Can you please help me identify the property that justifies the work between step 3 and step 4

Answer:

The half-life of a radioactive substance is the time it takes for half of the initial amount of the substance to decay. We can use the fact that 8% of the original amount remains after 100 days to determine the half-life of the isotope.

Let's assume that the initial amount of the substance is 1 unit (it could be any amount, but we're assuming 1 unit for simplicity). After one half-life, half of the original amount remains, or 0.5 units. After two half-lives, half of the remaining amount remains, or 0.25 units. After three half-lives, half of the remaining amount remains, or 0.125 units. We can see that the amount of substance remaining after each half-life is half of the previous amount.

We can use this information to set up the following equation:

0.08 = (1/2)^n

where n is the number of half-lives that have elapsed. We want to solve for n.

Taking the logarithm of both sides, we get:

log(0.08) = n*log(1/2)

Solving for n, we get:

n = log(0.08) / log(1/2) = 3.42

So the number of half-lives that have elapsed is approximately 3.42. Since we know that 100 days is the time for three half-lives (from the previous calculation), we can find the half-life by dividing 100 days by 3.42:

Half-life = 100 days / 3.42 = 29.2 days (rounded to one decimal place)

Therefore, the half-life of the radioactive isotope that leaked into the stream is approximately 29.2 days.

If Ivy had given 3 more people bonuses, Ivy would have rewarded 13 of the workforce, Ivy Corporation's workforce has 592 employees.

Let's assume that the total workforce at Ivy Corporation is represented by "x".

According to the problem statement, Ivy Corporation gave a bonus to 74 people. Therefore, the remaining non-bonus-receiving employees would be (x-74).

If Ivy had given 3 more people bonuses, then the number of employees that would receive the bonus would be (74+3)=77.

According to the problem, 77 is equal to 13% of the total workforce (x):

77 = 0.13x

We can solve for x by dividing both sides by 0.13:

x = 592

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This is a ratio problem and Sophia is expected to simplify the ratio by finding the smallest possible values and not compounding them by multiplying them by some values.

We can represent the given ratio as 60:36,

60/36

We proceed to reduce the fraction by dividing both the numerator and the denominator by a common factor say 6,

10/6

We can further reduce this with a common factor of 2

5/2

Thus, the ratio we have 5:2

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Answer:

Step-by-step explanation:

The product of powers of exponents cannot be used to simplify the binomial expansion

Given data ,

Let the binomial expansion be represented as A

A = ( 3z + y )³

According to the property of products of powers, exponents can be multiplied when a power is increased to a higher power.

The product of powers characteristic cannot be applied to the equation (3z + y)³. This is due to the fact that (3z + y)³ is a binomial raised to the power of 3, not just a power of a single word.

These terms cannot be simplified further using the product of powers property because they involve different variables or variable combinations.

In this case, it is more appropriate to expand the expression using the binomial expansion

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A) One way to write mixed fractions [tex]6\frac{2}{4} \\[/tex]  is 3 + 3 + [tex]\frac{1}{4}[/tex] +[tex]\frac{1}{4}\\[/tex]

B) Saturday: 3 to 4 hours work = trim bushes and weed garden

Sunday: 4 to 5 hour work = paint sheet and mow lawn

A) [tex]6\frac{2}{4} \\[/tex]  can be written as a simple fraction 6 + [tex]\frac{2}{4}[/tex]

This can be further broken into and written as

3 + 3 +  [tex]\frac{1}{4}[/tex] +[tex]\frac{1}{4}\\[/tex]

B) Saturday : 3 to 4 hours of work

Trim bushes + Weed garden

[tex]1\frac{1}{6} +2\frac{2}{6}[/tex]

1 + [tex]\frac{1}{6}[/tex] + 2+ [tex]\frac{2}{6}\\[/tex]

3 + [tex]\frac{3}{6}[/tex]

3 + [tex]\frac{1}{2}[/tex]

[tex]3\frac{1}{2}[/tex]

Sunday: 4 to 5 hours of work

Paint sheet + mow lawn

[tex]1\frac{3}{6} +3 \frac{4}{6}[/tex]

1 + 3 + [tex]\frac{3}{6} +\frac{4}{6}[/tex]

4 + [tex]\frac{8}{6}[/tex]

[tex]4\frac{8}{6}[/tex]

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The Given question is incomplete the complete question is :

Mr. mason likes to work around the yard during the weekends

ANSWER: 100% of 2000000000 is 2000000000

What is 100 Percent of 2000000000?

