For the function, f(x) = 8x + 5x, find the following. (a) f(5) D NUD K (b) f(-2) orea (c) f(4.2) muca (d) f(-4.2)

Answer:

Step-by-step explanation:

the answers are the points in the shaded region so plot the points and see which one is in the blue area so 3,-1

Answer:

A. (3,-1)

Step-by-step explanation:

In order to solve this you just have to search for the point in the graph, if the points are located in theline that the graph shows then they are actually a solution for the inequality shown, since the only point that is actually on the line that is shown in the graph is (3,-1) then that is the correct answer.

Answer:

x≤5

Step-by-step explanation:

-10(9-2x)-x≤2x-5

-90+20x-x≤2x-5

19x-2x≤90-5

17x≤85

x≤85/17

x≤5

Answer:

The equation of circle is [tex]x^2+y^2=65[/tex].

Step-by-step explanation:

It is given that the circle passes through the point (8,1) and center at the origin.

The distance between any point and the circle and center is called radius. it means radius of the given circle is the distance between (0,0) and (8,1).

Distance formula:

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Using distance formula the radius of circle is

[tex]r=\sqrt{\left(8-0\right)^2+\left(1-0\right)^2}=\sqrt{65}[/tex]

The standard form of a circle is

[tex](x-h)^2+(y-k)^2=r^2[/tex]         .... (1)

where, (h,k) is center and r is radius.

The center of the circle is (0,0). So h=0 and k=0.

Substitute h=0, k=0 and [tex]r=\sqrt{65}[/tex] in equation (1).

[tex](x-0)^2+(y-0)^2=(\sqrt{65})^2[/tex]

[tex]x^2+y^2=65[/tex]

Therefore the equation of circle is [tex]x^2+y^2=65[/tex].

The equation of a circle centered at origin and passing through the point (8,1) can be determined using principles of geometry. Calculate the radius using the Pythagorean theorem and then substitute it into the general equation of a circle (x-h)² + (y-k)² = r², where h and k are 0 since the circle is centered at the origin. The equation for the circle is x² + y² = 65.

The subject of this question is a circle in mathematics, particularly geometric principles. The given condition is that the circle's center is at the origin and it passes through the point (8,1). From our understanding of a circle, we know that the radius is the distance from the center to any point on the circle. Since our center is at the origin (0,0), the radius r can be calculated using the Pythagorean theorem as the distance from the origin to the point (8,1), which is sqrt((8-0)² + (1-0)²) = sqrt(65). Therefore, the equation of the circle in terms of x and y based on its center (h,k) and radius r is (x-h)² + (y-k)² = r². Given that the circle's center is at the origin, h and k equal to 0, which simplifies the equation to x² + y² = r², or in our case, x² + y² = 65.

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Answer:

[tex]v_1=(\frac{1}{10},-\frac{3}{10})[/tex]

[tex]v_2=(-\frac{1}{10},\frac{3}{10})[/tex]

Step-by-step explanation:

First we define two generic vectors in our [tex]\mathbb{R}^2[/tex] space:

By definition we know that Euclidean norm on an 2-dimensional Euclidean space [tex]\mathbb{R}^2[/tex] is:

[tex]\left \| v \right \|= \sqrt{x^2+y^2}[/tex]

Also we know that the inner product in [tex]\mathbb{R}^2[/tex] space is defined as:

[tex]v_1 \bullet v_2 = (x_1,y_1) \bullet(x_2,y_2)= x_1x_2+y_1y_2[/tex]

So as first condition we have that both two vectors have Euclidian Norm 1, that is:

[tex]\left \| v_1 \right \|= \sqrt{x^2+y^2}=1[/tex]

and

[tex]\left \| v_2 \right \|= \sqrt{x^2+y^2}=1[/tex]

As second condition we have that:

[tex]v_1 \bullet (3,1) = (x_1,y_1) \bullet(3,1)= 3x_1+y_1=0[/tex]

[tex]v_2 \bullet (3,1) = (x_2,y_2) \bullet(3,1)= 3x_2+y_2=0[/tex]

Which is the same:

[tex]y_1=-3x_1\\y_2=-3x_2[/tex]

Replacing the second condition on the first condition we have:

[tex]\sqrt{x_1^2+y_1^2}=1 \\\left | x_1^2+y_1^2 \right |=1 \\\left | x_1^2+(-3x_1)^2 \right |=1 \\\left | x_1^2+9x_1^2 \right |=1 \\\left | 10x_1^2 \right |=1 \\x_1^2= \frac{1}{10}[/tex]

Since [tex]x_1^2= \frac{1}{10}[/tex] we have two posible solutions, [tex]x_1=\frac{1}{10}[/tex] or [tex]x_1=-\frac{1}{10}[/tex]. If we choose [tex]x_1=\frac{1}{10}[/tex], we can choose next the other solution for [tex]x_2[/tex].

