Answers
The end behavior of f(x) is described as follows:
As x -> -∞, f(x) -> ∞.As x -> ∞, f(x) -> 0.How to obtain the end behavior of a function?The end behavior of a function is given by the limit of the function is the input x goes to either negative infinity or positive infinity.
The function for this problem is given as follows:
f(x) = 4(5/7)^x.
It is an exponential function with base with absolute value less than 1, hence the end behavior is given as follows:
Increases to the left of the graph -> As x -> -∞, f(x) -> ∞.Approaches the horizontal asymptote y = 0 at the right of the graph -> As x -> ∞, f(x) -> 0.More can be learned about the end behavior of a function at brainly.com/question/1365136
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Related Questions
If the distance between A (0,4) and B(3,a) is 5 units then Find the value of a.
Answers
Step-by-step explanation:
Using the distance formula, we know that the distance between points A and B is:
sqrt((3 - 0)^2 + (a - 4)^2) = 5
Simplifying:
sqrt(9 + (a - 4)^2) = 5
Squaring both sides:
9 + (a - 4)^2 = 25
Simplifying:
(a - 4)^2 = 16
Taking the square root of both sides (note that we can take the positive or negative square root since both will satisfy the equation):
a - 4 = ±4
Solving for a:
a - 4 = 4 or a - 4 = -4
a = 8 or a = 0
However, we need to check that these values of a actually make sense based on the given information. If a = 0, then the distance between A and B is:
sqrt((3 - 0)^2 + (0 - 4)^2) = sqrt(9 + 16) = 5
So a = 0 is a valid solution. If a = 8, then the distance between A and B is:
sqrt((3 - 0)^2 + (8 - 4)^2) = sqrt(9 + 16) = 5
So a = 8 is also a valid solution.
Therefore, the possible values of a are a = 0 or a = 8.
The value of a can be 0 or 8 if the distance between the given points is 5.
A straight line drawn between two points is the shortest distance between the two points. We can calculate the shortest distance by the distance formula.
Let A(0,4) = (x1,y1)
B(3,a) = (x2,y2)
Using distance formula,
[tex]D = \sqrt{(x2-x1)^2 + (y2-y1)^2}[/tex]
As distance = 5 units
Therefore,
[tex]\sqrt{(x2-x1)^2 + (y2-y1)^2} = 5[/tex]
[tex]\sqrt{(3-0)^2 + ((a-4)^2} = 5[/tex]
[tex]\sqrt{(3)^2 + (a^2 + 16 + - 8a)} = 5[/tex]
[tex]\(9 + 16 + a^2 - 8a = 25[/tex]
[tex]25 + a^2 -8a = 25[/tex]
[tex]a^2 - 8a = 0[/tex]
[tex]a(a-8) = 0[/tex]
a = 0 and a = 8
Therefore, the answer is 0 or 8.
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In circle S with m \angle RST= 30^{\circ}m∠RST=30
∘
and RS=13RS=13, find the area of sector RST. Round to the nearest hundredth.
Answers
The area of a sector of a circle is calculated as approximately 44.24 square units.
How to Find the Area of the Sector of a Circle?The area of the sector of a circle can be calculated by applying the given formula below:
Area of sector = ∅/360 * π * r², where: r is the radius of the circle and ∅ is the central angle or reference angle of the sector.
Given the parameters of the sector of the circle as:
Central angle (∅) = 30 degrees
Radius (r) = RS = 13 units
Plug in the values:
Area of sector = 30/360 * π * 13²
Area of sector ≈ 44.24 square units.
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Help me please I don’t know how to solve this.
