Help me solve this please

It becomes apparent that the initial population (or the population 5 days ago) is 3125 bacteria.

Logarithms arise in problems of exponential growth

and decay because they are inverses of exponential functions. Because of the Laws of Logarithms, they also turn out to be useful in the measurement of the loudness of sounds, the intensity of earthquakes, and other processes that occur in nature

We can write an exponential function to represent the situation.

We know that the current population is 100,000.

The population doubles each day.

The standard exponential function is given by P(t) = A (2)^t

Since our current population is 100,000, a = 100000.

Since our rate is doubling, r = 2.

So

P(t) = 100000 (2)^t

We want to find the population five days ago.

So, we can say that t = -5. The negative represent the number of days that has passed.

Therefore

P(-5) = 100000 (2)^-5 =100000*(1/32) =3125

However, we dealing within this context, we really can't have negative days. Although it works in this case, it can cause some confusion. So, let's write a function based on the original population.

We know that the bacterial population had been doubling for 5 days. Let A represent the initial population. So, our function is:

P(t) = A (2)^t

After 5 days, we reach the 100,000 population. So, when t = 5, P(t) = 100000:

100000 =A[tex]2^5[/tex]

And solving for A, we acquire     :[tex]\frac{100000}{2^5}[/tex]

                                                      =3125

So, our function in terms of the original day is:

P(t) = 3125 (2)^t

So, it becomes apparent that the initial population (or the population 5 days ago) is 3125 bacteria.

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Answer:

Find the value of xx in the equation below.

11=

11=

\,\,x -6

x−6

Step-by-step explanation:

The fractions is simplified to 8ac²/ 5

A fraction can be described as the part of a whole element, number or variable.

There are different types of fractions, which includes;

Examples of improper fractions; 5/3, 7/5

Examples of proper fractions; 3/4, 5/6

Examples of mixed fractions; 3 1/2, 4 5/6

Examples of simple fractions; 1/2 , 1/ 4

From the information given, we have that;

[tex]\frac{32a^2bc^3}{20abc}[/tex]

This is simplified to;

32 × a × a × b × c × c × c/ 20 × a × b × c

Divide the common terms

8ac²/ 5

Hence, the fractions is 8ac²/ 5

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Answer:

12

Step-by-step explanation:

3 + 4 + 5 = 12

The inequality to represent each situation is Harry > 60 inches, Sherry ≠ 4 years, and Hank ≥ $7.50.

a) Harry: This inequality states that Harry is taller than 60 inches. This is likely referring to Harry's height, as most people measure height in inches. This inequality can also be written as Harry ≥ 60 inches, emphasizing that Harry is at least 60 inches tall.

b) Sherry: This inequality states that Sherry is not equal to 4 years. This could be referring to Sherry's age, as it is common to measure age in years. This inequality can also be written as Sherry ≠ 4 years, emphasizing that Sherry is not exactly 4 years old.

c) Hank: This inequality states that Hank is greater than or equal to $7.50. This is likely referring to Hank's amount of money, as it is common to measure money in dollars. This inequality can also be written as Hank ≥ $7.50, emphasizing that Hank has at least $7.50.

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Answer:

Gain: $415

Loss: $10

Step-by-step explanation:

money earned - money spent = gain/loss

The gain is when the participant wins the telescope:

telescope = $425  ==> money earned

10 tickets * $1 = $10 for tickets  ==> money spent

$425 - $10 = $415

Loss is when the participant doesn't get the telescope:

$0 ==> money earned (No money earned)

10 tickets * $1 = $10 for tickets  ==> money spent

$0 - $10 = -$10 gained

-$10 gained ==> $10 lost

The width and height of the rectangle for which the perimeter of the window will be as small as possible are (3sqrt(2), 2sqrt(2)).

What is the Lagrange multiplier?

The Lagrange multiplier, λ, measures the increase in the objective function (f(x, y) that is obtained through a marginal relaxation in the constraint (an increase in k). For this reason, the Lagrange multiplier is often termed a shadow price.

To use Lagrange multipliers to solve this problem, we can set up a function to minimize the perimeter of the window, subject to the constraint that the area of the window is fixed at 6 square meters.

Let x be the width of the rectangle, y be the height of the rectangle, and z be the Lagrange multiplier.