Remembering,

[tex]y_1=-3x_1\\y_2=-3x_2[/tex]

The two vectors we are looking for are:

[tex]v_1=(\frac{1}{10},-\frac{3}{10})\\v_2=(-\frac{1}{10},\frac{3}{10})[/tex]

The two vectors in R2 with Euclidean Norm 1 that are orthogonal to (3,1) are (1/√10, -3/√10) and (-1/√10, 3/√10).

To find two vectors in R2 with Euclidean Norm 1 whose Euclidean inner product with (3,1) is zero, we need to look for vectors that are orthogonal to (3,1). The Euclidean inner product of two vectors (x, y) and (3,1) is calculated by (3x + y). To have an inner product of zero, we need 3x + y = 0. Also, we want the vectors to have a Euclidean Norm (or length) of 1, so we need to satisfy the equation x2 + y2 = 1.

Solving these two equations together, we get that y=-3x for orthogonality, and substituting this into the norm equation gives x2 + 9x2 = 1, or 10x2 = 1. This gives two solutions for x, which are x = 1/√10 or x = -1/√10. For y we get correspondingly y = -3/√10 or y = 3/√10.

The two vectors in R2 with Euclidean Norm 1 that are orthogonal to (3,1) are therefore (1/√10, -3/√10) and (-1/√10, 3/√10).

To calculate the amount of soil needed, you must first calculate the volume in cubic feet (20 feet x 3 feet x 4 feet = 240 cubic feet) and then convert that volume to cubic yards by dividing by 27 (240 cubic feet ÷ 27 = 8.89 cubic yards). So, approximately 8.89 cubic yards of soil is needed.

To find the volume needed to fill the planter, we use the formula to compute the volume of stack, which is length × width × height. The planter has a length of 20 feet, a width of 3 feet, and a height of 4 feet. So, multiplying these values: 20 feet × 3 feet × 4 feet = 240 cubic feet.

Finally, we need to convert cubic feet to cubic yards because soil is typically bought in cubic yards. Since 1 yard = 3 feet, 1 cubic yard = 3 feet × 3 feet × 3 feet = 27 cubic feet. Therefore, to convert 240 cubic feet to cubic yards, we divide that by 27: 240 cubic feet ÷ 27 = 8.89 cubic yards. So, you will need approximately 8.89 cubic yards of soil to fill the planter.

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To fill a planter that is 20 feet long, 3 feet wide, and 4 feet tall, we need 240 cubic feet of soil, which converts to approximately 8.888 cubic yards. Rounding up, we will need 9 cubic yards of soil.

Volume = Length × Width × Height

Here,

the dimensions are given as:

Length: 20 feet

Width: 3 feet

Height: 4 feet

So, the volume in cubic feet is:

Volume = 20 ft × 3 ft × 4 ft

            = 240 cubic feet

Next, we need to convert this volume from cubic feet to cubic yards.

Knowing that there are 3 feet in a yard,

we use the conversion factor:

1 cubic yard = 3 × 3 × 3

                    = 27 cubic feet

So, we divide the total cubic feet by the number of cubic feet in one cubic yard:

240 cubic feet ÷ 27

= 8.888 cubic yards

Since you typically can't have a fraction of a cubic yard in this context, we might round up to ensure you have enough soil, resulting in 9 cubic yards of soil needed.

Step-by-step explanation:

Consider the provided information.

For the condition statement [tex]p \rightarrow q[/tex] or equivalent "If p then q"

The rule for Contrapositive is: Negative both statements and interchange them. [tex]\sim q \rightarrow \sim p[/tex]

Part (A) If you are taller than 6 ft, then it is unpleasant for you to travel in economy class.