Answers
The equation is 2m + 4s = 16 and the table of values is solved
Given data ,
Let the equation be represented as A
Now , the value of A is
2m + 4s = 16
when m = 0
4s = 16
s = 4
when s = 3
2m + 12 = 16
2m = 4
m = 2
when m = -2
-4 + 4s = 16
4s = 20
s = 5
when s = 0
2m = 16
m = 8
Therefore , the table of values for m = { 0 , 2 , -2 , 8 } and the table of values for s = { 4 , 3 , 5 , 0 }
Hence , the equation is solved
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URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS
Answers
The statements that are true about the scatter plot graph are;
1. The line of best fit should have the same number of points above and below it.
2. There is a moderate positive correlation.
3. The line of best fit will have a positive slope.
What are some facts about scattered plot graphs that you should know?Some facts about scatter plot graphs:
Scatter plots are used to show the correlation or relationship between two variables. The variables are plotted on the x and y-axes, and each point represents a pair of values for the two variables.
Scatter plots can show different types of relationships between two variables, including positive, negative, or no correlation.
The shape of a scatter plot can provide insights into the strength and direction of the relationship between the two variables. For example, if the points on the scatter plot form a linear pattern, this indicates a strong correlation.
Scatter plots can be used to identify outliers or anomalies in the data that do not fit the general pattern of the relationship between the two variables.
The above answer is based on the following options provided in the picture.
Tick all the statements that applies
The y intercept of the line of best fit would be around 45
The line of best fit should have the same number of points above and below it
The slope of the line of best fit could be around -1/20000
The line of best fit must pass through at lest 2 points on the scattered plot
There is no correlation between happiness and income
There is a moderate positive correlation
The line of best fit will have a positive slope
As a person's income goes up, their happiness trends down
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Please help me with this word problem!
Suppose you lay exactly 160 feet of fencing around a rectangular garden. If the length of the garden is 3 times its width, find the dimensions of the garden.
Length: __ feet
Width: __ feet
Answers
Step-by-step explanation:
the perimeter (the way one time around it) of a rectangle is
2×length + 2×width
length = 3×width
2×(3×width) + 2×width = 160
6×width + 2×width = 160
8×width = 160
width = 160/8 = 20 ft
length = 3×width = 3×20 = 60ft
The point (8, −15) was reflected over an axis to (−8, −15). Which axis was it reflected over? Explain.
Answers
Answer: it is reflected by y-axis
Step-by-step explanation: as we know when point (x,y) reflected by y axis the value of y remains the same and the sign of x will be changed since the image is rotated horizontaly
Answer:
y-axis
Step-by-step explanation:
When a point is reflected over the y-axis, the x-coordinate of the point becomes the additive inverse of the original number.
Here, (8, -15) became (-8, -15). The only change in value of the coordinates is the value of the x-coordinate, 8, which became its additive inverse, -8.
Answer: y-axis
Answer: it is reflected by y-axis
Step-by-step explanation: as we know when point (x,y) reflected by y axis the value of y remains the same and the sign of x will be changed since the image is rotated horizontaly
Answer:
y-axis
Step-by-step explanation:
When a point is reflected over the y-axis, the x-coordinate of the point becomes the additive inverse of the original number.
Here, (8, -15) became (-8, -15). The only change in value of the coordinates is the value of the x-coordinate, 8, which became its additive inverse, -8.
Answer: y-axis
Break-Even Sales
BeerBev, Inc., reported the following operating information for a recent year (in millions):
Sales $6,512
Cost of goods sold $1,628
Gross profit $4,884
Marketing, general, and admin. expenses 592
Income from operations $ 4,292
Assume that BeerBev sold 37 million barrels of beer during the year, that variable costs were 75% of the cost of goods sold and 50% of marketing, general and administration expenses, and that the remaining costs are fixed. For the following year, assume that BeerBev expects pricing, variable costs per barrel, and fixed costs to remain constant, except that new distribution and general office facilities are expected to increase fixed costs by $21.09 million.
a. Compute the break-even sales (in barrels) for the current year. Round your answer to two decimal places. Enter your answers in millions.
