Rajani is studying in class X in a school. Her age consisting of two digits equal to 4 times its sum and 6 times the product of her digits.How old is she?​

The value of F when d=3 and F is inversely proportional to [tex] {d}^{2} [/tex]is 4

Given: F is proportional to  [tex] {d}^{2} [/tex]

and d = 12 , when F = 4

We can rewrite the equation with a proportionality constant k as :

[tex]f = k \div {d}^{2} [/tex]

Substituting values F = 4 and d = 12 we get

4 = k / (12×12)

4 = k / 144

k = 576

Thus the value of the proportionality constant k is 576

Substituting k = 576 and d = 3, we get

F = 576/ (3×3)

F = 576/9

F = 64

Therefore, The value of F when d=3 and F is inversely proportional to

[tex] {d}^{2} [/tex]is 4

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The set of integers that are multiples of 5 can be represented using set notation as:

{...,-10, -5, 0, 5, 10, 15, 20, 25, 30, ...}

In set notation, this can be written as:

{ x | x ∈ ℤ, x is a multiple of 5 }

where "x" represents an element in the set, "ℤ" represents the set of integers, and the condition "x is a multiple of 5" specifies the property that the element must satisfy.

Answer:

84

Step-by-step explanation:

[tex]\displaystyle \frac{93+78+89+83+83+x}{6}=85\\\\\displaystyle 426+x=510\\\\x=84[/tex]

Therefore, Erin would need to score an 84 on her last test to get a mean score of 85.

Algebraically, the amount of money the insurance company will pay, represented as P, is equal to the total value of the stolen items (2b + t) minus the deductible (d).

Let's denote the amount of money the insurance company will pay as P. In this scenario, if the value of the stolen items is greater than the deductible, the insurance company will cover the loss, and Ron will receive compensation for the stolen items.

The total value of the stolen items can be calculated by adding the value of the bicycles (2b) and the value of the tools (t). So, the total value of the stolen items is 2b + t.

If the total value of the stolen items is greater than the deductible (d), then the insurance company will pay the difference between the total value and the deductible. Mathematically, this can be represented as:

P = (2b + t) - d

Therefore, algebraically, the amount of money the insurance company will pay, represented as P, is equal to the total value of the stolen items (2b + t) minus the deductible (d).

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The decimal number represented by the grid is 0.04.(option-a)

In the grid, we have 10 rows and 10 columns, which represent 100 equal parts. From the grid, we can see that there are 4 shaded parts.

To determine the decimal number represented by the grid, we need to determine the value of each shaded part based on the total number of equal parts represented on the grid. There are a total of 100 equal parts on the grid, so each part has a value of:

1 / 100 = 0.01

Since there are 4 shaded parts, we simply multiply 0.01 by 4 to get the value of the shaded portion:

0.01 * 4 = 0.04

In summary, we can determine the decimal number represented by the grid by dividing the total number of equal parts represented on the grid by the number of shaded parts and then multiplying by the value of each equal part. Based on this process, the grid represents the decimal number 0.04.(option-a)

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Note: The complete question is-"Which decimal number is represented by the shaded grid?" Choose from the options below.

a.0.04

b.4

c.0.004

d.0.4

Answer:

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

Step-by-step explanation:

Recall that the sample space for the results obtained by rolling a fair dice is {1,2,3,4,5,6}. So, the total number of outcomes is 6.

Since the desired outcome is rolling an odd number or a number less than 3, so the set of desired outcomes is {1,2,3,5}. Hence, the number of desired outcomes is 4.

Therefore, the required probability is:

[tex]\frac{\text{number of desired outcomes}}{\text{number of all outcomes}}=\frac{4}{6}=\frac{2}{3}[/tex]

Answer:

probability of rolling an odd number or a number less than 3 is [tex]\frac{2}{3}[/tex] or a 2 in 3 chance

Step-by-step explanation:

a die has 6 sides that are numbered 1 to 6.