Here p is "you are taller than 6 ft, and q is "it is unpleasant for you to travel in economy class".

It is given that Your contrapositive must not contain explicit references to negation. Assume that the negation of "unpleasant" is "pleasant".

Contrapositive: If it is pleasant for you to travel in economy class then you are not taller than 6 ft then.

Part (B) "If x ≥ 0 and y ≥ 0 then xy ≥ 0" where x, y are real numbers.

Here p is "xy≥ 0, and q is "x ≥ 0 and y ≥ 0"

The negative of xy≥ 0 is xy<0, x ≥ 0 is x<0 and y ≥ 0 is y<0.

Remember negative means opposite.

Contrapositive: If xy < 0 then x<0 and y<0.

Final answer:

The contrapositive of a statement involves switching the positions of the subject and the complement, and negating both.

Explanation:

To rephrase the given statement in contrapositive form, we need to switch the positions of the subject and the complement, and negate both.

(a) The contrapositive of the statement 'If you are taller than 6 ft, then it is unpleasant for you to travel in economy class' is:

'If it is pleasant for you to travel in economy class, then you are not taller than 6 ft.'

(b) The contrapositive of the statement 'If x ≥ 0 and y ≥ 0 then xy ≥ 0' is:

'If xy < 0, then at least one of x or y is less than 0.'

Final answer:

To find the current average egg consumption, calculate 35% of the original consumption of 400 eggs, which is 140 eggs, and subtract that from the original to get 260. Therefore, the average consumption now is 260 eggs per person per year.

Explanation:

If we look back 40 years and find that egg consumption was 400 eggs per person per year, and there has been a 35% decrease in egg consumption, we can calculate the current average egg consumption. To do this, we find 35% of the original consumption:

Multiply 400 (original consumption) by 0.35 (35%) to find the decrease in consumption. This equals 140 eggs.

Subtract this decrease from the original consumption: 400 - 140 equals 260 eggs. Therefore, the average consumption of eggs per person per year in the U.S. today is 260.

These changes in dietary habits over the years mirror shifts in consumer tastes, as well as concerns about health and production costs, all of which can influence the demand for different food products.

Answer:

P(F | C) = 0.96

Step-by-step explanation:

Hi!

This is a problem on conditional probability. Lets call:

C = { cloudy day }

F = { foggy day }

Then F ∩ C = { cloudy and foggy day }

You are asked for P(F | C), the probability of a day being foggy given it is cloudy. By definition:

[tex]P(F|C)=\frac{P(F\bigcap C)}{P(C)}[/tex]

And the data you have is:

[tex]P(C) = 0.57\\P(F \bigcap C) =0.55[/tex]

Then: P(F | C) = 0.96

Answer:

There is not enough statistical evidence in the sample taken by the veterinarian to support his skepticism

Step-by-step explanation:

To solve this problem, we run a hypothesis test about the population proportion.

Proportion in the null hypothesis [tex]\pi_0 = 0.37[/tex]

Sample size [tex]n = 150[/tex]

Sample proportion [tex]p = 54/150 = 0.36[/tex]

Significance level [tex]\alpha = 0.05[/tex]

[tex]H_0: \pi_0 = 0.36\\H_a: \pi_0<0.36[/tex]

Test statistic [tex] = \frac{(p - \pi_0)\sqrt{n}}{\sqrt{\pi_0(1-\pi_0)}}[/tex]

Left critical Z value (for 0.01) [tex]Z_{\alpha/2}= -1.64485[/tex]

Calculated statistic = [tex]= \frac{(0.36 - 0.37)\sqrt{150}}{\sqrt{0.37(0.63)}} = -0.254[/tex]

[tex]p-value = 0.6003[/tex]

Since, test statistic is greater than critical Z, the null hypothesis cannot be rejected. There is not enough statistical evidence to state that the true proportion of pet owners who talk on the phone with their pets is less than 37%. The p - value is 0.79860.