Answers
Answer:
Step-by-step explanation:
To compute the break-even sales (in barrels) for the current year, we need to first determine the contribution margin per barrel, which is the amount left over from the selling price of each barrel of beer after variable costs are subtracted.
Variable costs per barrel can be calculated as 75% of the cost of goods sold per barrel, which is:
Variable cost per barrel = (75% x Cost of goods sold) / Barrels sold
Variable cost per barrel = (0.75 x $1,628 million) / 37 million barrels
Variable cost per barrel = $33.00
Similarly, the variable marketing, general, and administration expenses per barrel can be calculated as:
Variable marketing, general, and administration expenses per barrel = (50% x Marketing, general, and administration expenses) / Barrels sold
Variable marketing, general, and administration expenses per barrel = (0.50 x $592 million) / 37 million barrels
Variable marketing, general, and administration expenses per barrel = $8.00
Therefore, the contribution margin per barrel is:
Contribution margin per barrel = Selling price per barrel - Variable costs per barrel
Contribution margin per barrel = $6,512 million / 37 million barrels - $33.00 - $8.00
Contribution margin per barrel = $98.00
To compute the break-even sales (in barrels) for the current year, we can use the following formula:
Break-even sales (in barrels) = Fixed costs / Contribution margin per barrel
Fixed costs can be calculated as:
Fixed costs = Income from operations + Variable marketing, general, and administration expenses x Barrels sold - Contribution margin per barrel x Barrels sold
Fixed costs = $4,292 million + $8.00 x 37 million barrels - $98.00 x 37 million barrels
Fixed costs = $1,108 million
Substituting this into the formula, we get:
Break-even sales (in barrels) = $1,108 million / $98.00 per barrel
Break-even sales (in barrels) = 11.29 million barrels
Rounding this to two decimal places and converting to millions, we get:
Break-even sales (in barrels) = 11.29 million barrels = 11.3 million barrels (rounded)
12 1/2 - 4 7/10 a. 7 6/8 b. 7 8/10 c. 8 2/8 d. 8 3/10
Answers
Answer:
c. 8 2/8
Step-by-step explanation:
12 1/2 - 4 7/10 = 12 5/10 - 4 7/10 = 8 2/10 = 8 1/5 = 8.2
Therefore, the answer is option c. 8 2/8
Hope this helps!
Identify the triangle as acute, obtuse, or right by comparing the squares of the side lengths. Show your work and
explain the steps you used to solve.
14
21
16
Answers
We can see here that the triangle is a right-angle triangle.
What is a triangle?The fundamental geometric shape of a triangle has three straight sides and three angles. It is a triangular polygon with three edges.
We can see here that the triangle is a right-angle triangle because using Pythagoras Theorem, we will see the following:
Hypotenuse = c = 30
Adjacent = b = 24
Opposite = a = 18
c² = a² + b²
Hyp² = Opp² + Adj²
30² = 18² + 24²
324 + 576 = 900.
Thus, it is a right-angle triangle.
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There are approximately 7.48 liquid gallons in a cubic foot. If a cylindrical water tank holds 1,500 liquid gallons and has a radius of 3.4 feet, what is the approximate height of the water tank? Approximate using π = 3.14 and round to the nearest tenth.
Answers
In the cylinder , height of the water tank is 5.5 feet.
What is volume?
The space taken up by any three-dimensional solid constitutes a volume, to put it simply. A cube, cuboid, cone, cylinder, or sphere can be one of these solids. Cubic units are used to measure the volume of solids. The volume will be given in cubic metres, for instance, if the dimensions are given in metres.
Given that,
A cubic foot contains roughly 7.48 liquid gallons.
We have to find what is roughly the height of a cylindrical water tank with a capacity of 1,500 liquid gallons and a radius of 3.4 feet.
We know that,
7.48 liquid gallons are contained in one cubic foot.
The cylindrical water tank's radius is 3.4 feet.
1,500 liquid gallons are the maximum capacity.