The function to minimize is the perimeter of the window, which is P = 2x + 2y + (2 * sqrt(x^2 + y^2))

The constraint is the area of the window, which is A = xy + (x * sqrt(x^2 + y^2))/2 = 6

The Lagrange equation is

L(x,y,z) = P + z(A-6) = 2x + 2y + (2 * sqrt(x^2 + y^2)) + z(xy + (x * sqrt(x^2 + y^2))/2 - 6)

To find the critical points, we will take the partial derivatives of L(x,y,z) with respect to x, y, and z, and set them equal to zero:

∂L/∂x = 2 + zy + zx * (x/sqrt(x^2 + y^2)) = 0

∂L/∂y = 2 + zx + zy * (y/sqrt(x^2 + y^2)) = 0

∂L/∂z = xy + (x * sqrt(x^2 + y^2))/2 - 6 = 0

Solving these equations simultaneously will give us the critical points of the function.

We will get two critical points (x,y) = (3sqrt(2), 2sqrt(2)), (0,2sqrt(2))

We should eliminate the point where x=0 as it does not fulfill the condition of the problem ( the width should be greater than 0)

Hence, the width and height of the rectangle for which the perimeter of the window will be as small as possible are (3sqrt(2), 2sqrt(2)).

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The answer is a ≤ 30.

An inequality is a mathematical expression that compares two values or expressions using an inequality operator. In this case, the inequality would be a ≤ 30. This means that the area of the triangle must be less than or equal to 30 square inches.

This can be solved by finding the area of the triangle. To do this, we need to know the lengths of the sides of the triangle. Suppose we have sides p, q, and r. Then the area of the triangle can be found using the formula:

                       a = 1/2 × b × h

where q is the base of the triangle and r is the height. To ensure that the area of the triangle is no more than 30 square inches, the inequality must be satisfied. That is, 1/2 × b × h ≤ 30. This inequality can be rewritten in terms of the side lengths by substituting 1/2 × q × r for h. This gives us a ≤ 30.

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You roll a 100 sided die with sides 1-100. The optimal strategy is to stop and cash out only if the dice shows 87 or higher , else keep spending $1 to roll again n again, leading to expected winning of $87.35.

The 100-sided die Lou Zocchi created is known by the trademark "Zocchihedron," and it made its debut in 1985. It assumes the appearance of a ball with 100 flattened planes, unlike other polyhedran dice. It is also known as "Zocchi's Golfball."

optimal strategy is to stop and take dollar if roll > x, and to pay $1 and continue when roll <= x (as applicable to all/any turn/roll). We need to find x in order to maximize E.

Assuming optimal expected value to be E, once you cast the first roll, there are two possible scenario:

1) we get roll value > x with probability (100-x)/100, and decide to take that amount, expected winning = (x+1+100)/2 (that amount varies uniformly from x+1 to 100 in this case, take the average)

2) we get roll value <= x with probability x/100 and decide to continue, with future expected winning again = E and 1$ loss in fee. Total expected winning in this scenario = E-1

[math]E=\frac{(100-x)}{100}\times\frac{ (x+1+100)}{2}[/math][math] + \frac{x}{100}\times(E-1)[/math]

[math]E = \frac{(100-x)(101+x) - 2x}{200 - 2x}[/math]

maximizing E w.r.t. x,

dE/dx = 0

2x^2 - 400x + 19600 = 0

x = ( 400+/- sqrt(160000-4*2*19600) ) / 4

Since, 1<= x <= 100, therefore x = ( 400+ sqrt(160000-4*2*19600) ) / 4

x = 85.85786

We just have to calculate E(x=85) and E(x=86) and compare.

E((x=86) = 87.35 and E(x=85) = 87.33

so optimal strategy is to stop and cash out only if the dice shows 87 or higher , else keep spending $1 to roll again n again, leading to expected winning of $87.35.

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The number of terms in the expansion of [tex](2x^3y^2)(2x^2y^3)[/tex] is 12.

When expanding the product of two polynomials, the number of terms in the expansion is equal to the product of the number of terms in each polynomial. In this case, the first polynomial has 1 term (2x^3y^2) and the second polynomial has 1 term (2x^2y^3), so the number of terms in the expansion will be 1*1=1.

A polynomial function is one in which the variable's non-negative integer powers or positive integer exponents are the only parts of the equation. Examples of polynomial functions include quadratic, cubic, and other equations. An example of a polynomial with an exponent of 1 is 2x+5.