Answer with Step-by-step explanation:

Given

[tex]f(x)=\frac{1-cos(2x)}{1-cos(x)}\\\\\lim_{x \rightarrow 0}f(x)=\lim_{x\rightarrow 0}(\frac{1-(cos^2{x}-sin^2{x})}{1-cos(x)})\\\\(\because cos(2x)=cos^2x-sin^2x)\\\\\lim_{x \rightarrow 0}f(x)=\lim_{x\rightarrow 0}(\frac{1-cos^2x}{1-cos(x)}+\frac{sin^2x}{1-cosx})\\\\=\lim_{x\rightarrow 0}(\frac{(1-cosx)(1+cosx)}{1-cosx}+\frac{sin^2x}{1-cosx})\\\\=\lim_{x\rightarrow 0}((1+cosx)+\frac{sin^2x}{1-cosx})\\\\\therefore \lim_{x \rightarrow 0}f(x)=1[/tex]

Answer:

4/12 or 1/3

Step-by-step explanation:

If you have 12 equal sections of a clock face, and 1/12 or 1 section is done Monday, then 4/12 or 1/3 is done on Tuesday if you exclude Monday's section. It's 4/12 or simplified to be 1/3 because it is only asking you what Miki has painted on Tuesday not Monday and Tuesday combined. How you get 4/12 to be 1/3 is that you take both the top number, (numerator), and the bottom number, (demoninator), and you divide them by the greatest common factor for both. Which is 4, so 4 divided by 4 is 1, and 4 divided into 12 is 3, (3 x 4 = 12), and that's how you get 1/3 for a fraction.

Hope this helps! :)

Miki paints 4 out of 12 sections of the clock face on Tuesday which is 1/3rd of the clock face.

Miki divides the clock face into 12 equal sections and paints 4 sections on Tuesday.

To find the fraction of the clock face painted on Tuesday, we look at the number of sections painted on that day compared to the total number of sections.

Given that Miki paints 4 out of 12 sections, we write this as a fraction:

4 (sections painted on Tuesday) / 12 (total sections)

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

(4 / 4) / (12 / 4) = 1 / 3

Answer:

r=1 and t= -21/2.

Step-by-step explanation:

Two vectors are parallel if both are multiples. That is, for a vector (x,y), the parallel vector to (x,y) will be of the form k(x,y) with k a real number. Then,

a) (8r, -2) = 2(4r,-1). Then, we need to have that r=1, in other case the first component wouldn't be 4 or the second component wouldn't be -1 and the vector (8r,-2) wouldn't be parallel to (4, -1).

b) for the case of (8t, 21) we need -1 in the second component and 4 in the first component, then let t= -21/2 to factorize the -21 and get 4 in the fisrt component and -1 in the second component.

[tex](8\frac{-21}{2}, 21) = -21(\frac{8}{2}, -1) = -21(4,-1)[/tex]. In other case,  the vector (8t, 21) wouldn't be parallel to (4,-1).

Vector (8r,-2) is parallel to (4,-1) when r = 1, whereas (8t, 21) cannot be made parallel to it. To determine this, we look for a scalar multiple relation between the vectors.

The question asks for what value(s) of, if any, the given vector is parallel to (4,-1). To determine if two vectors are parallel, we need to see if one is a scalar multiple of the other, which means their components in each dimension multiply by the same scalar. Let's examine the given options:

(a) (8r,-2) is parallel to (4,-1) if there exists a scalar 'k' such that 4k = 8r and -1k = -2. By solving these equations, we find that k = 2 satisfies both, meaning if r = 1, the vector is parallel to (4,-1).

(b) (8t, 21) cannot be made parallel to (4,-1) through any scalar multiplication, as there's no single scalar that would simultaneously satisfy the required equations for both components.

Therefore, vector (8r,-2) is parallel to (4,-1) for r = 1, while vector (8t, 21) cannot be parallel to it under any circumstances.

Answer:

248.79 ft

Step-by-step explanation:

A projectile is fired from a cliff 220 ft above water at an inclination of 45 degrees to the horizontal, with a muzzle velocity of 65 ft per second.

[tex]h(x)=-\dfrac{32x^2}{65^2}+x+220[/tex]

For maximum value of x, h(x)≥0

[tex]-\dfrac{32x^2}{65^2}+x+220\geq0[/tex]

Solve quadratic equation for x

[tex]-\dfrac{1}{4225}(32x^2-4225x-929500)\geq0[/tex]

[tex]32x^2-4225x-929500\leq0[/tex]

Using quadratic formula,

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

where, a=32, b=-4225, c=-929500

[tex]x=\dfrac{4225\pm\sqrt{4225^2-4(32)(-929500)}}{2(32)}[/tex]

[tex]x\geq-116.75\text{ and }x\leq 248.79[/tex]

Hence, The maximum value of x will be 248.79 ft

Final Answer:

The maximum horizontal distance x of the projectile from the face of the cliff, at the peak of its trajectory, is approximately 33.01 feet.

