A country’s population in 1992 was 50 million. In 2992 it was 53 million. Estimate the population in 2010 using the exponential growth formula. Round your answer to the nearest formula.

ANSWER

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

EXPLANATION

Since the parabola has the (0, 0) and the directrix at y=2.

The equation is of the form

[tex] {(x - h)}^{2} = - 4p(y - k)[/tex]

where (h,k) is the vertex.

The vertex is half way between the directrix and the focus.

The vertex will be at (0,1)

and is the distance from the vertex to the focus which is 1.

[tex]{(x - 0)}^{2} = - 4(1)(y - 1)[/tex]

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

Speed is distance/time so

[(d/20)+(d/24)]/2= 24d+20d/24•20•2=44d/20•24•2=22d/20•24=11d/240

Answer:

  it depends

Step-by-step explanation:

If the tradeoff between giants and elves is linear, then about 76 elves can be admitted. If it is non-linear, we need more information about the tradeoff.

For example, elves may fit in the spaces between giants in such a way that they take up proportionately less room. Or, elves may repel giants in such a way that they take up proportionately more room.

See the attached graph for some possibilities.

_____

The linear equation can be written as ...

  g/160 + e/240 = 1

Then for g=109, we can solve for "e" as ...

  109/160 + e/240 = 1

  240(1 -109/160) = e = (3/2)(51) = 76.5

If the tradeoff is linear, 76 elves can be admitted.

Answer:

Option A. 5.8 miles

Step-by-step explanation:

It is given in the question that angle SCD = 45°  and ∠ SAD = 35°

Distance between Cooper and Drew = 10 miles

Let distance between Ship and Cooper is = x miles

Now we apply Sine rule in the Δ SCD

[tex]\frac{sin100}{10}=\frac{sin35}{x}[/tex]

[tex]\frac{0.9848}{10}=\frac{0.5736}{x}[/tex]

By cross multiplication

(0.9848)x = 10(0.5736)

[tex]x=\frac{5.736}{0.9848}=5.82[/tex] miles

Option A. 5.8 miles is the correct option.

Answer:

x=18

Step-by-step explanation:

The two triangles are similar. The ratio of the sides are the same.

20/20=40/(40+x)

Cross multiply and solve. Hint: the easiest way to solve is to simplify, then solve.

Answer:

8

Step-by-step explanation:

Let's set this up to see if we can simplify it a bit:

[tex](\frac{10^4*5^2}{10^3*5^3})^3[/tex]

Notice we have 4 tens on top and 3 on bottom, so we can eliminate the bottom 3 altogether and leave just one on top.  And we have 2 fives on the top and 3 on bottom, so we can eliminate the 2 on the top and leave one on the bottom.  Now that looks like this:

[tex](\frac{10}{5})^3[/tex]

10 divided by 5 is 2, and 2 cubed is 8

Final answer:

To find the value of the expression ((10⁴)(5²)/(10³)(5³))³, we simplify the powers of 10 and 5 and then raise the result to the power of 3, obtaining the final answer of 8.

Explanation:

To solve the expression ((10⁴)(5²)/(10³)(5³))³, we start by simplifying inside the parentheses. We use the properties of exponents to divide the powers of tens and fives separately:

So the inside of the parentheses simplifies to (10 * 1/5) which is 2. Then, we raise 2 to the power of 3 as indicated by the expression:

(2)^3 = 2*2*2 = 8

Therefore, the value of the expression ((10⁴)(5²)/(10³)(5³))³ is 8.

Answer:

Step-by-step explanation:

The length of EA is the difference between AD (6) and ED (5), so is ...

  6 - 5 = 1

That is, the distance ED is 5 times the distance EA.

The two triangles are similar, so the distance BC will be 5 times the distance BA:

  BC = 5·AB

  BC = 5·2 = 10 . . . . . substitute for length AB

  x +4 = 10 . . . . . . . . . substitute for length BC

  x = 10 -4 = 6 . . . . . . subtract 4

We already know BC = 10. Of course AC = AB + BC = 2+10 = 12.