100 percent *2000000000

= (100/100)*2000000000

= (100*2000000000)/100

= 200000000000/100 = 2000000000

Now we have: 100 percent of 2000000000 = 2000000000

Question: What is 100 percent of 2000000000?

We need to determine 100% of 2000000000 now and the procedure explaining it as such

Step 1: In the given case Output Value is 2000000000.

Step 2: Let us consider the unknown value as x.

Step 3: Consider the output value of 2000000000 = 100%.

Step 4: In the Same way, x = 100%.

Step 5: On dividing the pair of simple equations we got the equation as under

2000000000 = 100% (1).

x = 100% (2).

(2000000000%)/(x%) = 100/100

Step 6: Reciprocal of both the sides results in the following equation

x%/2000000000% = 100/100

Step 7: Simplifying the above obtained equation further will tell what is 100% of 2000000000

x = 2000000000%

Therefore, 100% of 2000000000 is 2000000000

The area of the triangle is 6x³y⁵

The space enclosed by the boundary of a plane figure is called its area.

A triangle is a polygon with three sides having three vertices.

There are different types of triangle, scalene triangle, equailteral triangle, isosceles triangle, right triangle e.t.c

The area of a triangle is expressed as ;

A = 1/2 bh

where b is the base and h is the height of the of the triangle.

Base = 3x²y²

height = 4x4y³

A = 1/2 × 3x²y² × 4x4y³

A = 1/2 × 12x³y⁵

A = 6x³y⁵

Therefore the area of the triangle is 6x³y⁵

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The area of the Triangle is [tex]48x^{3} y^{4}[/tex]

Triangle is a two-dimensional  three-sided polygon, which has three vertices, three sides and three angles. It is a shape formed when three straight lines meet.

How to determine this

Area of triangle = 1/2 base * height as given

Where area of triangle = ?

Base = [tex]3x^{2} y2[/tex]

i.e 3 * 2 [tex]x^{2} y[/tex]

Base, b = [tex]6x^{2} y[/tex]

Height = [tex]4x4y^{3}[/tex]

i.e [tex]4x[/tex] * [tex]4y^{3}[/tex]

Height,b = [tex]16xy^{3}[/tex]

Area of triangle = 1/2 * [tex]6x^{2} y[/tex] * [tex]16xy^{3}[/tex]

Area = 1/2 * 96* [tex]x^{2+1}[/tex] * [tex]y^{1+3}[/tex]

Area = 1/2 * 96 * [tex]x^{3}[/tex] * [tex]y^{4}[/tex]

Area = 48 * [tex]x^{3} y^{4}[/tex]

Area of the triangle = [tex]48x^{3} y^{4}[/tex]

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Answer:

Step-by-step explanation:

The function you provided, f(x) = -2.5 + 7, represents a linear function rather than a radical function. A linear function has a constant slope and a constant y-intercept.

The end behavior of a linear function is determined by its slope. In this case, the slope of the function is 0 since there is no term involving x. When the slope is 0, it means the function is a horizontal line.

The function f(x) = -2.5 + 7 represents a horizontal line at y = 4.5. As x approaches positive infinity (∞) or negative infinity (-∞), the value of y remains constant at 4.5. Therefore, the end behavior of this linear function is that y approaches 4.5 as x approaches both positive and negative infinity.

In conclusion, the end behavior of the function f(x) = -2.5 + 7 is that y approaches 4.5 as x approaches positive and negative infinity.

We have two explicit functions of x for y:

[tex]y = -1/2 + \sqrt{(x^2 - x + 1/4)} \\or\\y = -1/2 - \sqrt{(x^2 - x + 1/4)}[/tex]

To define y as an explicit function of x, we need to solve for y in terms of x in the given equation:

[tex]x + y + y^2 = x^2[/tex]

First, let's simplify the equation by moving all the terms to one side:

[tex]y^2 + y + (x - x^2) = 0[/tex]

Now, we can use the quadratic formula to solve for y:

[tex]y = (-b + \sqrt{(b^2 - 4ac)} ) / 2a[/tex]

where a = 1, b = 1, and [tex]c = x - x^2.[/tex]Substituting these values, we get:

[tex]y = (-1 + \sqrt{(1 - 4(x - x^2)} )) / 2[/tex]

Simplifying further:

[tex]y = (-1 + \sqrt{(1 - 4x + 4x^2)} ) / 2\\y = (-1 + \sqrt{(4x^2 - 4x + 1)} ) / 2\\y = (-1 + 2\sqrt{(x^2 - x + 1/4)} ) / 2\\y = -1/2 + \sqrt{(x^2 - x + 1/4)}[/tex]

Therefore, we have two explicit functions of x for y:

[tex]y = -1/2 + \sqrt{(x^2 - x + 1/4)} \\or\\y = -1/2 - \sqrt{(x^2 - x + 1/4)}[/tex]

Either of these expressions represents y as an explicit function of x.