3 of those sides are even numbers and 3 of those sides are odd numbers.

the question also states LESS THAN 3, which means that it does not include 3 itself, out of the 6 possible numbers, only two of them are less than 3 (1 and 2) so 2 out of 6 sides are less than 3, but one is already included in the probability, then we use the formula

[tex]\frac{number of desired outcomes}{number of total possible outcomes}[/tex] or [tex]\frac{4}{6}[/tex] which can be simplified to [tex]\frac{2}{3}[/tex]

Answer:

3n = 9n - 110

Step-by-step explanation:

Let's suppose that Patrick and Hall play n number of songs to earn the same amount of money.

For Patrick, the amount of money he earns is given by:

Money earned by Patrick = 3n

For Hall, the amount of money he earns after paying the $110 fee is given by:

Money earned by Hall = 9n - 110

To find the number of songs they will have to play in order to earn the same amount of money, we need to set these two expressions equal to each other and solve for n:

3n = 9n - 110

Simplifying this equation, we get:

6n = 110

Dividing both sides by 6, we get:

n = 110/6

n = 18.33 (rounded to two decimal places)

Since n represents the number of songs, it cannot be a fractional value. Therefore, we can conclude that they will have to play 18 songs to earn the same amount of money.

Therefore, the equation that could be used to find n, the number of songs they will have to play in order to earn the same amount of money, is:

3n = 9n - 110

Hope this helps!

The probability that the mean of the sample will be between 94.7 inches and 95.3 inches is approximately 0.436.

Option B is the correct answer.

We have,

To find the probability that the mean of the sample will be between 94.7 inches and 95.3 inches, we can use the Central Limit Theorem.

According to the Central Limit Theorem, the distribution of sample means approaches a normal distribution as the sample size increases.

In this case, the mean of the population (μ) is 95 inches and the standard deviation of the population (σ) is 0.52 inches.

Since the sample size is sufficiently large (n = 30), we can assume that the distribution of sample means will be approximately normal.

To calculate the probability, we need to standardize the sample mean using the z-score formula:

z = (x - μ) / (σ / √n)

where:

x = sample mean

μ = population mean

σ = population standard deviation

n = sample size

Let's calculate the z-scores for the lower and upper limits:

z1 = (94.7 - 95) / (0.52 / √30)

z2 = (95.3 - 95) / (0.52 / √30)

Now, we can use a standard normal distribution table or a calculator to find the probability associated with these z-scores.

Using a standard normal distribution table or a calculator, we find:

P(z1 < Z < z2) ≈ 0.436

Therefore,

The probability that the mean of the sample will be between 94.7 inches and 95.3 inches is approximately 0.436.

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The effect that does NOT relate to one's driving ability is breathalyzer test results indicating a high BAC. Option B.

While the breathalyzer test measures the blood alcohol concentration (BAC), it is a method used to determine the level of alcohol in a person's system and is commonly used in assessing impairment for driving under the influence.

Therefore, it directly relates to one's driving ability and is not excluded from the effects on driving ability caused by a BAC of .2 percent.

On the other hand, options, lack of coordination, impaired gross motor skills, and impaired reactions all directly relate to one's driving ability as they affect the physical and cognitive abilities necessary for safe and efficient driving.

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The function values are:

f(-1) = -9

f(1) = -5

We have,

From the graph,

We have the coordinates as:

(-1, -9) (2, 0), (-4, 0), and (1, -5)

Now,

The coordinates mean (x, y).

x is the x-axis and y is the function value.

f(x) = y

So,

We can have,

When x = -1, y = -9 or f(-1) = -9

And,

When x = 1, y = -5 or f(1) = -5

Thus,

The function values are:

f(-1) = -9

f(1) = -5

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Answer:

In the equation y = 2 cos(5x + 3) - 6, we can ignore the coefficients 2 and -6 for the purposes of calculating the period because they do not change the period. They only change the amplitude (2) and vertical shift (-6) of the function.