The cylindrical water tank's volume is calculated to hold 1500 liquid gallons ,
= [tex]\frac{1500}{7.48}[/tex]
= 200.53 ft³
Volume of the cylinder= πr²h
=> 200.53= 3.14 ×3.4×3.4×h
=> 200.53 = 3.14 × 11.56 × h
=> 200.53 = 36.29 × h
=> h= 5.5
=> h= 5.5 feet (Approximately)
Therefore, height of the water tank is 5.5 feet.
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Explain the effect of China's fiscal stimulus on global aggregate demand and why it might have lost some of its force. Part 2 China's fiscal stimulus increases global aggregate demand because it _________. This fiscal stimulus might have lost some of its force because _______. A. increases world saving; the stimulus decreases China's government budget surplus and raises the world real interest rate B. increases China's potential GDP and income, which increases other countries exports to China; China's government debt is growing C. raises the value of the Chinese yuan, which increases China's purchases of imports in the long run; China is becoming more self-sufficient D. increases China's GDP and imports, which increases other countries exports to China; trade restrictions limits China's imports
Answers
Answer:
China's fiscal stimulus increases global aggregate demand because it increases China's GDP and imports, which increases other countries' exports to China. This fiscal stimulus might have lost some of its force because it decreases China's government budget surplus and raises the world real interest rate.
Step-by-step explanation:
hope this helps
Willow owns 225 books. Of them, 5/9
are mysteries. How many of Willow's books
are mysteries?
A) 225
B) > 225
C) < 225
D) 0
(Side note:) please provide explanation if can!
Answers
(5/9) x 225 = 125
Therefore, there are 125 mystery books, which is greater than the total number of books (225).
A clothing retailer is interested in the average waist size of men. A sample is taken with the results given below.
please see the attached image for more info on the question
Answers
The confidence interval is given as (38.811,43.829)
What is a Margin of Error?In statistics, a margin of error refers to the degree of imprecision that can be projected in survey or polling outcomes due to random sampling involved in such studies.
This indicator serves as both the measure for the accuracy of results and an assurance of confidence in them. The percentage rate expressing the magnitude of the latter varies inversely with the former and depends on the sample size, desired level of confidence, and cumulative trends in feedback obtained from respondents.
Thus, surveys featuring minimal margins of error are considered substantively accurate while ones plagued by larger margins signal a lack thereof.
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How many yards are in four and one-half miles?
7,040 yards
7,920 yards
8,448 yards
8,800 yards
Answers
Answer:
7,920
Step-by-step explanation:
I took test and got 100%
What is the probability of flipping a coin 12 times and getting heads 5 times?
Round your answer to the nearest tenth of a percent.
OA. 12.1%
OB. 3.1%
OC. 19.3%
OD. 5.4%
Answers
Answer:
The correct answer is D. 5.4% after rounding my answer off to the nearest tenth of a percent.
i need the answer to this question
Answers
Answer:
Area of annulus is 40.85cm² to 2d.p
Step-by-step explanation:
Area of shaded part=Area of bigger circle -Area of smaler circle
Area of annulus= πR²- πr²= π(R²-r²)
A=3.142(7²-6²)
A=3.142(49-36)
A=3.142×13
A=40.846cm²
A=40.85cm² to 2d.p
What are the zeros in this function?
Answers
The zeros of the function are:
-¹/₂ ± √((3)/2) i, 1/3 and - 2
What are the zeros of the polynomial?We want to find the Zeros of the polynomial:
f(x) = 3x⁴ + 8x³ + 6x² + 3x - 2
By the rational root theorem, we know that any rational zeros of f(x) must be expressible in the form p/q for integers p,q with p a divisor of the constant term −2 and q a divisor of the coefficient 3 of the leading term.
