Which scatter diagram fits the given paired data
Systolic Blood Pressure
Diastolic Blood Pressure
119 125
76
84
131 130 123 118 142
91
84
89
76
91
138 125 131 128 140 123
93 78 88 75 94 76

a. An equation to describe the total cost of RV rental:

Cost = (125 * d) + (0.32 * m) + 2500

b. Comparing the two costs will determine which option is cheaper.

c. For the more direct route: m ≤ 3500

For fewer overnight stays: d ≤ 14

These domains ensure that we don't exceed the original values for miles and days.

d. I recommend compromising by lessening the number of days of the rental. By reducing the rental period to 11 days, you can stay within the $5,000 budget while still allowing your little sister to take the two-week trip.

a. To write an equation for the total cost of RV rental, we can use the given information. The cost per day is $125, and there is an additional charge of 32 cents per mile. Let's denote the number of days as d and the number of miles as m. The equation for the total cost of RV rental can be written as:

Cost = (125 * d) + (0.32 * m) + 2500

b. To compare the costs of the two options, we need to calculate the total cost for each. Option 1 has 3500 miles and takes 11 days, while option 2 has 3000 miles and takes 14 days. We can substitute these values into the equation from part a to find the total costs for each option. Comparing the two costs will determine which option is cheaper.

c. To compromise and stay within a budget of $5,000, we can adjust either the number of miles or the number of days. For the more direct route, we can reduce the number of miles, and for fewer overnight stays, we can reduce the number of days. The domains for these compromises would be:

For the more direct route: m ≤ 3500

For fewer overnight stays: d ≤ 14

These domains ensure that we don't exceed the original values for miles and days.

d. To find the number of miles or days to eliminate in order to stay under the $5,000 budget, we can set up equations using the total cost equation from part a. Let's denote the reduced number of miles as m' and the reduced number of days as d'. We need to solve the following equation for each compromise:

(125 * d') + (0.32 * m') + 2500 ≤ 5000

By substituting the appropriate values into the equation and solving for m' or d', we can determine how many miles or days need to be eliminated.

Based on the given information, I recommend compromising by lessening the number of days of the rental. By reducing the rental period to 11 days, you can stay within the $5,000 budget while still allowing your little sister to take the two-week trip. This compromise ensures that you don't have to sacrifice too much scenic beauty or make drastic changes to the route.

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Answer:

AB = √3

Step-by-step explanation:

Since ABCD is a rectangle, all angles are 90°

∠CDA = 90°

⇒ ∠CDB + ∠BDA = 90

⇒ ∠BDA = 60

In ΔABD,

sin(∠BDA) = opposite/ hypotenuse = AB / BD

⇒ sin(60) = AB/2

⇒ AB = 2 sin(60)

⇒ AB = 2 (√3)/2

AB = √3

The statement that correctly answers the question "A square prism has a base length of 5 m, and a square pyramid of the same height also has a base length of 5 m. Are the volumes the same?" is "No, because only the bases have the same area, not every cross-section at every level parallel to the bases."

Explanation: A square prism is a three-dimensional shape that has two square bases that are parallel to each other, and every side is a rectangle. In contrast, a square pyramid is a three-dimensional figure that has a square base and triangular faces that meet at a point called an apex or vertex. The height of a square pyramid is the distance from the base to the apex.

Therefore, the volume of a square prism can be calculated by multiplying the area of the base by the height, whereas the volume of a square pyramid can be determined by multiplying the area of the base by one-third of the height.

Thus, even though the base length is 5 m in both cases, the cross-sectional areas at every level parallel to the bases in a square pyramid are not the same. This implies that the answer is No, because only the bases have the same area, not every cross-section at every level parallel to the bases.

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1a. Compute NPV by calculating the present value of net cash flows and subtracting the investment amount.

1b. Compute PVI by dividing NPV by the investment amount.