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The model that represents the expression 87 - 42 is (d)

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

87 - 42

Using their place values, we have

87 = 8 tens 7 units

42 = 4 tens 2 units

This means that

87 - 42 = 8 tens 7 units - 4 tens 2 units

Subtract the tens

87 - 42 = 4 tens 7 units - 2 units

Subtract the units

87 - 42 = 4 tens 5 units

The model that represents the expression 87 - 42 is 4 tens 5 units

This is represented by model (d) bottom right

Hence, the model that represents the expression 87 - 42 is (d)

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Answer:

[tex](x-14)^2 +(y-22)^2 = (\sqrt{65})^2[/tex]

Step-by-step explanation:

If the two given points are the extremes of the diameter, the center of the circle has to be its middle point - that we can find by taking the average of the coordinates. The center thus sits in

[tex](\frac{22+6}2; \frac{21+23}2)[/tex] or [tex](14; 22)[/tex]. At this point we either find the length of the diameter and halve it, or the distance between the center and either point. Let's go for the diameter.

[tex]r=\sqrt{(22-6)^2+(21-23)^2}=\sqrt{16^2+2^2} = \sqrt {260}=2\sqrt{65}[/tex]. That makes our radius half of that. We can easily write the equation of the circle now:

[tex](x-14)^2 +(y-22)^2 = (\sqrt{65})^2[/tex]

Now, in theory you can improve it by multiplying it out and taking every term to the LHS, but I think it's good enough like that.

If the initial velocity is 75 feet per second, it will take approximately 5.125 seconds for the ball to hit the ground.

The given formula h= -16t²+vt+s represents the height (h) of an object thrown vertically in the air at time (t), with initial velocity (v) and initial height (s). In this case, we are given that the initial height of the softball is 4 feet and the initial velocity is 75 feet per second.

We want to find out how long it will take for the ball to hit the ground, which means we want to find the time (t) when the height (h) is 0.

Substituting the given values into the formula, we get:

0 = -16t² + 75t + 4

This is a quadratic equation in standard form, which we can solve using the quadratic formula:

t = (-b ± √(b² - 4ac)) / 2a

Where a=-16, b=75, and c=4. Substituting these values into the formula, we get:

t = (-75 ± √(75² - 4(-16)(4))) / 2(-16)

t = (-75 ± √(5625 + 256)) / (-32)

t = (-75 ± √(5881)) / (-32)

We can simplify the expression under the square root as follows:

√(5881) = √(49121) = 711 = 77

So we have:

t = (-75 ± 77) / (-32)

Simplifying further, we get two possible solutions:

t = 0.5 seconds or t = 5.125 seconds

Since the softball player hits the ball when it is 4 feet above the ground, we can disregard the solution t=0.5 seconds (which corresponds to when the ball is at its maximum height) and conclude that it will take approximately 5.125 seconds for the ball to hit the ground.

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The solutions to the quadratic equation 6x² = -3x + 1 to the nearest hundredth are -0.73 and 0.23.

Given the quadratic equation in the question:

6x² = -3x + 1

To solve the quadratic equation 6x² = -3x + 1, we can rearrange it into standard form, where one side is set to zero:

6x² + 3x - 1 = 0

Now we can solve the equation using the quadratic formula, which states that for an equation in the form ax² + bx + c = 0, the solutions for x are given by:

[tex]x = \frac{-b \±\sqrt{b^2-4ac} }{2a}[/tex]

Here; a = 6, b = 3, and c = -1.

Let's substitute these values into the quadratic formula:

[tex]x = \frac{-b \±\sqrt{b^2-4ac} }{2a}\\\\ x= \frac{-3 \±\sqrt{3^2-4\ *\ 6\ *\ -1} }{2*6}\\\\x = \frac{-3 \±\sqrt{9+24} }{12}\\\\x = \frac{-3 \±\sqrt{33} }{12}\\\\x = -0.73, \ x=0.23[/tex]

Therefore, the values of x are -0.73 and 0.23.