Therefore, the only possible rational zeros are:
±¹/₃, ±²/₃, ±1, ±2
Let's check x = 1/3 to get:
f(¹/₃) = 3(¹/₃)⁴ + 8(¹/₃)³ + 6(¹/₃)² + 3(¹/₃) - 2 = 0
f(-2) = 3(-2)⁴ + 8(-2)³ + 6(-2)² + 3(-2) - 2 = 0
Thus, x = 1/3 and - 2 are zeros of the function and (3x - 1) and (x + 2) are factors.
Using these factors and using the polynomial to divide them, we have that the remaining factor is: x² + x + 1
The zeros of x² + x + 1 are the Complex cube roots of 1, since
(x - 1)(x² + x + 1) = x³ - 1
using the quadratic formula, we have:
x = -¹/₂ ± √((3)/2) i
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multiply (−3x+4)(2x−1)
Answers
After multiplying (−3x+4) with (2x−1) we get a quadratic equation which is equal to −6x² + 11x − 4.
To multiply the expressions (−3x+4)(2x−1), we use the distributive property of multiplication:
(−3x+4)(2x−1) = −3x(2x) + 4(2x) − 3x(−1) + 4(−1)
Simplifying the above expression, we get:
= −6x² + 8x + 3x − 4
= −6x² + 11x − 4
Therefore, the product of (−3x+4)(2x−1) is −6x² + 11x − 4.
We can also use the FOIL method to multiply the two expressions:
(−3x+4)(2x−1) = −3x(2x) −3x(−1) + 4(2x) + 4(−1)
= −6x² + 3x + 8x − 4
= −6x² + 11x − 4
Either method can be used to get the same result.
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Adjacent angles. please don't get it wrong!
Answers
The adjascent angles to angle EDC is angle EDI and CDH.
What are adjascent angles?Two angles are said to be adjacent angles when they share the common vertex and side. The sum of adjascent angles is 180°. This means that adding two adjascent angles together will give 180°. Adjascent angles are also called supplementary angles.
Examples of adjascent angles in the diagram include;
AEB Is adjascent to AED
and also there are two angles that are adjascent to angle EDC. They are;
angle ED1 and CDH
Therefore the adjascent angles to angle EDC is angle EDI and CDH.
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Create a list of steps, in order, that will solve the following equation.
2
- 5 = 123
Solution steps:
Add 2 to both sides
Add 5 to both sides
Divide both sides by 2
Multiply both sides by 2
Subtract 5 from both sides
3
Subtract from both sides
Square both sides
Take the square root of both
sides
Answers
The graph of your equation is a parabola with x-intercepts at x = -8.75 and x = 7.25.
How to solvex = -8.75; x = 7.25
2(x + ¾)² - 5 = 123
Add 5 to each side: 2(x+ ¾)² = 128
Divide each side by 2: (x+ ¾)² = 64
Take the square root of each side: x + ¾ = 8 x + ¾ = -8
Subtract ¾ from each side: x = 8 - ¾ x = -8 - ¾
Convert to decimal fraction: x = 8 - 0.75 x = -8 – 0.75
Subtract: x = 7.25 x = -8.75
The graph of your equation is a parabola with x-intercepts at x = -8.75 and x = 7.25.
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The graph of your equation is a parabola with x-intercepts at x = -8.75 and x = 7.25.
How to solvex = -8.75; x = 7.25
2(x + ¾)² - 5 = 123
Add 5 to each side: 2(x+ ¾)² = 128
Divide each side by 2: (x+ ¾)² = 64
Take the square root of each side: x + ¾ = 8 x + ¾ = -8
Subtract ¾ from each side: x = 8 - ¾ x = -8 - ¾
Convert to decimal fraction: x = 8 - 0.75 x = -8 – 0.75
Subtract: x = 7.25 x = -8.75
The graph of your equation is a parabola with x-intercepts at x = -8.75 and x = 7.25.
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I need help completing the table and finding the velocity
Answers
We can conjecture that the value of the instantaneous velocity at t=3 is 81.6.
