In​ 2012, the population of a city was 6.81 million. the exponential growth rate was 1.61​% per year. ​a) find the exponential growth function. ​b) estimate the population of the city in 2018. ​c) when will the population of the city be 10 ​million? ​d) find the doubling time.

Given the lengths of the major and minor axes of an ellipse, one can derive the standard form equation of the ellipse. In this example, the lengths were 16 and 4, yielding a standard form equation of x²/4 + y²/64 = 1.

The subject of this question is about finding the standard form equation for an ellipse given the lengths of the major and minor axes. To find the equation of the ellipse, we need to identify the semi-major axis (a) which is half the length of the major axis, and the semi-minor axis (b), which is half the length of the minor axis.

In this case, the lengths of the major and minor axes are 16 and 4 respectively, which gives us a semi-major axis of 8 and a semi-minor axis of 2. The standard form equation of an ellipse with a vertical major axis is given by:

(x-h)²/b² + (y-k)²/a² = 1

Where (h, k) is the center of the ellipse. In this case, since the center of the ellipse is at the origin (0,0), the equation becomes:

x²/4 + y²/64 = 1

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Answer:

12

Step-by-step explanation:

The possible values of the third side of the triangle can be written as, 8 < c < 12.

A triangle is a two dimensional figure which consist of three vertices, three edges and three angles.

Sum of the interior angles of a triangle is 180 degrees.

Given a triangle.

Side lengths of the triangle are 10 centimeters, 2 centimeters, and c centimeters.

We have to find the possible values of c.

By the triangle Inequality Theorem,

10 - 2 < c < 10 + 2

8 < c < 12

Hence the possible values of c are 9, 10 and 11.

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The equation of the line perpendicular to y = 3x -7 that passes through the point (6, 8) is Option D. y = -1/3x + 10. The slope of the perpendicular line is -1/3. Using point-slope form, we determine the final equation.

To find the equation of a line perpendicular to y = 3x - 7 that passes through the point (6, 8), we must determine the slope of the perpendicular line first.

The slope of the given line is 3. The slope of a line perpendicular to this one is the negative reciprocal of 3, which is -1/3.

We now use the point-slope form of the linear equation, which is:

y - y₁ = m(x - x₁)

Substitute m = -1/3 and the point (6, 8):

y - 8 = -1/3(x - 6)

Distribute -1/3:

y - 8 = -1/3x + 2

Add 8 to both sides:

y = -1/3x + 10

Thus, the equation is Option D. -1/3x + 10.

Final answer:

To find the probability of at least 90 rolls being necessary for the total sum of all rolls to exceed 375, we need to consider the probabilities associated with each roll.

Explanation:

To find the probability of at least 90 rolls being necessary for the total sum of all rolls to exceed 375, we need to consider the probabilities associated with each roll.

The maximum possible sum from a single roll of a six-sided die is 6, so after 90 rolls, the maximum possible sum is 90 * 6 = 540. If the total sum exceeds 375, it means that at least one roll resulted in a value greater than 2.

To calculate the probability, we need to find the complement of the event that the total sum is less than or equal to 375, which is the event of the total sum being greater than 375. Let's assume that the probability of rolling a value greater than 2 is p. The probability of at least 90 rolls being necessary is 1 - (1 - p)^90.

Answer:

[tex]x=2\text{ (or) }x=11[/tex]

Step-by-step explanation:

We have been given factors of a polynomial as [tex](x-2)[/tex] and [tex](x-11)[/tex]. We are asked to find the values of x that would make the polynomial equal to zero.

We will use zero product property of polynomials to solve our given problem. The zero product property of polynomials states that if any of two factors of a polynomial is zero, then their product will be equal to zero.

Upon equating our given factors to zero, we will get:

[tex](x-2)(x-11)=0[/tex]

[tex](x-2)=0\text{ (or) }(x-11)=0[/tex]

[tex]x-2=0\text{ (or) }x-11=0[/tex]

[tex]x-2+2=0+2\text{ (or) }x-11+11=0+11[/tex]

[tex]x=2\text{ (or) }x=11[/tex]

Therefore, [tex]x=2\text{ (or) }x=11[/tex] will make the polynomial equal to zero.

Answer:

The answer is:   'no solution'

Step-by-step explanation:

The given equation is:   [tex]3\sqrt{5x-4}=3\sqrt{7x+8}[/tex]

Dividing both sides of the equation by 3, we will get.....

[tex]\sqrt{5x-4}=\sqrt{7x+8}[/tex]

Taking square on both sides.....

[tex](\sqrt{5x-4})^2=(\sqrt{7x+8})^2\\ \\ 5x-4=7x+8\\ \\ 5x-7x=8+4\\ \\ -2x=12\\ \\ x=\frac{12}{-2}=-6[/tex]

Plugging this [tex]x=-6[/tex] back to the given equation......

[tex]3\sqrt{5(-6)-4}=3\sqrt{7(-6)+8}\\ \\ 3\sqrt{-34}=3\sqrt{-34}[/tex]

As we are getting a negative number inside the square root, so the equation becomes imaginary. Thus [tex]x=-6[/tex] is a restricted value.

Hence, there is 'no solution' for this equation.

The equation of the conic section is [tex]\frac{145x\²}{24336} - \frac{145y\²}{169}=1[/tex]

The foci of the hyperbola are given as: (13,0) and (-13,0)

The asymptote is given as [tex]y =\pm 12x[/tex]

Divide both sides of [tex]y =\pm 12x[/tex] by x.

[tex]\frac yx = \pm 12[/tex]

Where y/x = a/b.

So, we have:

[tex]\frac ab = \pm 12[/tex]

Make a the subject

[tex]a = \pm 12b[/tex]

Recall that

[tex]c\²=a\²+b\²[/tex]

Where:

[tex]c = \pm13[/tex]

[tex]c\²=a\²+b\²[/tex] becomes

[tex](\pm 13)^2 = (\pm 12b)^2 +b^2[/tex]

[tex]169 = 144b^2 +b^2[/tex]

Evaluate like terms

[tex]169 = 145b^2[/tex]

Make b^2 the subject

[tex]b^2 = \frac{169}{145}[/tex]

Recall that [tex]a = \pm 12b[/tex]

Square both sides

[tex]a^2 = 144b^2[/tex]

Substitute  [tex]b^2 = \frac{169}{145}[/tex]

[tex]a^2 = 144 \times \frac{169}{145}[/tex]

[tex]a^2 = \frac{24336}{145}[/tex]

The equation of the conic section is represented as:

[tex]\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1[/tex]

Substitute known values

[tex]\frac{x^2}{24336/145} - \frac{y^2}{169/145} = 1[/tex]

Rewrite as:

[tex]\frac{145x\²}{24336} - \frac{145y\²}{169}=1[/tex]

Hence, the equation of the conic section is [tex]\frac{145x\²}{24336} - \frac{145y\²}{169}=1[/tex]

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The area of a given isosceles triangle with sides of 1 foot in length and a 45-degree angle between these sides is 0.5 square feet.

The area of an isosceles triangle can be calculated using the formula 1/2 base times height. But since we know that the triangle is isosceles and the angle between the equal sides is 45 degrees, this forms a 45-45-90 degree triangle which is a special kind of triangle. In a 45-45-90 degree triangle, the lengths of the sides are in the ratio 1:1:√2. Therefore, the length of the base (which also serves as the height in this case) will be the same length as the equal sides, 1 foot. Substituting these into the formula for area, we get Area = 1/2 * 1 ft * 1 ft = 0.5 square feet.

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