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Answer:

The answer is 176

Step-by-step explanation:

First lets find the area of rectangle by multiplying 16 by 8 to get 128.

Now lets find the area of each triangle. We know that one side is 6 cm and the base of both is 16.

So to find the base of one triangle we divide 16 by 2 to get 8.

Now we know that the base of one triangle is 8, so lets sub. that into the equation for a triangle. 1/2(b)(h) to get

1/2(6)(8)=24

Since there are two triangle 24+24=48 for area of both triangles

Now lets add that to the answer for the rectangle to get area for whole structure

128+48 = 176

So the answer is 176

HOPE THAT HELPS :)

If rectangle has an area of 114cm squared and a perimeter of 50 cm, the dimensions of the rectangle are approximately 5 cm by 22.8 cm.

Let's assume the length of the rectangle is "l" and the width is "w". We can start by using the formula for the area of a rectangle, which is A = lw. From the given information, we know that the area is 114cm².

So, we have:

lw = 114

Next, we can use the formula for the perimeter of a rectangle, which is P = 2l + 2w. From the given information, we know that the perimeter is 50cm.

So, we have:

2l + 2w = 50

We now have two equations with two variables, which we can solve using substitution or elimination. Let's use substitution by solving the first equation for l:

l = 114/w

We can then substitute this expression for l in the second equation:

2(114/w) + 2w = 50

Multiplying both sides by w to eliminate the fraction, we get:

228 + 2w² = 50w

Rearranging and simplifying, we get a quadratic equation:

2w² - 50w + 228 = 0

We can solve for w using the quadratic formula:

w = [50 ± √(50² - 4(2)(228))]/(2(2)) ≈ 11.4 or 5

Since the length and width must be positive, we can discard the solution w = 11.4. Therefore, the width of the rectangle is approximately 5 cm. We can then use the equation lw = 114 to solve for the length:

l(5) = 114

l ≈ 22.8

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Answer: the solutions to the equation are x = √(10/3) and x = -√(10/3).

Step-by-step explanation:

To solve the equation 10 - 9x^2 + 4x = -6x^2 + 4x, we can simplify it and then solve for x.

Rearranging the equation, we have:

10 - 9x^2 + 4x = -6x^2 + 4x

Combining like terms, we get:

10 - 9x^2 = -6x^2

Subtracting -6x^2 from both sides, we have:

10 - 9x^2 + 6x^2 = 0

Simplifying further, we get:

10 - 3x^2 = 0

To solve for x, we can isolate the term with x^2:

-3x^2 = -10

Dividing both sides by -3, we have:

x^2 = 10/3

Taking the square root of both sides, we get:

x = ±√(10/3)

Therefore, the solutions to the equation are x = √(10/3) and x = -√(10/3).

The approximate price of the new car in 2014 is:

B. $32,200

The general form of a linear equation is given by:

y = mx + c

where y is the future price of the car, x is the number of years, m is the rate of change of price and c is the initial price of the car

c = $27,700

m = ($29,200 - $27,700)/(2010 - 2008)

m = 1500/2

m = $750 per year

In 2014, x = 2014 - 2008 = 6 years

Substituting into y = mx + c:

y = 750(6) + 27,700

y = 4500 + 27700

y = $32,200

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The surface area for each of the cylinders is given as follows:

13. 126 yd².

14. 490 m².

15. 283 mm².

16. 297 cm².

The surface area of a cylinder of radius r and height h is given by the equation presented as follows, which combines the base area with the lateral area:

S = 2πrh + 2πr²

S = 2πr(h + r)

Item 13:

r = 2 yd and h = 8 yd, hence the surface area is given as follows:

S = 2π x 2(2 + 8)

S = 126 yd².

Item 14:

r = 6 m and h = 7 m, hence the surface area is given as follows:

S = 2π x 6(6 + 7)

S = 490 m².

Item 15:

r = 3 mm and h = 12 mm, hence the surface area is given as follows:

S = 2π x 3(3 + 12)

S = 283 mm².

Item 16:

r = 3.5 mm and h = 10 mm, hence the surface area is given as follows:

S = 2π x 3.5(3.5 + 10)

S = 297 cm².

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The percentage of buyers is approximately 68.26% of buyers of new houses paid between [tex]$149,000[/tex] and [tex]$151,000[/tex] .