However, the expression provided is not a polynomial, it is a number.

[tex]2x^3y^2 * 2x^2y^3 = 4x^5y^5[/tex]

It is just one term.

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Choose the variable whose numerical coefficients are smaller or have a smaller common multiple.

In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

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Answer:

No, they are not equivalent.

Step-by-step explanation:

The first expression, (3 6) 9, is equal to 9, while the second expression, 3 (6 9), is equal to 3. The two expressions are not equivalent because they do not produce the same result when the operations are carried out. The properties of associativity, commutativity, and identity do not apply to these expressions.

Answer:

no they are not equivalent

Step-by-step explanation:

The two expressions are not equivalent. This is because the distributive property does not apply in this case, as the parentheses on the left side of the expression have precedence over the parentheses on the right side. Therefore, the expression on the left side would be evaluated first, resulting in a different answer than the expression on the right side.

the cost of a broom is $62.50, and the cost of a mop is x + 15 = 62.5 + 15 = $77.50.

Therefore,

a) 1 mop cost $77.50

b) 1 broom cost $62.50

What is the system of equations?

A system of equations is a set of one or more equations involving a number of variables. The solutions to systems of equations are the variable mappings such that all component equations are satisfied in other words, the locations at which all of these equations intersect.

To find the cost of a mop and a broom, we can set up a system of equations using the information given:

Let x be the cost of a broom

Let y be the cost of a mop (which is $15 more than a broom)

From the problem, we know that:

y = x + 15 (since a mop costs $15 more than a broom)

5x + 7y = 855 (since 5 mops and 7 brooms cost a total of $855)

We can substitute the first equation into the second equation:

5x + 7(x + 15) = 855

5x + 7x + 105 = 855

12x = 750

x = 62.5

Hence, the cost of a broom is $62.50, and the cost of a mop is x + 15 = 62.5 + 15 = $77.50.

Therefore,

a) 1 mop cost $77.50

b) 1 broom cost $62.50

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The score interval that is within one standard deviation of the mean is

(70 , 80).

What is standard deviation?

The standard deviation is a statistic that expresses how much variance or dispersion there is in a group of numbers. While a high standard deviation suggests that the values are dispersed over a wider range, a low standard deviation suggests that the values tend to be close to the mean of the collection.

Some of the properties of the standard deviation are:

1. It cannot be negative

2. It is only employed to calculate the spread or dispersion around a data set's mean

3. It displays the degree of deviation  from the mean value

4. The larger the spread, the more standard deviation, with data of about the same mean.

Given,

 The mean of the test μ = 75

The standard deviation σ = 5

We are asked to find the scores within one standard deviation of the mean.

This means that the scores should be in the interval ( μ - σ , μ + σ )

μ - σ = 75 - 5 =70

μ + σ = 75 + 5 = 80

Therefore the scores should be in the interval = ( 70,80), which is within one standard deviation of the mean.

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If the equation that relates the Runner 2’s distance (in miles) with time (in minutes) is [tex]t=8.5d[/tex]  then the slope/rate of change of Runner 2 is 8.5 minutes for 1 mile  .

The three runners are training for the marathon ,

The equation that relates the Runner 2’s distance (in miles) with time (in minutes) is ⇒ [tex]t=8.5d[/tex] ,

to cover the distance of 2 miles Runner2 will take [tex]t= 8.5 \times 2[/tex] minutes ;

So , [tex]t=17[/tex] minutes ;

the first point will be (2,17) .

and to cover the distance of 4 miles the runner 2 will take [tex]t= 8.5 \times 4[/tex]

So , [tex]t=34[/tex] minutes ;

the second point will be (4,34) .

we have to find the rate of change/slope of runner 2  ,

that can be written as : [tex]Slope = \frac{34-17}{4-2}[/tex]

On simplifying ,

we get ;

[tex]Slope = \frac{17}{2}[/tex] ;

So , [tex]Slope = 8.5[/tex].

Therefore , the rate of change of Runner 2 is 8.5 .

The given question is incomplete , the complete question is

Three runners are training for a marathon. One day, they all run about ten miles, each at their own constant speed. The equation that relates the Runner 2’s distance (in miles) with time (in minutes) is "t = 8.5d". Find the slope/rate of change of Runner 2 ?

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Yes, isosceles triangles have three congruent angles.