The lengths of interest are x=6, BC=10, AC=12.

[tex]\boxed{d=2\sqrt{107}}[/tex]

For a rectangular prism whose side lengths are [tex]a,\:b\:and\:c[/tex] the internal diagonal can be calculated as:

[tex]d=\sqrt{a^{2}+b^{2}+c^{2}}[/tex]

So here, we know that:

[tex]a=l=18 \\ \\ b=w=10 \\ \\ c=h=2[/tex]

So:

[tex]d=\sqrt{l^{2}+w^{2}+h^{2}} \\ \\ d=\sqrt{18^{2}+10^{2}+2^{2}} \\ \\ d=\sqrt{324+100+4} \\ \\ d=\sqrt{428} \\ \\ \boxed{d=2\sqrt{107}}[/tex]

Answer : The value of diagonal of the rectangular solid is, 20.69 unit.

Step-by-step explanation :

First we have to calculate the side AC.

Using Pythagoras theorem in ΔABC :

[tex](Hypotenuse)^2=(Perpendicular)^2+(Base)^2[/tex]

[tex](AC)^2=(AB)^2+(BC)^2[/tex]

Given:

Side AB = l = 18

Side BC = w =  10

Now put all the values in the above expression, we get the value of side AC.

[tex](AC)^2=(18)^2+(10)^2[/tex]

[tex]AC=\sqrt{(18)^2+(10)^2}[/tex]

[tex]AC=\sqrt{324+100}[/tex]

[tex]AC=\sqrt{424}[/tex]

Now we have to calculate the side AD (diagonal).

Using Pythagoras theorem in ΔACD :

[tex](Hypotenuse)^2=(Perpendicular)^2+(Base)^2[/tex]

[tex](AD)^2=(AC)^2+(CD)^2[/tex]

Given:

Side AC = [tex]\sqrt{424}[/tex]

Side CD = h =  2

Now put all the values in the above expression, we get the value of side AD.

[tex](AD)^2=(\sqrt{424})^2+(2)^2[/tex]

[tex]AD=\sqrt{(\sqrt{424})^2+(2)^2}[/tex]

[tex]AD=\sqrt{424+4}[/tex]

[tex]AD=\sqrt{428}[/tex]

[tex]AD=20.69[/tex]

Thus, the value of diagonal of the rectangular solid is, 20.69 unit.

Answer:

k.lkkkkkkkkkkkkk

200 ft because 8x25 is 200

Answer:

Explanation:

1) Write the model for the volume and base area of the rectangular sotorage container:

       base width, w = x

       base length, l = 2x

        B = (2x) (x) = 2x²

        height = h

        V = 2x² h = 10 ⇒ h = 10 / (2x²)

2) Total area, A

        S₁ = (x) . (h) = (x) . 10 / (2x²) = 10 / (2x) = 5 / x

        S₂ = S₁ = 5 / x

        S₃ = (2x) . (h) = (2x) . 10 / (2x²) = 10 / x

        S₄ = S₃ = 10 / x

3) Cost

        $ 10 (2x²) = 20x²

$6 (S₁ + S₂ + S₃ + S₄) = 6 (5/x + 5/x + 10/x + 10/x ) = 6 ( 30/x) = 180/x

4) Cheapest container

Minimum cost ⇒ find the minimum of the function 20x² + 180 / x, which formally is done by derivating the function and making the derivative equal to zero.

Solve to find the value of x that makes the first derivative equal to zero:

Replace x = 1.65 in the equations of costs to find the minimum cost:

That is the final answer, already rounded to the nearest cent: $163.54

The question is about the optimization of the costs for constructing a rectangular container with a specific volume of 10 m^3.  This is done by creating equations for volume and cost, and then optimizing the cost using calculus.

The subject of this question is Optimization in Calculus. Here, we trying to minimize the cost of a rectangular storage container with given constraints. First, we need to setup equations for the volume, V = length x width x height and cost, as given. Since the volume of the box is fixed at 10 m3 and given that the length is twice the width we have:

We solve for h in terms of w from the volume equation and substitute it in the cost, we can then take the derivative of the cost equation and solve it to get the width. Lastly, we plug the width value in the cost equation to get the minimal cost.