Explanation:

The maximum value of x occurs at the peak of the projectile's flight, which we can find by analyzing the given quadratic equation:

[tex]\[ h(x) = -\frac{32x^2}{v^2} + x + h_0 \][/tex]

For our specific problem:
[tex]\[ h(x) = -\frac{32x^2}{(65)^2} + x + 220 \\\\\[ \text{where} \\\\\[ h_0 = 220 \text{ feet (initial height)}, \\\\\[ v = 65 \text{ ft/s (muzzle velocity)}, \\\\\[ \text{angle of inclination} = 45 \text{ degrees}. \\\\[/tex]
The quadratic equation represents a parabola opening downward (since the coefficient of x² is negative), and the x-coordinate of the vertex of this parabola will give us the maximum value of x.

The x-coordinate of the vertex (maximum value of x) for a parabola in the form ax² + bx + c is given by the formula [tex]\( -\frac{b}{2a} \).[/tex]

In our equation:

[tex]\[ a = -\frac{32}{v^2} = -\frac{32}{(65)^2} \\\\\[ b = 1 \\\\\[ c = h_0 = 220 \][/tex]
Now we substitute values of a and b into the formula for the x-coordinate of the vertex:

[tex]\[ -\frac{b}{2a} = -\frac{1}{2 \times ( -\frac{32}{(65)^2} )} \][/tex]

With the computations already given:

[tex]\[ \text{Maximum value of } x = 33.0078125 \text{ feet} \][/tex]
This means that the maximum horizontal distance x of the projectile from the face of the cliff, at the peak of its trajectory, is approximately 33.01 feet.

Answer:

$9142.46

Step-by-step explanation:

Use the compounded interest formula: [tex]A=P(1+\frac{r}{m} )^{m*t}[/tex]

Where

A is the accumulated amount after compounding (our unknown)

P is the principal ($7500 in our case)

r is the interest rate in decimal form (0.05 in our case)

m is the number of compositions per year (2 in our semi-annually case)

and t is the number of years (5 in our case)

[tex]A=P(1+\frac{r}{m} )^{m*t}= 7500 (1+\frac{0.05}{2} )^{2*5} =9142.4581996....[/tex]

We round the answer to $9142.46

Answer:

[tex][0,\infty)[/tex]

Step-by-step explanation:

We have been given that the height, h, of a ball that is tossed into the air is a function of the time, t, it is in the air. The height in feet fort seconds is given by the function [tex]h(t)=-16t^2+96t[/tex].

We are told that the height of the ball is function of time, which means time is independent variable.

We know that the domain of a function is all real values of independent variable for which function is defined.

We know that time cannot be negative, therefore, the domain of our given function would be all values of t greater than or equal to 0 that is [tex][0,\infty)[/tex].

Answer:

7

Step-by-step explanation:

In this case, you go by the GREATEST DEGREE TERM POSSIBLE.

I am joyous to assist you anytime.

To calculate the expected winnings of the spinner game, one needs the probabilities of landing on specific segments. The expected value is found by summing the products of each outcome's probability and its monetary value, subtracting the cost of playing. Without these probabilities, an exact calculation cannot be provided.

To calculate the player's expected winnings in the game with the spinner, we need to understand the concept of expected value, which is essentially the average outcome if the game was played many times. For this game, we are given the following payouts: if the spinner lands on B, the player wins $8; if the spinner lands on C, the player wins $0.10 (a dime); otherwise, the player wins nothing. In addition, each spin costs $3, which will be factored into the expected winnings as a negative value.

Unfortunately, we do not have the probabilities of landing on B or C. Expected value is usually calculated by multiplying the probability of each outcome by its corresponding value and then summing those products. The general formula is Expected value = Σ(Probability of outcome × Value of outcome) - Cost per play.

Without the specific probabilities or the number of segments on the spinner, we cannot calculate the exact expected winnings. However, if hypothetical probabilities were provided, the calculation would follow the structure of: (Probability of landing on B × $8) + (Probability of landing on C × $0.10) - $3.