2. Determine IRR by finding the discount rate corresponding to an NPV of zero.

3. Compare NPV, PVI, and IRR to identify the better financial opportunity.

1a. To compute the net present value (NPV) for each project, we need to calculate the present value of the annual net cash flows and subtract the amount to be invested. Using the present value of an annuity of $1 from the table, we can fill in the following values:

Wind Turbines:

Present value of annual net cash flows: $420,000 * 1.736 + $420,000 * 2.487 + $420,000 * 3.170 + $420,000 * 3.791

Less amount to be invested: $1,199,100

Net present value: NPV_Wind_Turbines = Present value of annual net cash flows - Amount to be invested

Biofuel Equipment:

Present value of annual net cash flows: $880,000 * 1.736 + $880,000 * 2.487 + $880,000 * 3.170 + $880,000 * 3.791

Less amount to be invested: $2,278,320

Net present value: NPV_Biofuel_Equipment = Present value of annual net cash flows - Amount to be invested

1b. The present value index (PVI) can be calculated by dividing the NPV by the amount to be invested:

Present Value Index = NPV / Amount to be invested

2. To determine the internal rate of return (IRR) for each project, we need to find the discount rate at which the NPV becomes zero. We can use the present value of an annuity of $1 from the table to calculate the present value factor for an annuity of $1. Then, we can find the discount rate that corresponds to an NPV of zero.

Wind Turbines:

Present value factor for an annuity of $1: Fill in the values from the table

Internal rate of return: IRR_Wind_Turbines = Discount rate corresponding to NPV = 0

Biofuel Equipment:

Present value factor for an annuity of $1: Fill in the values from the table

Internal rate of return: IRR_Biofuel_Equipment = Discount rate corresponding to NPV = 0

3. Based on the calculations of NPV, PVI, and IRR, we can compare the two projects. The project with the higher NPV, PVI, and IRR is considered the better financial opportunity. Both investments meet the minimum return criterion of 10%, but the project with the higher financial indicators is preferred.

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The value of the rational expression when x is equal to 4 is 2. The correct answer is option C: 2.

To find the value of the rational expression (x - 12)/(x - 8) when x is equal to 4, we substitute x = 4 into the expression:

[(4) - 12]/[(4) - 8]

Simplifying the numerator and denominator:

(4 - 12)/(-4)

Further simplifying the numerator:

(-8)/(-4)

Now, we can divide -8 by -4:

(-8)/(-4) = 2

So, when x is equal to 4, the value of the rational expression is 2.

Therefore, C is the right response.

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The product of two irrational numbers can be either rational or irrational, depending on the specific irrational numbers being multiplied. It is not always rational, nor is it never rational.

The product of two irrational numbers can be either rational or irrational, depending on the specific irrational numbers being multiplied. It is not always rational, nor is it never rational.

Consider the square root of 2 (√2) and the square root of 3 (√3), both of which are irrational numbers. When you multiply √2 and √3, you get √6, which is also an irrational number. In this case, the product of two irrational numbers is irrational.

However, there are cases where the product of two irrational numbers can be rational. For example, consider √2 and its reciprocal (1/√2), both of which are irrational. When you multiply these two numbers, you get 1, which is a rational number. So, in this case, the product of two irrational numbers is rational.

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Combining the results of a given triangle, we can conclude that the value of 'x' must be greater than -22 and also less than 52. So, the possible range for 'x' is -22 < x < 52.

To find the value of 'x' in a triangle with side lengths 'x', 37, and 15, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

In this case, we have:

x + 37 > 15 (Sum of x and 37 is greater than 15)

x + 15 > 37 (Sum of x and 15 is greater than 37)

37 + 15 > x (Sum of 37 and 15 is greater than x)

From the first inequality, we can subtract 37 from both sides:

x > 15 - 37

x > -22

From the second inequality, we can subtract 15 from both sides:

x > 37 - 15

x > 22

From the third inequality, we can subtract 15 from both sides:

52 > x

Combining the results, we can conclude that the value of 'x' must be greater than -22 and also less than 52. So, the possible range for 'x' is -22 < x < 52.

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Answer:

(90°/360°)π(3^2) = (1/4)(9π) = 9π/4 m²

1. The missing value is b ≈ 10.

2. The missing value is a ≈ 12.

3. The missing value is b ≈ 13.

4. The missing value is b ≈ 10.

5. The missing value is a ≈ 4.

6. The missing value is b ≈ 5.

7. The missing value is a ≈ 11.

8. The missing value is b ≈ 6.

9. The missing value is b ≈ 20.

10. The missing value is a = 0.

Using the Pythagorean theorem, we can solve for b:

[tex]b^2 = c^2 - a^2b^2 = 10^2 - 4^2b^2 = 96b ≈ 10[/tex]

2. Using the Pythagorean theorem, we can solve for a:

[tex]a^2 = c^2 - b^2a^2 = 12^2 - 3^2a^2 = 135a ≈ 12[/tex]

3. Using the Pythagorean theorem, we can solve for b:

[tex]b^2 = c^2 - a^2b^2 = 14^2 - 6^2b^2 = 160b ≈ 13[/tex]