We are given that the prices of the new homes are normally distributed with a mean of [tex]$150,000[/tex] and a standard deviation of $1000.

Using the 68-95-99.7 rule, we know that: approximately 68% of the data falls within one standard deviation of the mean approximately 95% of the data falls within two standard deviations of the mean, approximately 99.7% of the data falls within three standard deviations of the mean.

In order to determine the proportion of customers who spent between $149,000 and , we must first determine the z-scores for these values:

z1 = (149,000 - 150,000) / 1000 = -1 z2 = (151,000 - 150,000) / 1000 = 1

Now, we can determine the proportion of data that falls between z1 and z2 using the z-table or a calculator. The region to the left of z1 is 0.1587, and the area to the left of z2 is 0.8413, according to the z-table. Thus, the region bounded by z1 and z2 is:

0.8413 - 0.1587 = 0.6826

We can get the percentage of consumers who spent between by multiplying this by 100% is  [tex]$149,000[/tex]  and [tex]$151,000[/tex]:

0.6826 x 100% = 68.26%

Therefore, the standard deviation of customers who paid between is [tex]$149,000[/tex] and [tex]$151,000[/tex]  for this model of new homes.

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Using similar side theorem, the side with equivalent proportion to the given side is RQ/SQ

Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

In this problem, we can use this same theory to find the equivalent side of the given proportion.

OQ / PQ = RQ / SQ

The equivalent side is RQ/SQ

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The height of the plane is 5 miles.

To solve this problem, we can visualize it as a right triangle. The horizontal distance traveled by the plane forms the base of the triangle, which is 12 miles. The total distance traveled by the plane forms the hypotenuse of the triangle, which is 13 miles. We need to find the height, which corresponds to the vertical side of the triangle.

Using the Pythagorean theorem, we can calculate the height as follows:

height^2 + 12^2 = 13^2

height^2 + 144 = 169

height^2 = 169 - 144

height^2 = 25

Taking the square root of both sides, we get:

height = √25

height = 5

Therefore, the height of the plane is 5 miles.

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The expected value for a person buying a ticket is $0.35 rounded to the nearest cent.

The expected value is calculated considering the probabilities of winning each prize and the corresponding values of each prize.

Assuming:

P($25) as the probability of winning the $25 prize

P($10) as the probability of winning the $10 prize

There are 100 tickets sold, therefore, the probabilities can be found as follows:

P($25) = 1/100 (since there is only 1 $25 prize)

P($10) = 1/100 (since there is only 1 $10 prize)

The expected value (E), will then be:

E = P($25) * $25 + P($10) * $10

E = (1/100) * $25 + (1/100) * $10

E = $0.25 + $0.10

E = $0.35

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The value of the parameter p that satisfies the given conditions is 36.

Let the roots of the quadratic equation [tex]3x^2 - 24x + p = 0[/tex] be denoted by α and β, where α is the root that is triple the value of β.

Then we have:

α = 3β

The sum and product of the roots of the quadratic equation are given by:

α + β = 8 (from the coefficient of x in the linear term)

αβ = p/3 (from the constant term)

Substituting α = 3β in the first equation gives:

3β + β = 8

4β = 8

β = 2

Therefore, α = 6.

So the roots of the quadratic equation are α = 6 and β = 2.

The product of the roots is:

αβ = 6 × 2 = 12

From the equation αβ = p/3, we have:

p/3 = 12

p = 36

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Mrs. Garcia invested $2240 in the 8.5% account and $4091 in the 8% account.

Let x be the amount invested in the 8.5% account, and y be the amount invested in the 8% account. Since the total investment is $6331, we have x + y = 6331.

The total interest received is $517.68, which can be expressed as 0.085x + 0.08y = 517.68, where 0.085 and 0.08 are the decimal equivalents of the interest rates.

We can now solve this system of equations to find x and y. One possible method is to use substitution, where we solve for one variable in terms of the other from one of the equations, and substitute it into the other equation. From x + y = 6331, we have y = 6331 - x. Substituting this into the second equation, we get:

0.085x + 0.08(6331 - x) = 517.68

Simplifying and solving for x, we get:

0.005x + 506.48 = 517.68

0.005x = 11.2

x = 2240

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The range of possible values for the population parameter can be estimated using the margin of error, which is calculated as the critical value times the standard error.