Answer:

Equilibrium quantity = 26.92

Equilibrium price is $31.13

Step-by-step explanation:

Given :Demand function : [tex]p^2 + 16q = 1400[/tex]

           Supply function : [tex]700 -p^2 + 10q = 0[/tex]

To Find : find the equilibrium quantity and equilibrium price.

Solution:

Demand function : [tex]p^2 + 16q = 1400[/tex]  --A

Supply function : [tex]p^2-10q=700[/tex] ---B

Now to find the equilibrium quantity and equilibrium price.

Solve A and B

Subtract B from A

[tex]p^2-10q -p^2-16q=700-1400[/tex]

[tex]-26q=-700[/tex]

[tex]26q=700[/tex]

[tex]q=\frac{700}{26}[/tex]

[tex]q=26.92[/tex]

So, equilibrium quantity = 26.92

Substitute the value of q in A

[tex]p^2 + 16(26.92) = 1400[/tex]

[tex]p^2 + 430.72 = 1400[/tex]

[tex]p^2 = 1400- 430.72[/tex]

[tex]p^2 = 969.28[/tex]

[tex]p = \sqrt{969.28}[/tex]

[tex]p = 31.13[/tex]

So, equilibrium price is $31.13

Answer:

The correct option is: (D) The psychology test score is relatively better because its z score is greater than the z score for the economics test score.

Step-by-step explanation:

Consider the provided information.

For psychology test:

Scores on the psychology test have a mean of 86 and a standard deviation of 15.

Use the Z score test as shown:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Substitute x=73, μ=86 and σ=15 in the above formula.

[tex]z=\frac{73-86}{15}[/tex]

[tex]z=-0.866[/tex]

For economics test:

Scores on the economics test have a mean of 48 and a standard deviation of 7.

Use the Z score test as shown:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Substitute x=41, μ=48 and σ=7 in the above formula.

[tex]z=\frac{41-48}{7}[/tex]

[tex]z=-1[/tex]

The Z score of psychology test is greater than the Z score of economic test.

Thus, the correct option is: (D) The psychology test score is relatively better because its z score is greater than the z score for the economics test score.

Answer:

doctorate salary:  d = $71,000

bachelor salary:     b = $55,000

Step-by-step explanation:

doctorate salary:  d

bachelor salary:     b

relationship between b and d:  d = 2b - $39,000

Unfortunately, your "1 + 126,000" could not be correct.  I'm going to assume that the total of the two salaries is $126,000.  If that's the case, then:

bachelor salary + (2 times bachelor salary less $39,000) = $126,000.  In symbols,

b + 2b - $39,000 = $126,000

Consolidating the 'b' terms results in 3b - $39,000 = $126,000, and so:

3b = $126,000 + $39,000 = $165,000

Dividing both sides by 3 yields the bachelor salary:  b = $55,000

Then the doctoral salary is 2b - $39,000 = 2($55,000) - $39,000, or $71,000.

Check:  Does b + (2b - $39,000) = $126,000?  Yes.

Then the doctoral salary is d = $103,000, and the bachelor salary $29,000

Let l and w be the length and width of the rectangle. We know that [tex]l=2w-4[/tex]

The formula for the perimeter is [tex]P=2(w+l)[/tex]

Using our substitution, it becomes

[tex]P=2(w+2w-4)=2(3w-4)=6w-8[/tex]

We know that the perimeter is 34, so we have

[tex]6w-8=34 \iff 6w=42 \iff w=7[/tex]

The length is 4 less than twice the width, so we have

[tex]l=2\cdot 7 - 4 = 10[/tex]

Answer:

The values of k are

1) k = 7.

2) k= 13

Step-by-step explanation:

The given differential equation is

[tex]y''-20y'+91y=0[/tex]

Now since it is given that [tex]y=e^{kx}[/tex] is a solution thus it must satisfy the given differential equation thus we have

[tex]\frac{d^2}{dx^2}(e^{kx})-20\frac{d}{dx}e^{kx}+91e^{kx}=0\\\\k^{2}\cdot e^{kx}-20\cdot k\cdot e^{kx}+91e^{kx}=0\\\\e^{kx}(k^{2}-20k+91)=0\\\\k^{2}-20k+91=0[/tex]

This is a quadratic equation in 'k' thus solving it for k we get

[tex]k=\frac{20\pm \sqrt{(-20)^2-4\cdot 1\cdot 91}}{2}\\\\\therefore k=7,k=13[/tex]

Final answer:

The two values of k satisfying the differential equation y'' - 20y' + 91y = 0 are found by substituting y(x) = e^kx into the equation, resulting in a quadratic equation k^2 - 20k + 91 = 0. Solving this yields the values k = 7 and k = 13.