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X=93 degrees

Y=74 degrees

       10√2

and this simplifies to a = -----------

                                               2Answer:

Step-by-step explanation:

The cosine function links the angle (45°), the side a and the hypotenuse (10):

                     a

cos 45° = ------------

                     10                    

                                              1

and so a = 10 cos 45° = 10(------)  =  10√2 / 2  = a

                                             √2

Answer:

  x = 18

Step-by-step explanation:

The parts of the triangles are proportional, so you can write ...

  2x/80 = 27/60

Multiply by 120:

  3x = 54

Divide by 3:

  x = 18

2:6 so simplify it 1:3. This means the scale ratio is 1:3 for the figures. To find w, sub in the 3 from the smaller figure to the ratio so 3×3=9, 3:9 w=9

Answer:

w = 9

Step-by-step explanation:

Since the 2 polygons are similar then the ratios of corresponding sides are equal, that is

[tex]\frac{2}{6}[/tex] = [tex]\frac{3}{w}[/tex] ( cross- multiply )

2w = 18 ( divide both sides by 2 )

w = 9

Answer:

The maximum number of turns is 3

Step-by-step explanation:

The given function is

[tex]f(x)=x^4-x^3+3x+1[/tex]

The degree of this polynomial is 4.

If the degree of a given polynomial is n, then the polynomial has at least n-1 turns.

Therefore the number of turns of this 4th degree polynomial is at least 3.

Answer:

A is the closest one

Step-by-step explanation:

Remark

The general formula is

V = V_2 cones + V_cylinder.

Formula

That turns out to be

V = 2*(1/3)*pi*r^2*h1 +  pi*r^2*h

Givens

r = 7 feet

h = 22 feet

h1 = 12 feet

pi = 3.14

Solution

V = 2*(1/3)*3.14 * 7^2*12  +   3.14* 7^2 * 22

V = 1231.5 ft^3 + 3384.92

V = 4616.42

Answer:

4618.1 ft3

Step-by-step explanation:

That's the exact correct answer :)

The answer is 4 hours and 49 minutes HERE IS MY WORK

4 hours and 49 minutes

Answer:

  D. a constant function divided by the sum of a constant function and an exponential function

Step-by-step explanation:

It's all about the meaning of the math symbols used to create the function. The exponential function is called that because the variable (t) is in the exponent of the expression.

The "numerator" is the part of the expression above the "divided by" line. The "denominator" is the part below that line. When describing a fraction like this, we say, "<the numerator> divided by <the denominator>."

The numerator is a number with no math symbols or variables: it is a constant. The same is true of the left term of the denominator.

The plus sign between the denominator terms indicate these terms are added together. The result of that addition is called a "sum." The terms either side of the plus symbol are those that the symbol is indicating the sum of.

So, we can describe the expression as ...

  a constant function divided by the sum of a constant function and an exponential function

The logistic equation is a model of population growth, correctly described as a constant function divided by the sum of a constant function and an exponential function. This matches option D from the choices provided.

The logistic equation is a model of population growth where the growth rate decreases as the population approaches its carrying capacity. The equation structure is generally given as a constant function divided by the sum of a constant function and an exponential function. Thus, the correct description from the options provided would be D. A constant function divided by the sum of a constant function and an exponential function.

As an example, the basic form of a logistic function is P(t) = c / (1 + ae-bt). Here c is the constant function representing the maximum achievable population and ae-bt is an exponential function that models the population growth over time 't'.

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A destructive earthquake happens once per year.

The exponential equation would be f(x) = 1-e^-x

The probability of going 3 months out of a year would be

P(X≥3/12)  ( divide 3 months by 12 months per year).

Now x equals 3/12

Now you have

P = 1-(1-e^-3/12)

= e^-1/4

= 0.7788

The probability that at least 3 months elapse would be 0.7788

(Round answer as needed).

The probability that at least 3 months elapse before the first earthquake of destructive magnitude occurs is; 0.7788

We are told that a destructive earthquake happens once per year.