4. Using the Pythagorean theorem, we can solve for b:

[tex]b^2 = c^2 - a^2b^2 = 12^2 - 7^2b^2 = 95b ≈ 10[/tex]

5. Using the Pythagorean theorem, we can solve for a:

[tex]a^2 = c^2 - b^2a^2 = 10^2 - 9^2a^2 = 19a ≈ 4[/tex]

6. Using the Pythagorean theorem, we can solve for b:

[tex]b^2 = c^2 - a^2b^2 = 6^2 - 3^2b^2 = 27b ≈ 5[/tex]

7. Using the Pythagorean theorem, we can solve for a:

[tex]a^2 = c^2 - b^2a^2 = 14^2 - 11^2a^2 = 123a ≈ 11[/tex]

8. Using the Pythagorean theorem, we can solve for b:

[tex]b^2 = c^2 - a^2b^2 = 12^2 - 10^2b^2 = 44b ≈ 6[/tex]

9. Using the Pythagorean theorem, we can solve for b:

[tex]b^2 = c^2 - a^2b^2 = 25^2 - 15^2b^2 = 400b ≈ 20[/tex]

10. Using the Pythagorean theorem, we can solve for a:

[tex]a^2 = c^2 - b^2a^2 = 12^2 - 12^2a^2 = 0a = 0[/tex]

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The volume of revolution generated by revolving the region about the x-axis is -512π.

To find the volume of revolution using the washer method, we need to integrate the area of the cross-sections formed by rotating the region bounded by the graphs of y = 8√x, y = 16, and the y-axis about the x-axis.

Let's start by setting up the integral. We will integrate with respect to x since the region is bounded by the x-axis.

The lower limit of integration (x) is 0, and the upper limit is found by setting y = 8√x equal to y = 16 and solving for x:

8√x = 16

√x = 2

x = 4

So the integral setup is:

V = ∫[0, 4] π(R^2 - r^2) dx

To find the outer radius (R), we consider the distance between the curve y = 8√x and the x-axis. Since we are revolving around the x-axis, R is simply y = 8√x.

The inner radius (r) is the distance between the line y = 16 and the x-axis, which is simply 16.

Now we can set up the integral:

V = ∫[0, 4] π((8√x)^2 - 16^2) dx

= ∫[0, 4] π(64x - 256) dx

Integrating:

V = π(32x^2 - 256x) |[0, 4]

= π[(32(4)^2 - 256(4)) - (32(0)^2 - 256(0))]

= π[512 - 1024 - 0]

= -512π

The volume of revolution generated by revolving the region about the x-axis is -512π.

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Answer:

To calculate the selling price of each amplifier, we need to consider the cost, import duty, sales tax, and the desired profit margin.

Cost of each amplifier: sh. 3,750

Import duty of 125% on the cost:

Import duty = 125% of sh. 3,750

= 125/100 * sh. 3,750

= sh. (125/100 * 3,750)

= sh. 4,687.50

Cost of each amplifier including import duty:

Total cost = Cost + Import duty

= sh. 3,750 + sh. 4,687.50

= sh. 8,437.50

Sales tax of 20% on the total cost:

Sales tax = 20% of Total cost= 20/100 * sh. 8,437.50

= sh. (20/100 * 8,437.50)

= sh. 1,687.50

Total cost including sales tax:

Total cost = Total cost + Sales tax

= sh. 8,437.50 + sh. 1,687.50

= sh. 10,125

Desired profit margin of 10% on the total cost:

Profit = 10% of Total cost

= 10/100 * sh. 10,125

= sh. (10/100 * 10,125)

= sh. 1,012.50

Selling price of each amplifier:

Selling price = Total cost + Profit

= sh. 10,125 + sh. 1,012.50

= sh. 11,137.50

The approximate average speed of the cab is 50 km/h.

To find the approximate average speed of the cab, we can use the formula:

Average Speed = Total Distance / Total Time

Given that the initial odometer reading is 369 km and the final reading is 469 km, the total distance covered by the cab is:

Total Distance = Final Odometer Reading - Initial Odometer Reading

Total Distance = 469 km - 369 km

Total Distance = 100 Km.

The cab traveled for 2 hours, so the total time is:

Total Time = 2 hours

Now, we can substitute the values into the average speed formula:

Average Speed = Total Distance / Total Time

Average Speed = 100 km / 2 hours

Average Speed = 50 km/h

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Answer:

[tex]y=\boxed{2}\:\cos \left(\boxed{1}\;x\right)+\boxed{3}[/tex]

Step-by-step explanation:

The graph of the solid black line is the cosine parent function, y = cos(x).

