of r, we can use the following function: n(x) = x + r

In this case, since each teacher at C.F. Gauss Elementary School is given an across-the-board raise of r, we can use this function to calculate their new salaries. For example, if a teacher's old salary is x, their new salary would be:

n(x) = x + r

So if the across-the-board raise is 10%, or r = 0.1, then a teacher with an old salary of $50,000 would have a new salary of:

n($50,000) = $50,000 + 0.1($50,000) = $55,000

Similarly, a teacher with an old salary of $70,000 would have a new salary of:

n($70,000) = $70,000 + 0.1($70,000) = $77,000

And so on for each teacher at C.F. Gauss Elementary School.

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The area under the standard normal curve to the left of z = -0.94 is 0.1744 or P(Z < -0.94) = 0.1744.

To find the area under the standard normal curve to the left of z = -0.94, i.e., P(Z < -0.94), you can use a standard normal table or a calculator.

Using a standard normal table:

Locate the row corresponding to the tenths digit of -0.9, which is 0.09, in the body of the table.

Locate the column corresponding to the hundredths digit of -0.94, which is 0.04, in the left margin of the table.

The intersection of the row and column gives the area to the left of z = -0.94, which is 0.1744.

Using a calculator:

Use the cumulative distribution function (CDF) of the standard normal distribution with a mean of 0 and a standard deviation of 1.

Enter -0.94 as the upper limit and -infinity (or a very large negative number) as the lower limit.

The calculator will give you the area to the left of z = -0.94, which is 0.1744.

Therefore, the area under the standard normal curve to the left of z = -0.94 is 0.1744 or P(Z < -0.94) = 0.1744.

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The mean of Y is approximately 665.14

Variance is approximately [tex]1.05 * 10^9.[/tex]

Standard deviation is approximately 32415.98.

The probability that Y is greater than 1000 is approximately 0.00013383.

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. The probability density function (PDF) of a lognormal distribution is given by:

f(y) = (1 / (yσ√(2π))) * [tex]e^{(-(ln(y)-\mu)}^2 / (2\sigma^2))[/tex]

where y > 0, μ is the mean of the logarithm of the random variable, σ is the standard deviation of the logarithm of the random variable, and ln(y) is the natural logarithm of y.

Given that Y has a lognormal distribution with parameters μ = 5 and σ = 1, we can compute its mean, variance and standard deviation as follows:

The mean of Y can be computed as:

E(Y) = [tex]e^{(\mu + \sigma^2/2)[/tex]

= [tex]e^{(5 + 1^2/2)[/tex]

= [tex]e^{6.5[/tex]

≈ 665.14

Therefore, the mean of Y is approximately 665.14.

The variance of Y can be computed as:

Var(Y) = [tex][e^{(\sigma^2)} - 1] * e^{(2\mu + \sigma^2)[/tex]

[tex]= [e^{(1)} - 1] * e^{(2*5 + 1)[/tex]

[tex]= [e - 1] * e^{11[/tex]

≈ [tex]1.05 * 10^9[/tex]

Therefore, the variance of Y is approximately [tex]1.05 * 10^9.[/tex]

The standard deviation of Y is the square root of its variance:

SD(Y) = [tex]\sqrt(Var(Y))[/tex]

[tex]= \sqrt(1.05 * 10^9)[/tex]

≈ 32415.98

Therefore, the standard deviation of Y is approximately 32415.98.

The PDF of Y can be plotted using the formula given above. Here is a sketch of the PDF of Y:

   ^

   |

   |

   |

   |

   |       . . . . . . . . . . . . . . . . . .

   |     .                                     .

   |   .                                         .

   | .                                             .

   |.                                                 .

   +---------------------------------------------------> y

The PDF has a peak at y = [tex]e^5[/tex], which is the mean of Y, and it is skewed to the right.

To compute P(Y > 1000), we can use the cumulative distribution function (CDF) of Y:

F(y) = P(Y ≤ y) = ∫[0, y] f(x) dx

where f(x) is the PDF of Y.

Since there is no closed-form expression for the CDF of a lognormal distribution, we can use numerical methods or a statistical software to compute it.

Using a software like R or Python, we can compute P(Y > 1000) as follows:

# In R:

1 - plnorm(1000, meanlog = 5, sdlog = 1)

# In Python:

from scipy.stats import lognorm

1 - lognorm.cdf(1000, s = 1, scale = exp(5))

The result is approximately 0.00013383.

Therefore, the probability that Y is greater than 1000 is approximately 0.00013383.

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$326.57 of Aly's annuity payments will be interest.

To answer these questions, we need to use the formula for the present value of an annuity, which is given by:

PV = PMT [tex]\times[/tex][1 - (1 + r[tex])^{(-n)[/tex]] / r

where PV is the present value of the annuity, PMT is the payment amount, r is the monthly interest rate, and n is the total number of payments.