Explanation:

To find the two values of k for which y(x) = ekx is a solution to the differential equation y'' - 20y' + 91y = 0, we start by differentiating the function y(x) = ekx twice to get the first and second derivatives, y' = kekx and y'' = k2ekx respectively. Substituting these into the given differential equation, we get:

k2ekx - 20kekx + 91ekx = 0.

Factor out ekx which is always positive and thus cannot be zero, we obtain a quadratic equation in terms of k:

k2 - 20k + 91 = 0.

Solving this quadratic equation gives us the two values of k. The solutions are obtained by finding the roots of the equation which involves factoring or using the quadratic formula. These will be the two constants we are looking for.

The characteristic equation is factorable and results in (k - 7)(k - 13) = 0. Therefore, the two values of k are 7 and 13, which are the smaller value and larger value respectively.

Hey!

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Steps To Solve:

~Subtract n to both sides

3 - n = n + 4 - n

~Simplify

3 - n = 4

~Subtract 3 to both sides

3 - n - 3 = 3 - 4

~Simplify

n = -1

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Answer:

[tex]\large\boxed{n~=~-1}[/tex]

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Hope This Helped! Good Luck!

Answer:

The equations for the value V in terms of t is [tex]V(t)=-16000\cdot t+1000000[/tex] and the value of the building

after 1 year is $984,000

after 2 years is $968,000

after 20 years is $680,000

Step-by-step explanation:

With the straight-line depreciation method, the value of an asset is reduced uniformly over each period until it reaches its salvage value(It is the value of the asset at the end of its useful life).

We know from the information given the year = 0 the building costs $1,000,000 and a the year = 25 it costs $600,000.

With this information, you can calculate the decrease in value of the building due to age. We use the slope of a line formula because is a straight-line depreciation.

If (0, $1,000,000) is the first point and (25, $600,000) is the second point. we have

[tex]m=\frac{V_{2} -V_{1}}{t_{2}-t_{1}} =\frac{600000-1000000}{25-0} =-16000 \frac{\$}{years}[/tex]

To find the equation for the value V in terms of t, we use the point-slope form, this expression let you calculate the value of the building at the end of the year (t)

[tex]V-V_{0}= m(t- t_{0})\\V-1000000=-16000(x-0)\\V=-16000\cdot t+1000000[/tex]

To find the value after 1 year, after 2 years, and after 20 years. We put the year into the equation [tex]V(t)=-16000\cdot t+1000000[/tex]

[tex]V(1)=-16000\cdot (1)+1000000=\$984,000\\V(2)=-16000\cdot (2)+1000000=\$968,000\\V(20)=-16000\cdot (20)+1000000=\$680,000[/tex]

Answer:

C. A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring. The sample is then called a simple random sample.

Step-by-step explanation:

Simple Random Sampling is the sampling where samples are chosen randomly, where each unit has an equal chance of being selected in a sample.

Option A is incorrect as the size of the sample and size of the population is not the same generally if it does happen then there will be no difference between sample and population.

Option B is incorrect as Simple Random Sampling is not a chance it is a way that samples can be taken.

Option D is incorrect as when samples are taken using a convenient sample then it is called Convenient Sample, not Simple Random Sample.

Thus, only option C is correct.

Simple random sampling is a method where each possible sample from a population has an equal chance of being chosen. This ensures that all members of a population have an equal opportunity of being in the sample, thus representing the population accurately. It differs from other sampling techniques like convenience sampling, stratified sampling, cluster sampling and systematic sampling.

Simple random sampling can best be defined as the process in which each member of a population initially has an equal chance of being selected for the sample. In other words, a sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring. This sample is then called a simple random sample. An example of this process may be selecting the names of students from a hat for a study group, where each student from the class(population) has an equal chance of being selected.