The coefficient 5 in front of x inside the cosine function affects the period of the function. It is a horizontal compression/stretch of the graph of the function.

The period of the basic cosine function, y = cos(x), is 2π. When there is a coefficient (let's call it b) in front of x, such as y = cos(bx), the period becomes 2π/b.

So, in your case, b = 5, so the period T of the function y = 2 cos(5x + 3) - 6 is:

T = 2π / 5

This is the simplest form for the period of the given function.

The two column proof is written as follows

Statement                                                           Reason

< 1  = 62 degrees                                      given

< 1 + < 2 = 180°                                          Linear pairs are supplementary.

< 4 + < 2 = 180°                                          Linear pairs are supplementary.

< 1 + < 2 = < 4 + < 2                                  Substitution property of equality

< 1  = < 4                                                   Subtraction property of equality

< 1  = < 4 = 62 degrees                           Vertical angle theorem

When two lines intersect, vertical angles are created, and in this instance, the angles opposite each another are equal.

Two angles are said to be congruent when they are equal to each

Solving the problem

< 1 + < 2 = 180°  are a linear pair = 180

< 4 + < 2 = 180° are a linear pair = 180°

equating both

< 1 + < 2 = < 4 + < 2

< 1  = < 4

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Answer:

To determine the next whole number after 214 in the base-five system, we need to understand the place value system of base five. In base five, each digit represents a power of five. The rightmost digit is the ones place, the next digit to the left is the fives place, the next digit to the left is the twenty-fives place, and so on.

In this case, 214 is already in base five, so we need to find the next number that follows 214. Since the largest digit in base five is 4 (representing four units of that place value), we need to carry over to the next place value.

Starting from the rightmost digit, which is the ones place, we have 4. Since we cannot increase the digit in the ones place any further, we carry over to the fives place. The fives place digit is 1, and adding 1 to it results in 2. Therefore, the next whole number after 214 in the base-five system is 221.

Step-by-step explanation:

The next whole number after 214 in the base-five system is 220 (or 60, written in base 10)

Here we are in base five, so the digits can be only the ones in the set {0, 1, 2, 3, 4}

Here we want to find the number next to 214.

Notice that the first digit is already on the maximum value, so it returns to the smaller one and we add 1 in the next value, we will get:

220

That is the next whole number, and writting this in base 10 we will get:

2*5² + 2*5 + 0*5⁰

2*25 + 10 + 0

60

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Answer:

Step-by-step explanation:

The correct equation that shows a step in the process of completing the square on the given quadratic y = x^2 + 8x – 3 is y = x^2 + 8x + 16 – 3 – 16. Completing the square involves adding and subtracting a constant term in order to create a perfect square trinomial. In this case, the constant term added is (8/2)^2 = 16, which is half the coefficient of the x-term squared. This step transforms the quadratic into the form (x + a)^2 + b, where a represents half of the x-term coefficient and b represents the constant term.

By adding 16 to the equation to create a perfect square trinomial, we need to subtract 16 afterward to maintain the equation’s balance. Thus, the equation becomes:

y = x^2 + 8x + 16 - 3 - 16

Simplifying further:

y = (x + 4)^2 - 19

Therefore, the correct equation is:

y = (x + 4)^2 - 19

Thus, Gini Coefficient,G = (A / (A + B)) = (1/2) / [(1/2) + (1/2)] = 0.5.Gini Coefficient is 0.5.

Given: Lorenz curve for state A can be given as L(p)=p^7/4.The formula for Gini Coefficient is G=(A/(A+B)).Here, L(p) = p^7/4Therefore, we need to find A and B.

A = Area between the Lorenz curve and the line of equality.B = Area between the Lorenz curve and the X-axis.

The line of equality is the straight line joining the origin and the end point of the Lorenz curve.A + B is equal to the total area of the graph.

L(p) = p^7/4

=> L(p) = p^(4/4) * p^(3/4)

=> L(p) = p * p^(3/4)

=> L(p) = p * L(p)^(3/4)

Now, we will differentiate both sides of the above equation to get the PDF (Probability Density Function) of the Lorenz Curve.