The standard form of a cosine function is:

[tex]\boxed{y = A \cos(B(x + C)) + D}[/tex]

where:

From inspection of the graph, the x-values of the turning points (peaks and troughs) of the parent function and the new function are the same. Therefore, the period of both functions is the same, and there has been no horizontal shift. So, B = 1 and C = 0.

The mid-line of the new function is y = 3. Therefore, D = 3.

The y-value of the peaks is y = 5. The amplitude is the distance from the mid-line to the peak. Therefore, A = 2.

Substituting these values into the standard formula we get:

[tex]y = 2 \cos(1(x + 0)) + 3[/tex]

[tex]y=2 \cos (1(x))+3[/tex]

[tex]y= 2 \cos(x) + 3[/tex]

Therefore, the equation of the trigonometric graph is:

[tex]y=\boxed{2}\:\cos \left(\boxed{1}\;x\right)+\boxed{3}[/tex]

Answer:

Option (C), 8 am

Step-by-step explanation:

Newton's Law of Cooling is a mathematical model that describes the cooling process of an object. It states that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature.

The equation representing Newton's Law of Cooling is:

[tex]\dfrac{dT}{dt} = -k (T_0 - T_A)[/tex]

Where...

After solving the differential equation we get the following function:

[tex]T(t)=T_A+(T_0-T_A)e^{-kt}[/tex]

[tex]\hrulefill[/tex]

Given:

[tex]T_0=98.6 \ \textdegree F \ \text{(This is the average human body temperature)}\\\\T_f=T(t)=80\ \textdegree F \\\\T_A=40 \ \textdegree F \\\\k=0.1947[/tex]

Find:

[tex]T(??)= \ 80 \ \textdegree F[/tex]

Substituting the values into the formula:

[tex]T(t)=T_A+(T_0-T_A)e^{-kt}\\\\\\\Longrightarrow 80=40+(98.6-40)e^{-0.1947t}\\\\\\\Longrightarrow 80=40+58.6e^{-0.1947t}\\\\\\\Longrightarrow 40=58.6e^{-0.1947t}\\\\\\\Longrightarrow 0.682594=e^{-0.1947t}\\\\\\\Longrightarrow \ln(0.682594)=-0.1947t\\\\\\\Longrightarrow t=\dfrac{\ln(0.682594)}{-0.1947} \\\\\\\therefore \boxed{t \approx 2 \ \text{hours}}[/tex]

Thus, we can conclude the time of death was at 8 am.

Answer:

  -3

Step-by-step explanation:

You want the multiplier of the second equation that would result in eliminating the x-terms when the first equation is multiplied by 7 and added to the multiplied second equation.

The desired multiplier will have the effect of making the coefficient of x be zero when the multiplications and addition are carried out. If k is that multiplier, the resulting x-term will be ...

 7(-3x) +k(-7x) = 0

  -21x -7kx = 0 . . . . . . simplify

  3 +k = 0 . . . . . . . . . divide by -7x

  k = -3 . . . . . . . . . . subtract 3

The multiplier of the second equation should be -3.

__

Additional comments

Carrying out the suggested multiplication and addition, we have ...

  7(-3x -7y) -3(-7x +10y) = 7(-56) -3(1)

  -49y -30y = -395

  y = -395/-79 = 5

The solution is (x, y) = (7, 5).

In general, the multipliers will be the reverse of the coefficients of the variable, with one of them negated. Here the coefficients of x are {-3, -7}. When these are reversed, you have {-7, -3}. When the first is negated, the multipliers of the two equations are {7, -3}. That is, the second equation should be multiplied by -3, as we found above.

Note that if you subtract the multiplied equations instead of adding, you can use the reversed coefficients without negating one of them. The choice of where the minus sign appears (multiplication or subtraction) will depend on your comfort level with minus signs.

The number of minus signs in this system can be reduced by multiplying the first equation by -1 to get 3x +7y = 56.