Thus, the exponential equation in this scenario is;

f(x) = 1 - e⁻ˣ

Thus, the probability that at least 3 months elapse before the first earthquake of destructive magnitude occurs is given as;

P(X ≥ ³/₁₂)  since 12 months make a year

This will be;

P(X ≥ ³/₁₂) = 1 - (1 - e^(⁻³/₁₂))

P(X ≥ ³/₁₂) = e^-1/4

P(X ≥ ³/₁₂)= 0.7788

In conclusion, the probability that at least 3 months elapse would be 0.7788

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The first line of music shows a glide reflection from all the lines.

A glide reflection is one of the types of reflection in 2-dimensional geometry.

It is a symmetry figure that consists of a reflection over a line and then translates along the line.

In this figure, the first option shows the exact symmetry operation and reflection. but the second and third line doesn't represent a glide reflection.

In the second option, the reflection does not take place, the music lines are the same before and after.

In the third option, the music lines are just interchanged which doesn't prove the glid reflection.

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Answer:

102 students

Step-by-step explanation:

Note that 65% and 71% are both 1 standard deviation from the mean (71%).  According to the empirical rule, 68% of scores lie within 1 std. dev. of the mean.

68% of 150 students would be 0.68(150 students) = 102 students

68% of 150 students would be 0.68 (150 students) is, 102 students.

According to the question:

65% and 71% are both 1 standard deviation from the mean (71%).

According to the empirical rule, 68% of scores lie within 1 std. dev. of the mean.

68% of 150 students would be 0.68(150 students) = 102 students.

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Answer:

x = 2

Step-by-step explanation:

Answer:

x = 2

Step-by-step explanation:

use the Vertex form, Y=a(x-h)^2+k, to determine the values of a, h, and k.

a=2

h=2

k=5

since the values of a are positive the parabola opens up

Find the vertex (h,k)

(2,5)

Find p from the vertex and the focus

1/8

find the focus

(2,41/8)

Find the axis of symmetry by finding the line that passes though the Vertex and the focus

x=2

Or also a line that goes through the middle but that is just my opinion!

Answer:

See below

Step-by-step explanation:

The given rational function is;

[tex]y=\frac{(x-3)(x+2)}{(x+4)(x-4)(x+2)}[/tex]

The given function is not continuous where the denominator is equal to zero.

[tex](x+4)(x-4)(x+2)=0[/tex]

The function is discontinuous at [tex]x=-4,x=4,x=-2[/tex]

b) The point at x=-2 is a removable discontinuity(hole) because (x+2) is common to both the numerator and the denominator.

The point at x=-4 and x=4 are  non-removable discontinuities(vertical asymptotes)

c) The equation of the vertical asymptotes are x=-4 and x=4

To find the equation of the horizontal asymptote, we take limit to infinity.

[tex]\lim_{x\to \infty}\frac{(x-3)(x+2)}{(x+4)(x-4)(x+2)}=0[/tex]

The horizontal asymptote is y=0

C) no solution

use desmos.com to help with these problems

Answer: Option C

C) No solution

Step-by-step explanation:

To graph both lines identify the cut points with the axes

For [tex]2x = -y + 7[/tex]

The intersection with the x axis is:

[tex]2x = -(0) +7\\\\x = 3.5[/tex]

The intersection with the y axis is:

[tex]2 (0) = -y +7\\\\y = 7[/tex]

Draw a line that cuts the y-axis in 7 and the x-axis in 3.5

For [tex]2y = -4x + 10[/tex]

The intersection with the x axis is:

[tex]2 (0) = -4x +10\\\\4x = 10\\\\x = 2.5[/tex]

The intersection with the y axis is:

[tex]2y = -4 (0) +10\\\\y = 5[/tex]

Draw a line that cuts the y-axis in 5 and the x-axis in 2.5

The intersection of both lines will be the solution of the system. Observe the attached image

The lines are parallel, so they never intercept. Therefore the system has no solution

There are two possible ways for X=1: first light is green and second is red OR first light is red and second is green. The probabilities for these two options are to be summed: 0.6*0.4 + 0.4*0.6 to give the probability of exactly one light being green, namely 0.48.