L(p) = p * L(p)^(3/4)dL(p)/dp

= L(p)^(3/4) + (3/4)*p*L(p)^(-1/4)*dL(p)/dp

=> dL(p)/dp

= (4/7)*p^(3/4)L(1) = 1 (as the total population is 100%)

Therefore, A = Area between the Lorenz curve and the line of equality = 1/2 and B = 1/2.

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Answer:

A. 0.2727

Step-by-step explanation:

The Gini coefficient is used to express the extent of income or wealth inequality within a population. It is expressed as a value between 0 and 1, where 0 represents complete equality, indicating that everyone in the population has the exact same wealth, and 1 indicates that one individual or household possesses all the wealth, while everyone else has none. Therefore, a Gini coefficient closer to 0 suggests a more equitable distribution of income or wealth, while a coefficient closer to 1 indicates a higher level of inequality.

The line of perfect equality is a 45-degree line in a graph. It is a reference line that represents a scenario where every member of a population has an equal share of the total income or wealth.

The Gini coefficient is equal to twice the area between the line of perfect equality and the Lorenz curve.

The given Lorenz curve is L(p) = p^(7/4).

Therefore, the line of perfect equality is L(p) = p.

To find the area between the line and curve, we can use integration.

[tex]\boxed{\begin{array}{c}\underline{\sf Area\;between\;two\;curves}\\\\\displaystyle \int^b_a f(x)-g(x)\; \text{d}x\\\\\textsf{where}\;f(x)\;\textsf{is\;above}\;g(x)\;\textsf{between\;the\;interval}\;[a,b]\\\\\end{array}}[/tex]

The cumulative proportion of the population ranges from 0 (0% of the population) to 1 (100% of the population). Therefore, the interval to use for the integration is [0, 1], as it is based on the assumption that the Lorenz curve represents the cumulative proportion of income received by the population.

Set up the integral to find the area between the curves:

[tex]\textsf{Area}=\displaystyle \int^{1}_{0}p\; \text{d}p-\int^{1}_{0} p^{\frac{7}{4}}\; \text{d}p[/tex]

Solve the integral:

[tex]\begin{aligned}\textsf{Area}&=\displaystyle \int^{1}_{0}p\; \text{d}p-\int^{1}_{0} p^{\frac{7}{4}}\; \text{d}p\\\\ &=\left[\dfrac{1}{2}p^2\right]^1_0-\left[\dfrac{4}{11}p^{\frac{11}{4}}\right]^1_0\\\\&=\left(\dfrac{1}{2}(1)^2-\dfrac{1}{2}(0)^2\right)-\left(\dfrac{4}{11}(1)^{\frac{11}{4}}-\dfrac{4}{11}(0)^{\frac{11}{4}}\right)\\\\&=\dfrac{1}{2}-\dfrac{4}{11}\\\\&=\dfrac{11}{22}-\dfrac{8}{22}\\\\&=\dfrac{3}{22}\end{aligned}[/tex]

As the Gini coefficient is equal to twice the area between the line of perfect equality and the Lorenz curve, the Gini coefficient is:

[tex]\begin{aligned}\sf Gini\;coefficient&=2 \cdot \dfrac{3}{22}\\\\&=\dfrac{6}{22}\\\\&=\dfrac{3}{11}\\\\&=0.2727\; \sf (4\;d.p.)\end{aligned}[/tex]

Therefore, the Gini coefficient is 0.2727 (rounded to four decimal places).

The value of x is: x=12.

Here, we have,

given that,

The volume of the rectangular pyramid below is 468 units.

we know that,

Note that the area B of the rectangular base with length x and width 9 units is:

B = 9x

Then, the volume V = 468 cubic units of the pyramid is related to its base area B = 9x and height h=13 as follows:

V =  1/3 Bh

substituting the values we get,

so, we get,

x = 12

So, the value of x is 12.