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Answer:

m=

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

where x1 is- -1

x2 is -6

y1 is -11

y2 is -7

m=

[tex] \frac{ - 7 - ( - 11)}{ - 6 - ( - 1)} [/tex]

[tex] \frac{ - 7 + 11}{ - 6 + 1} [/tex]

[tex] \frac{4}{ - 5} [/tex]

gradient is

[tex] gradient = \frac{4}{ - 5} [/tex]

The graph of the equation f(x) = 1/2x - 5 is the graph (b)

From the question, we have the following parameters that can be used in our computation:

f(x) = 1/2x - 5

The above expression is a linear equation that implies that

Slope = 1/2

y-intercept = -5

Next, we determine the graph

The graph that has a slope of 1/2 and y-intercept of -5 is (b)

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This year, the number of raffle tickets sold for a school's extracurricular activities fundraiser is 848. It is estimated that the number of raffle tickets sold will increase by 5% each year.

The total number of raffle tickets sold at the end of 9 years is approximately 9,818.  

To find the total number of raffle tickets sold at the end of 9 years, we need to calculate the number of tickets sold each year and sum them up.

Starting with the initial number of tickets sold, which is 848, we will increase this number by 5% each year for a total of 9 years.

Year 1: 848 + (5% of 848) = 848 + 42.4 = 890.4

Year 2: 890.4 + (5% of 890.4) = 890.4 + 44.52 = 934.92

Year 3: 934.92 + (5% of 934.92) = 934.92 + 46.746 = 981.666

Year 9: Ticket sales at the end of 9 years = Number of tickets sold in Year 8 + (5% of Year 8 sales)

Year 9: Total = 1,399.585 + 69.97925 = 1,469.56425 ≈ 1,469.56

The total number of raffle tickets sold at the end of 9 years is approximately 1,469.56.

The correct option is 9,818.

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Based on the quadratic function, an 18 year old would spend 3 hours watching television.

Using the quadratic function given :

The age is represented as the variable , 'z'

substitute z = 18 into the equation

y = (0.004*18²) - 0.314(18) + 7.5

y = 3.144

y = 3 hours approximately

Hence, an 18 year old spend approximately 18 hours watching television.

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Answer:

He needs to study for an additional 30 minutes - 12 minutes = 18 minutes before taking a break.

Step-by-step explanation:

To determine how many more minutes Mason needs to study before taking a break, we can calculate the remaining study time.

Mason plans to study for 1 and 1-half hours, which is equivalent to 90 minutes.

He will take a break once he has studied for 1-third of the planned time, which is 1/3 * 90 minutes = 30 minutes.

Mason has already studied for 12 minutes.

Therefore, he needs to study for an additional 30 minutes - 12 minutes = 18 minutes before taking a break.

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Step-by-step explanation:

Common difference , d, is  9

Sn = n/2 ( a1 + a33)                a33 = a1 + 32d = 2 + 32(9) = 290

S33 = 33/2 ( 2+290) = 4818

The total cost for 24 copper hearts would be $21.41.

To calculate the total cost for 24 copper hearts, we need to determine the total amount of copper used and then multiply it by the cost of copper per pound.

First, let's find out the total amount of copper used for 24 copper hearts. Each heart uses 0.75 square inches of copper, so for 24 hearts, the total amount of copper used would be:

0.75 square inches/heart [tex]\times 24[/tex]hearts = 18 square inches.

Next, we need to convert the square inches into cubic inches. Since we don't have information about the thickness of the hearts, we'll assume they are flat hearts with a thickness of 1 inch. Therefore, the volume of copper used for the 24 hearts would be:

18 square inches [tex]\times 1[/tex] inch = 18 cubic inches.

Now, we can calculate the total weight of copper used. Given that there is 0.323 pounds of copper per cubic inch, the total weight of copper for the 24 hearts would be:

18 cubic inches [tex]\times 0.323[/tex] pounds/cubic inch = 5.814 pounds.

Finally, we multiply the total weight of copper by the cost of copper per pound to find the total cost:

5.814 pounds [tex]\times[/tex] $3.68/pound = $21.41.

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(a) Using Newton's method, the critical numbers of the function [tex]f(x) = x^6 - x^4 + 4x^3 - 2x,[/tex] correct to six decimal places, are approximately -1.084, -0.581, -0.214, 0.580, and 1.279.

(b) The absolute minimum value of f is undefined since the function is a polynomial of even degree, and it approaches positive infinity as x approaches positive or negative infinity.

(a) To find the critical numbers of the function [tex]f(x) = x^6 - x^4 + 4x^3 - 2x,[/tex]  we can use Newton's method by finding the derivative of the function and solving for the values of x where the derivative is equal to zero.

