Answer:
-4 is the coefficient or -4x whatever the answers are
Step-by-step explanation:
the coefficient in mathematics is basically whatever the number is infront of a variable in an expression, equation, etc.
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? A. Yes, because the heights are the same, and the cross-sectional areas at every level parallel to the bases are also the same. B. Yes, because the figures are congruent. C. No, because only the bases have the same area, not every cross section at every level parallel to the bases. D. No, because the heights are not the same.
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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Find the sum of the first 33 terms of the following series, to the nearest
integer.
2, 11, 20,...
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
Select the correct answer.
The number of hours that 20 people spent watching television per day, in relation to age, is graphed. This quadratic equation represents the model
for the set of data.
y = 0.004z²0.314z + 7.5
Based on the model, approximately how much time does an 18-year-old spend watching television each day?
O A.
OB.
O C.
O D.
3 hours
2 hours
7.5 hours
0.5 hour
Based on the quadratic function, an 18 year old would spend 3 hours watching television.
Using the quadratic function given :
y = 0.004z²-0.314z + 7.5The 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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Please help what is the slope of the line?
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.
How can you use the transformations to prove 2 triangles are congruent
Answer:
There are three main ways to prove that two triangles are congruent using transformations:
1. Congruence by translation: If there is a translation that maps one triangle onto the other, then the triangles are congruent.
2. Congruence by reflection: If there is a line of reflection that maps one triangle onto the other, then the triangles are congruent.
3. Congruence by rotation: If there is a rotation that maps one triangle onto the other, then the triangles are congruent.
In order to use these transformations to prove congruence, you need to show that all corresponding parts (angles and sides) are congruent after the transformation is applied. This can be done either algebraically, using coordinate geometry, or by providing a clear visual representation of the transformations that are applied.
For example, if you want to prove that two triangles, ABC and DEF, are congruent by translation, you need to show that there is a vector that, when added to the coordinates of any point on triangle ABC, produces the coordinates of the corresponding point on triangle DEF. Once you have shown that all three sides and angles are congruent after the translation, you can conclude that the triangles are congruent.
Net Present Value Method, Internal Rate of Return Method, and Analysis
The management of Advanced Alternative Power Inc. is considering two capital investment projects. The estimated net cash flows from each project are as follows:
Year Wind Turbines Biofuel Equipment
1 $420,000 $880,000
2 420,000 880,000
3 420,000 880,000
4 420,000 880,000
Present Value of an Annuity of $1 at Compound Interest
Year 6% 10% 12% 15% 20%
1 0.943 0.909 0.893 0.870 0.833
2 1.833 1.736 1.690 1.626 1.528
3 2.673 2.487 2.402 2.283 2.106
4 3.465 3.170 3.037 2.855 2.589
5 4.212 3.791 3.605 3.352 2.991
6 4.917 4.355 4.111 3.784 3.326
7 5.582 4.868 4.564 4.160 3.605
8 6.210 5.335 4.968 4.487 3.837
9 6.802 5.759 5.328 4.772 4.031
10 7.360 6.145 5.650 5.019 4.192
The wind turbines require an investment of $1,199,100, while the biofuel equipment requires an investment of $2,278,320. No residual value is expected from either project.
Required:
1a. Compute the net present value for each project. Use a rate of 10% and the present value of an annuity of $1 in the table above. If required, use the minus sign to indicate a negative net present value. If required, round to the nearest whole dollar.
Wind Turbines Biofuel Equipment
Present value of annual net cash flows $fill in the blank 1 $fill in the blank 2
Less amount to be invested $fill in the blank 3 $fill in the blank 4
Net present value $fill in the blank 5 $fill in the blank 6
1b. Compute a present value index for each project. If required, round your answers to two decimal places.
Present Value Index
Wind Turbines fill in the blank 7
Biofuel Equipment fill in the blank 8
2. Determine the internal rate of return for each project by (a) computing a present value factor for an annuity of $1 and (b) using the present value of an annuity of $1 in the table above. If required, round your present value factor answers to three decimal places and internal rate of return to the nearest whole percent.