Assuming a 95% confidence level, the critical value is approximately 1.96. The standard error for a sample proportion can be calculated as:

SE = sqrt[(p * (1 - p)) / n]

Where p is the sample proportion and n is the sample size. Substituting the values given in the question, we get:

SE = sqrt[(0.65 * 0.35) / n]

We do not know the sample size, so we cannot calculate the standard error exactly. However, we can use a rule of thumb that states that if the sample size is at least 30, we can use the normal distribution to estimate the margin of error.

With a sample proportion of 0.65, the margin of error can be estimated as:

ME = 1.96 * sqrt[(0.65 * 0.35) / n]

We do not know the sample size, so we cannot calculate the margin of error exactly. However, we can use the rule of thumb that a margin of error of about ±5% is typical for a 95% confidence level.

Using this margin of error, we can construct the following range of possible values for the population parameter:

0.65 ± 0.05

This range can be expressed as (0.6, 0.7), which corresponds to option A.

Therefore, the correct answer is option A) (0.6, 0.69).

Step-by-step explanation:

3. Let P(t) be the number of people infected by the virus at time t (in days). We can model the situation with the following exponential function:

P(t) = 400 * 1.25^t

Here, 400 represents the initial number of infected people, and 1.25 represents the growth factor, since the virus is spreading to 25% more people each day.

To find the number of people infected after t days, we can substitute t = (log(3000) - log(400)) / log(1.25) into the equation:

P(t) = 400 * 1.25^t

P(t) = 400 * 1.25^((log(3000) - log(400)) / log(1.25))

P(t) ≈ 2,343

Therefore, approximately 2,343 people are infected when the total number of infections reaches 3000.

4. Let P(t) be the population of the town at time t (in years). We can model the situation with the following exponential function:

P(t) = 10,800 * 0.975^t

Here, 10,800 represents the initial population in 2002, and 0.975 represents the decay factor, since the population is decreasing at a rate of 2.5% each year.

To find when the population reaches half the 2002 value, we can set P(t) = 5,400 and solve for t:

5,400 = 10,800 * 0.975^t

0.5 = 0.975^t

log(0.5) = t * log(0.975)

t ≈ 28.2

Therefore, the population will reach half the 2002 value in approximately 28.2 years, which corresponds to the year 2030.

Answer:

3) 9.03 days

4)  27.38 years

Step-by-step explanation:

To model the spread of the virus over time, we can use an exponential function in the form:

[tex]\large\boxed{P(t) = P_0(1 + r)^t}[/tex]

where:

Given the virus has infected 400 people in the town and is spreading to 25% more people each day:

Substitute these values into the formula to create a function for P in terms of t:

[tex]P(t) = 400(1 + 0.25)^t[/tex]

[tex]P(t) = 400(1.25)^t[/tex]

To find how many days it will take for 3000 people to be infected, set P(t) equal to 3000 and solve for t:

[tex]\begin{aligned}P(t)&=3000\\\implies 400(1.25)^t&=3000\\(1.25)^t&=7.5 \\\ln (1.25)^t&=\ln(7.5)\\t \ln (1.25)&=\ln(7.5)\\t &=\dfrac{\ln(7.5)}{\ln (1.25)}\\t&=9.02962693...\end{aligned}[/tex]

Therefore, it will take approximately 9.03 days for the virus to infect 3000 people, assuming the daily growth rate remains constant at 25%.

Note: After 9 days, 2980 people would be infected. After 10 days, 3725 people would be infected.

[tex]\hrulefill[/tex]

To model the population of the town over time, we can use an exponential function in the form:

[tex]\large\boxed{P(t) = P_0(1 - r)^t}[/tex]

where:

Given the initial population was 10,800 and the population has decreased at a rate of 2.5% each year:

Substitute these values into the formula to create a function for P in terms of t:

[tex]P(t) = 10800(1 -0.025)^t[/tex]

[tex]P(t) = 10800(0.975)^t[/tex]

To find how many days it will take for the population to halve, set P(t) equal to 5400 and solve for t:

[tex]\begin{aligned}P(t)&=5400\\\implies 10800(0.975)^t&=5400\\(0.975)^t&=0.5 \\\ln (0.975)^t&=\ln(0.5)\\t \ln (0.975)&=\ln(0.5)\\t &=\dfrac{\ln(0.5)}{\ln (0.975)}\\t&=27.3778512...\end{aligned}[/tex]

Therefore, it will take approximately 27.38 years for the population to reach half the 2002 value, assuming the annual decay rate remains constant at 2.5%.