An isosceles triangle is a triangle with at least two congruent sides and two congruent angles. The two congruent sides and two congruent angles form an isosceles triangle. The two congruent angles are known as the base angles, and they are opposite to the two congruent sides. The third angle in an isosceles triangle must also be congruent to the two base angles in order for the triangle to be true. This is because the angles of a triangle must always sum up to 180 degrees. Therefore, isosceles triangles have three congruent angles. These angles are all equal in measure and have the same shape. The angles of each isosceles triangle can be different from other isosceles triangles, but the angles are still congruent to each other. Therefore, isosceles triangles have three congruent angles.

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1, 15, 29, 43, 57 , the common difference is constant which is 14 and formula 1 + 14(n-1)

37, 34, 31, 29, 26 the common difference is constant which is -3 and formula is 37 - 3(n-1)

-4, 5, 14, 23, 32, the common difference is constant which is 9 and formula is -4 + 9(n-1).

The series where the common difference between any two next words stays constant is known as an arithmetic sequence. Let's review what a sequence is. A collection of numbers that follow a pattern is called a sequence. As an illustration, the numbers 1, 6, 11, 16,... are an arithmetic sequence because they follow a pattern in which the preceding term is multiplied by 5 to produce each subsequent number. There are two formulae for arithmetic sequences.

The equation to determine the nth term in an arithmetic series

The formula for calculating the sum of an arithmetic sequence's first n terms

There are two definitions for an arithmetic sequence. It is a "series where the differences between every two succeeding terms are the identical" or "each term in an arithmetic sequence is formed by adding a fixed number (positive, negative, or zero) to its preceding term."

Option 1, 4 and 6 are in Arithmatic Sequence as

In 1:

1, 15, 29, 43, 57 , the common difference is constant which is 14 and formula 1 + 14(n-1)

again in 4 :

37, 34, 31, 29, 26 the common difference is constant which is -3 and formula is 37 - 3(n-1)

and in 6

-4, 5, 14, 23, 32, the common difference is constant which is 9 and formula is -4 + 9(n-1).

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[tex]2x = 25-3x[/tex]

Simplify:

[tex]2x=-3x+25[/tex]

Add 3x to both sides:

[tex]2x+3x=-3x+25+3x[/tex]

[tex]5x=25[/tex]

Divide both sides by 5:

[tex]\dfrac{5x}{5} =\dfrac{25}{5}[/tex]

[tex]\fbox{x = 5}[/tex]

If 400 people attend the convention,

the food will last for 3 days.

Two quantities a and b are said to be in inverse proportion if an increase in quantity a, there will be a decrease in quantity b, and vice-versa. In other words, the product of their corresponding values should remain constant.  Sometimes, it is also known as inverse variation.

That is, if ab = k, then a and b are said to vary inversely. In this case, if b1, b2 are the values of b corresponding to the values a1, a2 of a, respectively then a1 b1 = a2 b2 or a1/a2 = b2 /b1

Given,

Food for 300 people will Last for 4 days.

Attending people = 400

Let the food for 400 will last for x days.

According to the question,

300 × 4 = 400 × x

x = 300 × 4 /400

x = 3

Hence, the food will last 3 days,

if 400 people attend the convention.

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5 * root(x ^ 2, 3) is the equivalent expression.

Simply multiplying the radicands and placing the result behind the radical sign is all that is required to multiply two square roots. The square root of the product of the radicands, in other words, is equal to the product of two square roots.

Only if the numbers under the radical sign are equal can you add or subtract square roots themselves. The original number will still be in the radical sign after adding or subtracting the coefficients (numbers in front of the radical sign). Implement the suggested action.

5 * root(x ^ 2, 3) is the equivalent expression.

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A form of reasoning called induction is the process of forming general ideas and rules based on your experiences and observations.

The process of welcoming newly hired employees and assisting them in adjusting to their new positions and working surroundings is known as induction. Beginning a new work can be stressful, so new hires require assistance adjusting to their new environment.

The action or process of inducting someone to a position or organization.

We have given that,

A form of reasoning called is the process of forming general ideas and rules based on your experiences and observations.

As a result, the process of formulating general hypotheses and rules based on your observations and experiences is known as induction.

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Answer:

5 kg.

Step-by-step explanation:

2 kg for $0.96 is more expensive than 5 kg for $1.95

The nature of roots of quadratic equation x² = -2x + 1 is real and distinct.