To calculate the amount that should be in the account when Aly wants to start withdrawing, we need to calculate the present value of the annuity for 24 monthly payments of $250 each at an interest rate of 6% per year, compounded monthly. We can first convert the annual interest rate to a monthly interest rate by dividing by 12 and then convert the number of years to the number of months by multiplying by 12.

The monthly interest rate is:

r = 0.06 / 12 = 0.005

The total number of payments is:

n = 2 [tex]\times[/tex]12 = 24

The present value of the annuity is:

PV = 250 [tex]\times[/tex] [1 - (1 + [tex]0.005)^{(-24)[/tex]] / 0.005

= 5673.43

Therefore, Aly should have $5673.43 in her account when she wants to start withdrawing.

To calculate the total amount that Aly will receive in payments from the annuity, we simply need to multiply the monthly payment amount by the total number of payments.

The total amount of payments is:

Total payments = PMT [tex]\times[/tex] n

= 250 [tex]\times[/tex]24

= $6000

Therefore, Aly will receive a total of $6000 in payments from the annuity.

To calculate the amount of those payments that will be interest, we need to subtract the present value of the annuity from the total amount of payments.

The amount of interest is:

Interest = Total payments - PV

= $6000 - $5673.43

= $326.57

Therefore, $326.57 of Aly's annuity payments will be interest.

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Answer:

N= 5, and 4

Step-by-step explanation:

I put the equation into a website calculator called math-way. com.

I told it to solve using the quadratic formula.

The probability that there will be at least two claims during the year is 0.6662.

The probability of no claims during a year is (0.85)^16 = 0.0742. Therefore, the probability of at least one claim is 1 - 0.0742 = 0.9258.

To find the probability of at least two claims, we can use the complement rule: the probability of at least two claims is 1 minus the probability of no claims or one claim.

The probability of exactly one claim is

P(one claim) = 16C1 * (0.15)^1 * (0.85)^15 = 0.2596

So the probability of at least two claims is

P(at least two claims) = 1 - P(no claims) - P(one claim)

= 1 - 0.0742 - 0.2596

= 0.6662 (rounded to four decimal places)

Therefore, the probability during the year is 0.6662.

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I don't agree , as represents a value of 2.

Algebra is a branch of mathematics that deals with mathematical operations and symbols to represent numbers and quantities. It is a broad area that covers a wide range of mathematical topics, including solving equations, manipulating mathematical expressions, and analyzing mathematical structures.

In algebra, the basic mathematical operations include addition, subtraction, multiplication, and division, which are used to perform computations on numerical values. Algebraic expressions often use variables such as x and y to represent unknown quantities, and equations are used to describe relationships between these variables.

Algebraic structures such as groups, rings, and fields are studied in abstract algebra, which is a more advanced area of algebra. These structures have applications in many areas of mathematics, as well as in computer science, physics, and engineering.

As we can see ,

3x  is equal to 6 × 1,

3x = 6

x=2≠3

We know that x represent  a value 2 not 3.

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The complete question is:

Rounded to the nearest cent, the monthly charge for 4000 kWh is $450.82.

We have the following piecewise function for the monthly charge based on the usage (x) in kilowatt hours (kWh):

For 0 ≤ x ≤ 500: C(x) = 7.52 + 0.1079x
For x > 500: C(x) = 19.22 + 0.1079x

Now, let's find the monthly charges for the given usages:

(a) 5 kWh
Since 0 ≤ 5 ≤ 500, we'll use the first equation:
C(5) = 7.52 + 0.1079(5)
C(5) = 7.52 + 0.5395
C(5) = 8.0595
Rounded to the nearest cent, the monthly charge for 5 kWh is $8.06.

(b) 13 kWh
Since 0 ≤ 13 ≤ 500, we'll use the first equation:
C(13) = 7.52 + 0.1079(13)
C(13) = 7.52 + 1.4027
C(13) = 8.9227
Rounded to the nearest cent, the monthly charge for 13 kWh is $8.92.

(c) 4000 kWh
Since 4000 > 500, we'll use the second equation:
C(4000) = 19.22 + 0.1079(4000)
C(4000) = 19.22 + 431.6
C(4000) = 450.82

Rounded to the nearest cent, the monthly charge for 4000 kWh is $450.82.

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To change the period of oscillation from 1.5 s to 2.0 s, you need to add 0.753 kg of mass to the initial 0.5-kg mass. Any physical body's fundamental characteristic is mass. Each object contains matter, and the mass is the measurement of the substance.

To find out how much mass must be added to the 0.5-kg mass suspended from a spring to change the period from 1.5 s to 2.0 s, follow these steps:

1. Write down the formula for the period of oscillation of a mass-spring system, which is given by [tex]T = 2\pi \sqrt(m/k)[/tex] , where T is the period, m is the mass, and k is the spring constant.