What is the average velocity over an interval?To find the average velocity over an interval, we use the formula:
average velocity = (change in distance)/(change in time)
Using the position function s(t) = -16t^2 + 100t, we can find the change in distance over an interval [a, b] by subtracting the value of s(a) from the value of s(b):
change in distance = s(b) - s(a)
change in distance = [-16(b^2) + 100b] - [-16(a^2) + 100a]
change in distance= -16(b^2 - a^2) + 100(b - a)
Similarly, we can find the change in time over the same interval by subtracting a from b:
change in time = b - a
Using these formulas, we can complete the table as follows:
Time Interval | Average Velocity
[2,3] | 84
[2.9,3] | 82
[2.99,3] | 81.6
[2.999, 3] | 81.6
[2.9999,3] | 81.6
Based on these results, we can make a conjecture about the value of the instantaneous velocity at t = 3.
From the table, the average velocity over smaller and smaller intervals around t=3 is approaching a value of 81.6.
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An adult patient has come into the burn unit during your shift. You take an inventory of the location of their burns in the chart below.
Part of Body Burned?
(1 = yes, 0 = no)
Head 1
Left Arm 1
Right Arm 1
Upper Front Torso 0
Upper Back Torso 0
Lower Front Torso 1
Lower Back Torso 1
Upper Left Leg 0
Upper Right Leg 0
Lower Left Leg 0
Lower Right Leg 1
Given that their weight/mass is 50.9 kg, determine how much fluid they should receive per hour in the first 8 hours of their care. Answer to the nearest hundredth of a liter (two decimal places). ( Please help me by showing work, thank you!)
Answers
Answer:
To determine the fluid replacement rate for a burn patient in the first 8 hours, we can use the Parkland formula:
Fluid (in liters) = 4 mL × body weight in kg × % total body surface area (TBSA) burned
First, we need to calculate the TBSA burned. We can use the Rule of Nines to estimate this:
Head: 9%
Left Arm: 9%
Right Arm: 9%
Upper Front Torso: 0%
Upper Back Torso: 0%
Lower Front Torso: 18%
Lower Back Torso: 18%
Upper Left Leg: 0%
Upper Right Leg: 0%
Lower Left Leg: 0%
Lower Right Leg: 9%
Total TBSA burned = 9 + 9 + 9 + 18 + 18 + 9 = 72%
Now we can plug in the values into the Parkland formula:
Fluid (in liters) = 4 mL × 50.9 kg × 72% = 14,694.72 mL
To convert mL to liters, we divide by 1000:
Fluid (in liters) = 14,694.72 mL ÷ 1000 = 14.69 L
This is the total amount of fluid the patient needs in the first 8 hours. To determine the hourly rate, we divide by 8:
Hourly fluid rate = 14.69 L ÷ 8 = 1.84 L/hour
Therefore, the patient should receive approximately 1.84 liters of fluid per hour in the first 8 hours of their care.
Thabangs transport fee to school increased from 800 to 1150. Determine the percentage increase?
Answers
Answer:
percentage increase: 43,75%
Step-by-step explanation:
If we want to solve thai as an equation, we can write:
800 + (x/100) * 800 = 1150
we multiply: 800 + 800x/100 = 1150
we simplify: 800+ 8x = 1150
we bring the unknown to the left, and numbers to the right:
8x = 1150 - 800
8x = 350
we find x by dividing each part by 8
x= 43,75
(note that the x/100 in the first equation is the same thing as writing x%, so 800 plus a percentage of 800 is equal to 1150)
a rectangle has a perimeter of 48 in. the length and width are scaled by a factor of 2.5. what is the perimeter of the resulting rectangle?
Answers
If a rectangle has a perimeter of 48 in. the length and width are scaled by a factor of 2.5, the perimeter of the resulting rectangle is 225 inches.
Let L be the length of the original rectangle and W be the width. Then, the perimeter of the original rectangle is P = 2L + 2W = 48 inches.