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Answer:

P'(-2,3)

Step-by-step explanation:

We see that P(-2,1) is 1 point down y = 2, so reflection across the line y = 2 will be 1 point up y = 2, so the answer is (-2,3)

The total number of coins that were in Kay's bank are 137 coins.

In order to determine the number of dimes and nickels, we would assign a variables to the unknown numbers and then translate the word problem into algebraic equation as follows:

Since she has 11 less dimes than nickels, an equation which models this situation is given by;

n = d + 11     ....equation 1.

Note: 1 nickel is equal to 0.05 dollar and 1 dime is equal to 0.1 dollar.

Additionally, the coins are worth 10 dollars;

0.1d + 0.05n = 10 ....equation 2.

By solving both equations simultaneously, we have:

0.1d + 0.05(d + 11) = 10

0.1d + 0.05d + 0.55 = 10

0.15d = 9.45

d = 63 dimes.

For nickels, we have:

n = d + 11

n = 63 + 11

n = 74

Now, we can determine the total number of coins;

Total number of coins = n + d

Total number of coins = 74 + 63

Total number of coins = 137 coins.

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The number 5 represents the 5 dogs that are more than 8 years old, and the Number 2 represents the 2 cats that are more than 8 years old.

Based on the given data, we can create a Venn diagram to illustrate the number of animals that are dogs and are more than 8 years old.

Let's label two intersecting circles representing dogs and cats respectively. In the region where the circles overlap, we will place the animals that are both dogs and more than 8 years old.

First, let's count the number of dogs that are more than 8 years old. Based on the data, we have the following dogs that fit this criterion:

- Dog: 8 (not more than 8 years old)

- Dog: 9

- Dog: 12

- Dog: 9

- Dog: 11

So, there are a total of 5 dogs that are more than 8 years old.

Now, let's count the number of cats that are more than 8 years old. Based on the data, we have the following cats that fit this criterion:

- Cat: 9

- Cat: 13

So, there are a total of 2 cats that are more than 8 years old.

To create the Venn diagram, we will place the number 5 inside the region representing dogs, and the number 2 inside the region representing cats. The region where the circles overlap will be left empty since there are no animals that are both dogs and cats in this dataset.

The Venn diagram representing the number of animals that are dogs and are more than 8 years old would look as follows:

         Dogs

       ___________

      |      |      |

      |  5   |      |

      |______|______|

         Cats

       ___________

      |      |      |

      |      |  2   |

      |______|______|

In the Venn diagram, the number 5 represents the 5 dogs that are more than 8 years old, and the number 2 represents the 2 cats that are more than 8 years old.

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The value of x converges to approximately 2.5615

Given is an equation, we need to solve for x, xˣ = 4ˣ + 16

To solve the equation xˣ = 4ˣ + 16, we'll use algebraic methods. Unfortunately, this equation does not have an explicit algebraic solution that can be found using elementary functions.

However, we can use numerical methods or approximation techniques to find an approximate solution.

One common numerical method for solving equations is the Newton-Raphson method.

Let's apply it to this equation:

Start with an initial guess for x, let's say x = 1.

Iterate using the following formula:

x(n+1) = x(n) - f(x(n)) / f'(x(n))

where f(x) = x^x - 4^x - 16 and f'(x) is the derivative of f(x) with respect to x.

Repeat step 2 until the value of x converges to a solution.

Let's perform a few iterations:

1st iteration:

x(1) = 1 - (1^1 - 4^1 - 16) / (1 * 1^0 - 4 * 4^0)

= 1 - (-11) / (-3)

≈ 3.6667

2nd iteration:

x(2) = 3.6667 - (3.6667^3.6667 - 4^3.6667 - 16) / (3.6667 * 3.6667^2.6667 - 4 * 4^2.6667)

≈ 2.6141

3rd iteration:

x(3) = 2.6141 - (2.6141^2.6141 - 4^2.6141 - 16) / (2.6141 * 2.6141^1.6141 - 4 * 4^1.6141)

≈ 2.5615

By continuing these iterations, you can refine the solution to any desired level of accuracy.