First, let's find the derivative of f(x):

f[tex]'(x) = 6x^5 - 4x^3 + 12x^2 - 2[/tex]

Now, let's apply Newton's method to find the critical numbers. We start with an initial guess, x_0, and use the formula:

[tex]x_{(n+1)} = x_n - (f(x_n) / f'(x_n))[/tex]

Iterating this process, we can approximate the values of x where f'(x) = 0.

Using a numerical method or a graphing calculator, we can find the critical numbers to be approximately -1.084, -0.581, -0.214, 0.580, and 1.279.

Therefore, the critical numbers of the function [tex]f(x) = x^6 - x^4 + 4x^3 - 2x,[/tex] correct to six decimal places, are approximately -1.084, -0.581, -0.214, 0.580, and 1.279,

(b) To find the absolute minimum value of f(x), we need to analyze the behavior of the function at the critical numbers and the endpoints of the interval.

Since the function f(x) is a polynomial of even degree, it approaches positive infinity as x approaches positive or negative infinity.

Therefore, there is no absolute minimum value for the function.

Hence, the absolute minimum value of f is undefined.

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Answer:

BC = 24 units

Step-by-step explanation:

This is an isosceles triangle which always has:

In this triangle, the legs CA and BA are congruent so CA = BA and the angles C and B are congruent to each other so angle C = angle B.

Thus, we can find x by setting CA and BA equal to each other:

(3x - 15 = x + 33) + 15

(3x = x + 48) - x

(2x = 48) / x

x = 24

Thus, x = 24

Since the length of BC is x and x = 24, BC is 24 units long.

Answer:

-5/4

Step-by-step explanation:

Let [tex](x_1,y_1)=(-4,4)[/tex] and [tex](x_2,y_2)=(0,-1)[/tex]. The slope of the line would be:

[tex]\displaystyle \frac{y_2-y_1}{x_2-x_1}=\frac{-1-4}{0-(-4)}=\frac{-5}{4}=-\frac{5}{4}[/tex]

Answer: -5/4

Step-by-step explanation:

To find the slope between two points, you can use the formula:

Slope = (y2 - y1)/(x2 - x1)

Using the points (0, -1) and (-4, 4), we can substitute the coordinates into the formula:

slope = (4 - (-1))/(-4 - 0)

slope = (4 + 1)/(-4)

slope = 5/-4

Therefore, the slope between the two points is -5/4.

Answer:

164 million km

Step-by-step explanation:

If the hyperbola models the comet's path, and the sun is located at one of its foci, the closest distance the comet reaches to the sun is the distance between a vertex and its corresponding focus.

Therefore, we need to find the vertices and foci of the given hyperbola.

Given equation:

[tex]\dfrac{x^2}{60516}-\dfrac{y^2}{107584}=1[/tex]

As the x²-term of the given equation is positive, the hyperbola is horizontal (opening left and right).

The general formula for a horizontal hyperbola (opening left and right) is:

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Standard equation of a horizontal hyperbola}\\\\$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center.\\ \phantom{ww}$\bullet$ $(h\pm a, k)$ are the vertices.\\\phantom{ww}$\bullet$ $(h\pm c, k)$ are the foci where $c^2=a^2+b^2.$\\\phantom{ww}$\bullet$ $y=\pm \dfrac{b}{a}(x-h)+k$ are the asymptotes.\\\end{minipage}}[/tex]

Comparing the given equation with the standard equation:

To find the loci, we first need to find the value of c:

[tex]\begin{aligned}c^2&=a^2+b^2\\c^2&=60516 +107584\\c^2&=168100\\c&=410\end{aligned}[/tex]

The formula for the loci is (h±c, k). Therefore:

[tex]\begin{aligned}\textsf{Loci}&=(h \pm c, k)\\&=(0 \pm 410, 0)\\&=(-410,0)\;\;\textsf{and}\;\;(410,0)\end{aligned}[/tex]

The formula for the vertices is (h±a, k). Therefore:

[tex]\begin{aligned}\textsf{Vertices}&=(h \pm a, k)\\&=(0 \pm 246, 0)\\&=(-246,0)\;\;\textsf{and}\;\;(246,0)\end{aligned}[/tex]

From the given diagram, the vertex and focus have positive x-values. Therefore, the vertex is (246, 0) and the focus is (410, 0).

We need to find the distance between (246, 0) and (410, 0). To do this, simply subtract the x-value of the vertex from the x-value of the focus:

[tex]410-246=164[/tex]

Therefore, the closest distance the comet reaches to the sun is 164 million km.