Wind Turbines Biofuel Equipment
Present value factor for an annuity of $1 fill in the blank 9 fill in the blank 10
Internal rate of return fill in the blank 11 % fill in the blank 12 %
3. The net present value, present value index, and internal rate of return all indicate that the
is a better financial opportunity compared to the
, although both investments meet the minimum return criterion of 10%.
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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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. Find the total number of raffle tickets sold at the end of 9 years.
Select the correct answer below:
9,158
9,351
9,818
10,666
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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The product of two irrational numbers is
rational. (Sometimes,Never,always)?
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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Find the slope of the lines graphed below (-1,-11) and (-6,-7)
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]
Please explain how to do this and what the answer is. The answer with best explaination gets Brainliest
Answer:
102
Step-by-step explanation:
we substitute x by 7
7^2+9(7)-10
49+63-10
=102
An import company brought into the country some amplifiers that cost sh. 3,750-each. The government imposed an import duty of 125% and a sales tax of 20 %. If the company decided to make a 10% profit on the sales, calculate the selling price of each amplifier.
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
which graph represents this function
f(x)=1/2x-5
help would be appreciated
The graph of the equation f(x) = 1/2x - 5 is the graph (b)
How to determine the graph of the equationFrom 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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What is the mean and median reasoning, As the last one is incorrect
The best measure of center of the data is (a) mean; because the data are close together
How to determine the best measure of center of the dataFrom the question, we have the following parameters that can be used in our computation:
The dataset of 10 values
Where we can see that there are no outliers present in the dataset
By definition, outliers are extreme values.
Since there are no outliers, it means that the mean is the best measure of center
This is because the mean is affected by the presence of outliers and since no outlier is present, we use the mean
From the list of options, we have the mean value to be 42.536
Hence, the true statement is (a)
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Which system has the same solution as the system of equations shown?
3x + 2y = -5
2x + 3y = 5
Step-by-step explanation:
To find a system with the same solution as the given system, we can multiply both sides of both equations by a nonzero constant, which will result in a system that is equivalent to the original one.
For example, let's multiply the first equation by 2 and the second equation by 3:
First equation (multiplied by 2):
6x + 4y = -10
Second equation (multiplied by 3):
6x + 9y = 15
The new system of equations is:
6x + 4y = -10
6x + 9y = 15
This system has the same solution as the original system because it's just a scalar multiple of the original system.
A basket of cucumbers contains 10 cucumbers that were grown using conventional methods and 22 cucumbers that were grown using organic methods. If a customer randomly selects 5 cucumbers, what is the probability they select two conventional cucumbers and 3 organic cucumbers?
3.
Your family is planning a road trip stretching from coast to coast for this summer. The route and the time frame are nearly set; now you need to plan out the finances. Your parents have decided that rental of an RV will be cheaper than staying in hotels, but they would like an estimate on the total cost. Can you help them?
a. To rent an RV, the following costs apply: $125 per day, plus 32 cents per mile. Additionally, to drop off the RV on the other side of the country, there is an extra fee of $2,500. Write an equation to describe the total cost of RV rental.
b. Your parents have two options for their road trip plans. The first option stretches over 3500 miles and includes fewer stops but more beautiful scenery. It will take about a week and a half (11 days). The second option stretches over just 3000 miles, but it includes more overnight stops and will therefore take two weeks (14 days). Which of these two options is cheaper?
c. Your little sister really wants to take the two-week trip, but your parents really want to keep the RV rental cost under $5,000. You can compromise by either taking a more direct route (lessening the miles) or by stopping for less overnight stays (lessening the days of the rental). What would the domains be for these two compromises? Justify why you think your domains are correct.
d. Write and solve equations to find how many miles or how many days you would have to eliminate in order to stay under the $5,000 budget. Explain each step as you solve your equations. Finally, make a recommendation to your parents about which compromise you think is best.
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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What is the value of the rational expression below when x is equal to 4?
x-12
X-8
O A. -2
о B. 8
о C. 2
OD. -8
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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A jewelry company makes copper heart pendants. Each heart uses 0.75in® of copper and there is o.323 pound of copper per cubic inch. If copper costs $3.68 per pound, what is the total cost for 24 copper hearts?