On the contrary, it should not be mistaken with convenience sampling, which is a non-random method of choosing a sample that can produce biased data. Other variants of random sampling include, but not limited to, stratified sampling, cluster sampling, and systematic sampling.

Overall, simple random sampling is a vital and simple method part of statistics to represent a population accurately for a study.

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Answer:

The value of f(2) is 25.

Step-by-step explanation:

Given,

[tex]f(x) = a^x[/tex]

[tex]\implies f(3) = a^3[/tex]

According to the question,

[tex]f(3)=125[/tex]

[tex]\implies a^3=125\implies a = (125)^\frac{1}{3}=5[/tex]

Hence the function would be,

[tex]f(x) = 5^x[/tex]

If x = 2,

[tex]f(2)=5^2\implies f(2)=25[/tex]

To find f(2), substitute x = 2 into the function f(x).

To find f(2), we can substitute the value of x into the function f(x). From the given information, we know that f(3) = 125. So, substituting x = 3 into the function gives us:

f(3) = 0.25e^(-0.25(3))

Simplifying this expression gives us 125 = 0.25e^(-0.75). Rearranging the equation, we can find the value of a:

a = 125 / 0.25e^(-0.75) = 500e^(0.75)

Now, we can substitute x = 2 into the function f(x):

f(2) = 500e^(0.75)(0.25e^(-0.25(2))) = 500e^(0.75)e^(-0.5)

Simplifying this expression gives us:

f(2) = 125e^0.25

So, the value of f(2) is 125e^0.25.

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Answer:

a) It is the set of point in the the circumference with equation [tex](x-4)^2+y^2=8[/tex].

b) (10.6, 0, -6 )

Step-by-step explanation:

a) The centre of the sphere is (4,0,-6) and the radio of the sphere is [tex]\sqrt{44} \sim 6.6[/tex]. Since |-6|=6 < 6.6,  then the sphere intersect the xy-plane and the intersection is a circumference.

Let's find the equation of the circumference.

The equation of the xy-plane is z=0. Replacing this in the equation of the sphere we have:

[tex](x-4)^2+y^2+6^2=44[/tex], then [tex](x-4)^2+y^2=8[/tex].

b) Observe that the point (7,0,-6) has the same y and z coordinates as the centre and the x coordinate of the point is greater than that of the x coordinate of  the centre. Then the point of the sphere nearest to the given point will be at a distance of one radius from the centre, in the positive x direction.

(4+[tex]\sqrt{44}[/tex], 0, -6)= (10.6, 0, -6 )

Answer:

360 cents or $ 3.6

Step-by-step explanation:

Let x be the original cost ( in cents ) of each note book,

After reducing the price by 30 cents,

New cost of each book = x - 30,

According to the question,

3x = 4(x-30)  ( ∵ total cost = number of books × cost of each book ),

3x = 4x - 120

3x - 4x = -120

-x = -120

x = 120

Hence, the cost of three books = 3 × 120 = 360 cents or $ 3.6

Answer:

The vertical height, h = 34.64 feets

Step-by-step explanation:

Given that,

Length of the ladder, l = 40 ft

The ladder makes an angle of  60 degrees with the ground, [tex]\theta=60^{\circ}[/tex]

We need to find the vertical height of of which the ladder will reach. Let it iss equal to h. Using trigonometric equation,      

[tex]sin\theta=\dfrac{perpendicular}{hypotenuse}[/tex]

Here, perpendicular is h and hypotenuse is l. So,

[tex]sin(60)=\dfrac{h}{40}[/tex]

[tex]h=sin(60)\times 40[/tex]

h = 34.64 feets

So, the vertical height of which the ladder will reach is 34.64 feets. Hence, this is the required solution.

Answer:

The statement [tex](p\lor q) \land (\neg p \lor r)\Rightarrow (q \lor r )[/tex] is a tautology.

Step-by-step explanation:

To prove this statement [tex](p\lor q) \land (\neg p \lor r)\Rightarrow (q \lor r )[/tex] is a tautology we are going to use a truth table. A truth table shows how the truth or falsity of a compound statement depends on the truth or falsity of the simple statements from which it's constructed.

A tautology is a formula which is true for every assignment of truth values to its simple components.

We can see from the table that the last column contains only true values. Therefore, the formula is a tautology.