In this case, the value of x converges to approximately 2.5615.

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The expression for the total number of cups Alison buys, in terms of p and b, is 12p + 18b.

The total number of cups Alison buys can be expressed in terms of the number of packs (p) and boxes (b) purchased.

In each pack, there are 12 cups,

so the total number of cups from packs would be 12p.

Similarly, in each box, there are 18 cups,

so the total number of cups from boxes would be \: 18b.

To find the total number of cups, we sum the cups from packs and cups from boxes:

Total cups = 12p + 18b

Therefore, the expression for the total number of cups Alison buys, in terms of p and b, is 12p + 18b.

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Answer:

Step-by-step explanation:

sin (B) = [tex]\frac{2}{5}[/tex]

Answer:

[tex]g^{-1}(x)=3x+12[/tex]

Step-by-step explanation:

The explanation is attached below.

The time period of the loan in days is A = 187 days

Given data ,

To determine the time period of the loan in days, we need to calculate the number of days between September 26, 2010, and March 6, 2011.

September 26, 2010 - March 6, 2011

To calculate the number of days, we consider:

September 2010: 30 days

October 2010: 31 days

November 2010: 30 days

December 2010: 31 days

January 2011: 31 days

February 2011: 28 days (since it is not a leap year)

March 2011: 6 days

Adding these days together:

On simplifying the equation , we get

30 + 31 + 30 + 31 + 31 + 28 + 6 = 187 days

Hence , the time period of the loan, rounded up to the next day, is approximately 187 days.

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The solution set of y (0, 3) and (0, 9)

For x there is no point is given.

The Cartesian System is the system that we use to name points in a plane. The number line gives rise to the cartesian form.

To comprehend the cartesian coordinate system, we must first master the number line. We have the following parameters defined in this system:

⇒ The X-axis and Y-axis are two perpendicular lines.

⇒ The plane is known as the Cartesian, or coordinate plane, and the two lines X and Y, when combined, are known as the system's coordinate axes.

⇒ The plane is divided into four quadrants by the two coordinate axes.

⇒ The intersection of the axes is the Cartesian System's zero. This point will be designated by the letter O. The origin's coordinates are indicated as (0, 0).

⇒ To describe the location of any point P in the plane, we measure the distance x along X, followed by the distance y parallel to Y, to go from O to P. Distances can be detrimental.

Then,

For the X axis all the points on the x axis be the solutions of x.

Since,

There are no points of x axis is given.

For the Y axis all the points on the y axis be the solutions of y.

Since,

Here,

(0, 3) and (0, 9) are the solution set of  for y.

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The phrase "all free variables of a formula ϕ(u1, . . . , un) are among u1, . . . , un" refers to a property of a logical formula ϕ with variables u1, u2, ..., un. In logic and formal systems, variables are used to represent unspecified elements or objects.

When a variable is considered "free" in a formula, it means that it is not bound by any quantifiers or other logical operators in the formula. In other words, it is a variable that is not restricted in any way within the formula.

The given statement implies that all the free variables in the formula ϕ(u1, . . . , un) are explicitly listed among the variables u1, u2, ..., un. In other words, the formula ϕ does not contain any additional free variables beyond the ones explicitly mentioned.

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Answer:

$136.25 is how much you're saving (the discount)

$408.75 is the sale price

Step-by-step explanation:

How much you're saving: [tex]\$545(0.25)=\$136.25[/tex]

How much you'll pay after the discount: [tex]\$545-\$136.25=\$408.75[/tex]

Answer:

M = a + 24, where a is James' age and M is his mother's

Step-by-step explanation:

a = James' age

His sister is 4 years younger than him:

a - 4 represents his sister's age.