P(X = 1) = 0.48

"Probability is a branch of mathematics which deals with finding out the likelihood of the occurrence of an event."

"It is a probability distribution that gives only two possible results in an experiment, either success or failure. "

[tex]P(x)=n_C_x\times (p)^{x}\times (q)^{n-x}[/tex], here n is the total number of outcomes, 'p' is the probability of success and 'q' is the probability of failure

For given example,

The probability that traffic lights on a certain road will be green is 0.6

This means, the probability  of success (p) = 0.6

So, the probability of failure (q) = 1 - p

q = 0.4

When a driver on that road approaches two traffic lights in a row, X is the number of traffic lights that are green.

here, n = 2, p = 0.6, q = 0.4, x = 1

Using the formula of Binomial distribution the required probability is,

[tex]P(x=1)=2_C_1\times (0.6)^{1}\times (0.4)^{2-1}\\\\P(x=1)=\frac{2!}{1!\times (2-1)!}\times 0.6\times 0.4^1\\\\P(x=1)=2\times 0.6\times 0.4\\\\\bold{P(x=1)=0.48}[/tex]

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Answer:

Option D.

Step-by-step explanation:

Given that (x-3) is a factor of  P(x)=x^3-7x^2+15x-9. If we divide the P(x) by (x-3) we will get a second grade polynomial, which is easier to factorize.

Dividing x^3-7x^2+15x-9 by (x-3), the answer is: x^2 - 4x + 3 with a remainder of zero.

Now, to factorize x^2-4x+3 we just need to find two numbers that equal -4 when added and 3 when multiplied. These two numbers are -1 and -3.

So the complete factorization of P(x) is: (x-3)(x-1)(x-3)

Which is option D.

Answer: D. (x-3)(x-3)(x-1)

Step-by-step explanation:

Answer:

False

Step-by-step explanation:

Using the trigonometric identities

• sin²x + cos²x = 1

• secx = [tex]\frac{1}{cosx}[/tex]

Consider the left side

sec²x(1 - sin²x)

= [tex]\frac{1}{cos^2x}[/tex] × cos²x = 1

The correct identity is

sec²x(1 - sin²x) = 1

Answer:

h = 10.92 m.

Step-by-step explanation:

The small triangle on the left is similar to the whole triangle, so:

12/20 = h/18.2

h = 12*18.2 / 20

h = 10.92 m.

Answer:

Part 1) option a. [tex]y=(x+1)^{2}[/tex]

Part 2) option c. [tex]y(x)=10x+1[/tex]

Part 3) option a. Yes , d=-2

Part 4) option b.  [tex]y=2x+4[/tex]

Part 5) option b. [tex]m=-2[/tex]

Part 6) option c. [tex]y=4x+14[/tex]

Part 7) option c. [tex]y=4x+5[/tex]

Part 8) option a. y=2x-1 and y=x+1

Step-by-step explanation:

Part 1)

we know that

If a ordered pair satisfy a function, then the function pass through the ordered pair

Verify each function with the points (1,4), (2,9) and (3,16)

case a) we have

[tex]y=(x+1)^{2}[/tex]

For x=1, y=4

[tex]4=(1+1)^{2}[/tex]

[tex]4=4[/tex] ----> is true

For x=2, y=9

[tex]9=(2+1)^{2}[/tex]

[tex]9=9[/tex] ----> is true

For x=3, y=16

[tex]16=(3+1)^{2}[/tex]

[tex]16=16[/tex] ----> is true

therefore

The function pass through the three points

case b) we have

[tex]y=(x+3)^{2}[/tex]

For x=1, y=4

[tex]4=(1+3)^{2}[/tex]

[tex]4=16[/tex] ----> is not true

therefore

The function not pass through the three points

case c) we have

[tex]y=7x-5[/tex]

For x=1, y=4

[tex]4=7(1)-5[/tex]