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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help me on this question ive been stuck on this
MATH QUESTION HELP PLS!
Stephen predicted that he would sell 50 cakes at his school bake sale. However, only 45 were sold. What was Stephen's percent error?
Show that y₁(t) = e^ãt cos(μt) and
y₂(t) = e^ãt sin(μt)
are a fundamental set of solutions and state the general solution.
The functions y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) are a fundamental set of solutions because they are linearly independent and satisfy the given homogeneous linear differential equation, allowing for the formation of the general solution.
To show that y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) are a fundamental set of solutions, we need to demonstrate two things: linear independence and satisfaction of the given homogeneous linear differential equation.
First, let's consider linear independence. We can prove it by showing that there is no constant c₁ and c₂, not both zero, such that c₁y₁(t) + c₂y₂(t) = 0 for all t.
Now, let's verify that y₁(t) and y₂(t) satisfy the homogeneous linear differential equation. If the given differential equation is of the form ay''(t) + by'(t) + cy(t) = 0, we can substitute y₁(t) and y₂(t) into the equation and verify that it holds true.
Once we have established linear independence and satisfaction of the differential equation, we can state that the general solution to the homogeneous linear differential equation is given by y(t) = c₁y₁(t) + c₂y₂(t), where c₁ and c₂ are arbitrary constants. This general solution represents the linear combination of the fundamental set of solutions.
In summary, y₁(t) = e^ãt cos(μt) and y₂(t) = e^ãt sin(μt) form a fundamental set of solutions for the given differential equation, and the general solution is given by y(t) = c₁y₁(t) + c₂y₂(t).
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(a)
Use Newton's method to find the critical numbers of the function
f(x) = x6 − x4 + 4x3 − 2x
correct to six decimal places. (Enter your answers as a comma-separated list.)
x =
Incorrect: Your answer is incorrect.
(b)
Find the absolute minimum value of f correct to four decimal places.
(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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PLEASE HELPPPPPPP NEED NOW
Answer:
BC = 24 units
Step-by-step explanation:
This is an isosceles triangle which always has:
two legs that are congruent to each other (i.e., equal),and two angles that are congruent to each other.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.
A comet follows a hyperbolic path in which the sun is located at one of its foci. If the equation... 100 pts
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:
h = 0k = ka² = 60516 ⇒ a = 246b² = 107584 ⇒ b = 328To 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.
The diagonal of rectangle ABCD measures 2 inches in length. What is the length of line segment AB?
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
Use the washer method to find the volume of revolution generated by revolving the region bounded by the graphs of y = 8√x,
y = 16, and the y-axis about the x-axis.
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
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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The radius of a circle is 3 meters. What is the area of a sector bounded by a 90° arc?
Give the exact answer in simplest form.
Answer:
(90°/360°)π(3^2) = (1/4)(9π) = 9π/4 m²
Lesson 24 Review
Directions: Follow the directions in Part A and Part B to complete the assignment.
Part A
Directions: Find the missing value in the following right triangles.
Note: use your calculator and round all answers to whole numbers.
1. a=4, b=?. c=10
2. a=?, b=3, c= 12
3. a=6. b=? c= 14
4. a=7.
b=?.
C= 12
5. a=?. b=9.
C= 10
6. a=3. b=?.
c=6
7. a=?, b= 11, c=14
8. a=10. b=?. c= 12
9. a=15, b=?, c=25
10. a =?, b= 12, c=12
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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Write the equation of the trigonometric graph.
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:
A is the amplitude (height from the mid-line to the peak).2π/B is the period (horizontal distance between consecutive peaks).C is the phase shift (horizontal shift - positive is to the left).D is the vertical shift (the mid-line is y = D).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]
Mason plans to study for 1 and 1-half hours. Once he has studied for 1-third of the planned time, he will take a break. Mason has been studying for 12 minutes.
Question
How many ,begin emphasis,more,end emphasis, minutes does Mason need to study before he takes a break? Enter the answer in the box.
Response area with 1 text input box
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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