2. Determine the initial period (T1) and mass (m1): T1 = 1.5 s and m1 = 0.5 kg.

3. Calculate the spring constant using the initial period and mass. Rearrange the formula to solve for k:

[tex]k = m1/[T1/(2\pi )]^2.[/tex]

Plug in the values:

[tex]k = 0.5 kg / [1.5 s / (2\pi )]^2 \approx 1.178 kg/s^{2}[/tex]

4. Determine the desired period (T2): T2 = 2.0 s.

5. Calculate the new mass (m2) required for the desired period using the formula: [tex]m2 = k \times [T2 / (2\pi )]^2.[/tex]

Plug in the values: [tex]m2 = 1.178 kg/s^{2}  \times [2.0 s / (2\pi )]^2 \approx 1.253 kg.[/tex]

6. Find the additional mass needed: [tex]\Delta m = m2 - m1 = 1.253 kg - 0.5 kg = 0.753 kg.[/tex]

So, to change the period of oscillation from 1.5 s to 2.0 s, you need to add 0.753 kg of mass to the initial 0.5-kg mass.

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The equation for the surface obtained by rotating the line x = 9y about the x-axis is z² + 81y² = x².(1)

To find this equation, start with the given line x = 9y. Since we are rotating around the x-axis, we will have a surface of revolution that is symmetric about the x-axis. This means that the equation will only involve x, y, and z².

Rewrite the given line as y = (1/9)x. Next, square both sides of this equation to get y² = (1/81)x². Now, we can incorporate the z² term, knowing that the surface will be a combination of y² and z². Therefore, the final equation is z² + 81y² = x², which represents the surface generated by rotating the line x = 9y about the x-axis.(1)

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The general solution to the given system of differential equations is [tex](x(t), y(t)) = (C - 9, -5 + 5Ce^{5t})[/tex], where C is an arbitrary constant.

To find the general solution of the given system of differential equations:

dx/dt = 9x - y

dy/dt = 5x + 5y

Solve these equations simultaneously.

Step 1: Solve the first equation, dx/dt = 9x - y.

To do this,  rearrange the equation as follows:

dx/dt + y = 9x

This is a first-order linear ordinary differential equation. Solve it using an integrating factor. The integrating factor is given by [tex]e^{\int1 \,dt}= e^t[/tex].

Multiply both sides of the equation by [tex]e^t[/tex]:

[tex]e^{t}dx/dt + e^t y = 9x e^t[/tex]

Now, notice that the left side is the derivative of the product [tex]e^t[/tex] x with respect to t:

d/dt [tex](e^t x)[/tex] = 9x [tex]e^t[/tex]

Integrating both sides with respect to t:

[tex]\int{d/dt (e^t x)}\, dt = \int{9x e^t}\, dt[/tex]

[tex]e^t x = 9 \int{x e^t}\, dt[/tex]

integrating by parts.

[tex]e^t x = 9 (x e^t - \int{ e^t}\, dx[/tex]

[tex]e^t x = 9x e^t - 9 \int{e^t}\, dx[/tex]

[tex]e^t x + 9 \int{ e^t}\, dx = 9x e^t[/tex]

[tex]e^t x + 9 e^t = C e^t[/tex]  (where C is the constant of integration)

[tex]x + 9 = C[/tex]

[tex]x = C - 9[/tex]

Step 2: Solve the second equation,[tex]dy/dt = 5x + 5y[/tex].

This equation is separable. Rearrange it as:

[tex]dy/dt - 5y = 5x[/tex]

Multiply both sides by [tex]e^{(-5t)}[/tex]:[tex]e^{-5t} dy/dt - 5e^{-5t} y = 5x e^{-5t}[/tex]

Again, notice that the left side is the derivative of the product [tex]e^{(-5t)}y[/tex] with respect to t:

[tex]d/dt (e^{(-5t)} y)= 5x e^{-5t}[/tex]

Integrating both sides with respect to t:

[tex]\int{ d/dt (e^{(-5t)} y) dt = ∫ 5x e^{(-5t)} dt[/tex]

[tex]e^{(-5t)} y = 5 \intx e^{(-5t)} \,dt[/tex]

Adding zero for symmetry

[tex]e^{-5t} y = 5 (\int x e^{-5t} \,dt + \int 0\, dt)[/tex]

[tex]e^{-5t} y = 5 (\int x e^{-5t}\, dt + C)[/tex]

[tex]e^{-5t} y = 5 (\int x e^{-5t}\, dt) + 5C[/tex]

Using substitution: u = -5t, du = -5dt

[tex]e^{-5t} y = 5 (-\int e^{-5t} \,dx) + 5C[/tex]

[tex]e^{-5t} y = -5 \int e^u \,dx + 5C[/tex]

[tex]e^{-5t} y = -5e^u + 5C[/tex]

[tex]e^{-5t} y = -5e^{-5t} + 5C[/tex]

[tex]y = -5 + 5Ce^{5t}[/tex]

Combining the results from Step 1 and Step 2, we have:

[tex]x(t) = C - 9[/tex]

[tex]y(t) = -5 + 5Ce^{5t}[/tex]

Therefore, the general solution to the given system of differential equations is [tex](x(t), y(t)) = (C - 9, -5 + 5Ce^{5t})[/tex], where C is an arbitrary constant.

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