[tex]4=2[/tex] ----> is not true

therefore

The function not pass through the three points

Part 2)

Let

y------> the number of laps

x-----> the number of hours

we know that

The linear equation that represent this situation is

[tex]y(x)=10x+1[/tex]

Part 3) we have

{4,2,0,-2,-4,-6,...}

Let

a1=-4

a2=2

a3=0

a4=-2

a5=-4

a6=-6

we know that

a2-a1=2-4=-2 -----> a2=a1-2

a3-a2=0-2=-2 ----> a3=a2-2

a4-a3=-2-0=-2 -----> a4=a3-2

a5-a4=-4-(-2)=-2----> a5=a4-2

a6-a5=-6-(-4)=-2----> a6=a5-2

therefore

Is an arithmetic sequence, the common difference is -2

Part 4) we know that

The y-intercept of the graph is (0,4)

The x-intercept of the graph is (-2,0)

therefore

the function is [tex]y=2x+4[/tex]

because

For x=0 -----> y=2(0)+4 -----> y=4

For y=0 ----> 0=2x+4 --------> x=-2

Part 5) we know that

The formula to calculate the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

we have

[tex]A(3,5)\ B(2,7)[/tex]

substitute the values

[tex]m=\frac{7-5}{2-3}[/tex]

[tex]m=-2[/tex]

Part 6) we know that

The equation of the line into slope point form is equal to

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]m=4[/tex]

[tex]point(-3,2)[/tex]

substitute the values

[tex]y-2=4(x+3)[/tex]

Convert to slope intercept form

[tex]y=4x+12+2[/tex]

[tex]y=4x+14[/tex]

Part 7) we know that

If two lines are parallel, then their slopes are the same

The equation of the given line is [tex]y=4x-2[/tex]

so

The slope of the given line is [tex]m=4[/tex]

therefore

The line [tex]y=4x+5[/tex] is parallel to the given line

Because the slope is equal to [tex]m=4[/tex]

Part 8) we know that

If a ordered pair is a solution of a system of equations, then the ordered pair must satisfy both equations of the system

Verify each case for (2,3)

case a)

y=2x-1 -----> equation 1

y=x+1 -----> equation 2

Substitute the value of x and the value of y in each equation and then compare the results

Verify equation 1

3=2(2)-1

3=3 -----> is true

Verify equation 2

3=2+1

3=3 -----> is true

therefore

The point (2,3) is a solution of the system of equations case a

case b)

y=2x+1 -----> equation 1

y=x-1 -----> equation 2

Substitute the value of x and the value of y in each equation and then compare the results

Verify equation 1

3=2(2)+1

3=5 -----> is not true

therefore

The point (2,3) is not a solution of the system of equations case b

case c)

y=4x-5 -----> equation 1

y=2x -----> equation 2

Substitute the value of x and the value of y in each equation and then compare the results

Verify equation 1

3=4(2)-5

3=3 -----> is true

Verify equation 2

3=2(2)

3=4 -----> is not true

therefore

The point (2,3) is not a solution of the system of equations case c

k = coefficient of similarity for lengths

k ^ 2 = coefficient of similarity for surfaces

k ^ 3 = coefficient of similarity for volumes

[tex]\displaystyle\bf\\ \text{Coefficient of similarity for lengths}=~k=\frac{1}{4}\\\\\implies~\text{Coefficient of similarity for surfaces}=k^2=\left(\frac{1}{4}\right)^{\b2}=\boxed{\bf\frac{1}{16}}[/tex]

When both the radius and slant height of a right cone are multiplied by 1/4, the effect on the surface area is that it is multiplied by 1/16.

In the case of a right cone, the total surface area is given by the formula πr(r + l), where r is the radius and l is the slant height. If both the radius and slant height are multiplied by 1/4, then the new surface area becomes π(1/4r)((1/4r)+(1/4l)) = π(1/16)r(r+l). This tells us our new surface area is 1/16 of the original surface area.

Therefore, the correct effect on the surface area when both the radius and slant height of a right cone are multiplied by 1/4 is that the surface area is multiplied by 1/16.

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