The standard form of a quadratic equation is:

ax² + bx + c = 0

The discriminant is defined as:

D = b² - 4ac

Using  the value of discriminant, we can determine the nature of roots of a quadratic equation.

If:

D > 0, the quadratic equation has two distinct real roots or the roots are real and distinct.

D = 0, the quadratic equation has 1 real equal roots or the roots are unique

D < 0, the quadratic equation has complex conjugate roots or the roots are imaginary.

In this case, the quadratic equation is:

x² = -2x + 1

transform it into the standard form:

x² + 2x - 1 = 0

Calculate the discriminant:

D = 2²  - 4 (-1) = 8

Since D > 0, the roots are two distinct real roots.

Your question is incomplete, but most probably your question was:

What is the nature of roots of quadratic equation x² = -2x + 1?

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The possible formula for the depth of water in tank is , D = 2.8 + 8 sin(π t/4) where t is time in hours.

Sinusoidally function, is defined as one of triagnometric function with a smooth, repetitive oscillation. "Sinusoidal" comes from "sine", function. We can write its equation in the following form y = A·sin(B(x - C)) + D

where, A --> amplitude

B--> period in form of pi

C--> phase or horizontal shift

D --> vertical shift

We have, the depth of water in a tank oscillates sinusoidally.

The smallest depth of water in tank = 6.6 feet

The largest depth of water in tank = 9.4 feet

We have to determine the formula of depth of water in tank as a function of time in hours .

For sinusoidal function,

Amplitude , A = average depth

= ( largest depth + smallest depth)/2

= (6.6 + 9.4)/2

= 16.0/2 = 8

Now, The period of 8 hours so, B = 2π/8

=> B = 45° = π/4

Also, upward shift , D is 9.4 - 6.6 = 2.8

Horizontal shift, C = 0

So, required function is D = 2.8 + 8 sin(π t/4) .

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Answer:80

Step-by-step explanation:by5 will be this and 20 by 10 mulltiply 60 by 9

Answer:660,000

Step-by-step explanation:

principle= 24,000

rate=5.5%

time = 5years

simple interest= P×R×T

. =24000×5.5×5

. = 660,000

For given right triangle, the length of hypotenuse x = 4/√3

Here we have been given a 30° - 60°- 90° right triangle.

A side opposie to angle 60 degrees measures 2 units.

And x is the hypotenuse of right tirnagle.

To find the value of x, consider sine of angle 60 degrees.

sin(60°) = (opposite side of angle 60°) / hypotenuse

From tirgonometric table, sin(60°) = √3 / 2

So, above equation becomes,

√3/ 2 = 2/x

x = 2 × (2/√3)

x = 4/√3

We can see that the value of x is in in simplest radical form with a rational denominator.

Therefore, x = 4/√3

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Option c: y = -x + 5 represents scatter plot 1, Option g: The relationship is not linear represents scatter plot 2, Option a: y = 3x represents scatter plot 3, and Option f: y = x + 3 represents scatter plot 4.

What is a scatter plot?

The graphs that show the association between two variables in a data set are called scatter plots. It displays data points either on a Cartesian system or a two-dimensional plane.

The given equations are -

a. y = 3x

b. y = 4x - 2

c. y = -x + 5

d. y = -5x

e. y = -2x - 4

f. y = x + 3

g. The relationship is not linear.

The first scatter plot is going from positive y-axis to positive x-axis in a linear form. The graph for the equation that goes from positive y-axis to positive x-axis is y = -x + 5.

The second scatter plot is a curved parabola going and not a linear plot. So, the scatter plot is said to be a non-linear scatter plot.

The third scatter plot is going from positive x-axis to infinity in a linear form. The graph for the equation that goes from x-axis to infinity is y = 3x.

The fourth scatter plot is going from positive y-axis to infinity in a linear form. The graph for the equation that goes from positive y-axis to infinity is y = x + 3.

Therefore, scatter plot 1 is represented by equation y = -x + 5, scatter plot 2 is not linear, scatter plot 3 is represented by equation y = 3x and scatter plot 4 is represented by equation y = x + 3.

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Answer:

15% off means take $3.75 off the regular price

Step-by-step explanation:

$25(0.15) = $3.75

$25 - $3.75 = $21.75  sale price