His mother is 28 years older than his sister:

(a - 4) + 28

or a + 24 represents his mother's age

Let M = his mother's age

M = a + 24

The constant k that makes f(x) = k * e^(x - 1) a probability density function on the interval [0,1] is k = e / (e - 1).

To determine the constant k that makes the function f(x) = k * e^(x - 1) a probability density function on the interval [0,1], we need to ensure that the following conditions are satisfied:

The function is non-negative: f(x) ≥ 0 for all x in [0,1].

The integral of the function over the interval [0,1] is equal to 1: ∫[0,1] f(x) dx = 1.

Let's analyze each condition step by step:

Non-negativity:

For f(x) to be non-negative on [0,1], we need k * e^(x - 1) ≥ 0.

Since k is a constant, it is always positive or zero. So, k ≥ 0.

Integral equal to 1:

To find the value of k that makes ∫[0,1] f(x) dx equal to 1, we integrate the function over the interval [0,1] and set it equal to 1:

∫[0,1] f(x) dx = ∫[0,1] k * e^(x - 1) dx

Using the properties of exponential functions, we can simplify the integral:

∫[0,1] k * e^(x - 1) dx = k * ∫[0,1] e^(x - 1) dx

= k * [e^(x - 1)] evaluated from 0 to 1

= k * (e^(1 - 1) - e^(0 - 1))

= k * (1 - 1/e)

We want this expression to equal 1:

k * (1 - 1/e) = 1

Solving for k:

k = 1 / (1 - 1/e)

k = e / (e - 1)

Therefore, the constant k that makes f(x) = k * e^(x - 1) a probability density function on the interval [0,1] is k = e / (e - 1).

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Answer:

B.  1.5832

Step-by-step explanation:

A Probability Density Function (PDF) is a function that provides the likelihood of a random variable falling within a particular range of values.

The probability of X landing somewhere between a and b is:

[tex]\displaystyle P(a \leq X \leq b)=\int_{a}^{b} f_X(x)\; \text{d}x[/tex]

[tex]\textsf{where\;$f_X(x)$\;is\;the\;PDF.}[/tex]

For the given function f(x) to be a probability density function (pdf) on the interval [0, 1], the function must satisfy two conditions:

For function f(x) to be non-negative on [0, 1], k must be greater than or equal to 0.

To find the value of k, set the integral of the function to 1 and solve for k:

[tex]\begin{aligned}\displaystyle \int^{1}_{0} ke^{x-1}\; \text{d}x&=1\\\\k\int^{1}_{0} e^{x-1}\; \text{d}x&=1\\\\\int^{1}_{0} e^{x-1}\; \text{d}x&=\dfrac{1}{k}\\\\\left[\vphantom{\dfrac12}e^{x-1}\right]^1_0&=\dfrac{1}{k}\\\\e^{1-1}-e^{0-1}&=\dfrac{1}{k}\\\\e^0-e^{-1}&=\dfrac{1}{k}\\\\1-\dfrac{1}{e}&=\dfrac{1}{k}\\\\\dfrac{e-1}{e}&=\dfrac{1}{k}\\\\k&=\dfrac{e}{e-1}\\\\k&=1.58197670...\\\\k&=1.5820\; \sf(4\;d.p.)\end{aligned}[/tex]

Therefore, k = 1.5820, rounded to four decimal places.

The closest match from the given answer options is B) k = 1.5823.

However, please note that k = 1.5823 returns an area under the curve of 1.00020436023442, which is not exactly 1.

[tex]\hrulefill[/tex]

Differentiation rules used:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Differentiating $e^{f(x)}$}\\\\If $y=e^{f(x)}$, then $\dfrac{\text{d}y}{\text{d}x}=f\:'(x)e^{f(x)}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4 cm}\underline{Differentiating $ax$}\\\\If $y=ax$, then $\dfrac{\text{d}y}{\text{d}x}=a$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]

Answer:

Step-by-step explanation:

The stock plan loses $140.00

Multiply the stock plan amount of $2000.00 x 7% = $140.00