If we scale the length and width by a factor of 2.5, we get a new length of 2.5L and a new width of 2.5W. The perimeter of the new rectangle would be:
P' = 2(2.5L) + 2(2.5W)
= 5L + 5W
To find the new perimeter, we need to find the new values of L and W. Since the length and width are scaled by the same factor, we can write:
2.5L = kL
2.5W = kW
where k is the scaling factor.
Since the new rectangle is scaled by a factor of 2.5, k = 2.5. Therefore:
L' = 2.5L = 2.5(12) = 30 inches
W' = 2.5W = 2.5(6) = 15 inches
The new perimeter is:
P' = 5L' + 5W'
= 5(30) + 5(15)
= 225 inches
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URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS
Answers
Answer:
1) 26
2) 118
3) 144
4) 58
5) 38
6) 96
7) 84
8) 156
9) 240
Step-by-step explanation:
Draw a Venn diagram to confirm my answers.
35% of the 240 students were interested in athletics, so .35 × 240 = 84 students were interested in athletics. Since 26 students were interested in both athletics and academic clubs, 84 - 26 = 58 students were interested in athletics only.
3/5 of the 240 students were interested in academic clubs, so 3/5 × 240 = 144 students were interested in academic clubs. Since 26 students were interested in both athletics and academic clubs,
144 - 26 = 118 students were interested in academic clubs only. From this information, you should be able to complete the two-way table.
2
If 3 12, what is the value of x?
O 4
08
O 18
24
Answers
Answer:
8
Step-by-step explanation:
12 ÷ 3 = 4
2 x 4 = 8
Hope I could help :)
PLS HELP!!
hassan is finding the quotient of (2+3i) and (-4-i)
Answers
Use the Quadratic Formula to solve 2x2 + 6x = –3. Which of the following gives the solutions to the nearest hundredth?
A. 2.37 and 0.63
B. 2.37 and –0.63
C. –2.37 and 0.63
D. –2.37 and –0.63
Answers
Answer:
First, we need to put the equation in standard form, which is ax^2 + bx + c = 0. So, we have 2x^2 + 6x + 3 = 0.
Using the Quadratic Formula, x = (-b ± sqrt(b^2 - 4ac)) / 2a, we can plug in the values for a, b, and c:
x = (-6 ± sqrt(6^2 - 4(2)(3))) / 2(2)
x = (-6 ± sqrt(36 - 24)) / 4
x = (-6 ± sqrt(12)) / 4
Simplifying further, we have:
x = (-6 ± 2sqrt(3)) / 4
x = (-3 ± sqrt(3)) / 2
To the nearest hundredth, the solutions are:
A. 2.37 and 0.63
Which graph shows the image of ABC after a rotation about the origin
Answers
The graph that shows the image of ABC after a rotation about the origin is the graph d
The graph that shows the image of ABCRotation about the origin is a transformation in which a point or object is rotated around the origin (0, 0) by a certain angle. The origin is the fixed point of the rotation, and all other points move in a circular path around the origin.
To perform a rotation about the origin, you need to know the angle of rotation, the coordinates of the point or object being rotated, and the direction of rotation (clockwise or counterclockwise).
Using the above as a guide, we have the following:
The graph after a rotation about the origin is the graph (d)
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Suppose that 10 years ago you bought a home for $110,000, paying 10% as a down payment, and financing the rest at 9% interest for 30 years.
a. Let's consider your existing mortgage:
i. How much money did you pay as your down payment?
ii. How much money was your mortgage (loan) for?
iii. What is your current monthly payment?
iv. How much total interest will you pay over the life of the loan?
b. This year, you check your loan balance. Only part of your payments has been going to pay down the loan; the rest has been going towards interest. You see that you still have $88,536 left to pay on your loan. Your house is now valued at $150,000.
i. How much of the loan have you paid off? (i.e., how much have you reduced the loan balance by? Keep in mind that interest is charged each month - it's not part of the loan balance.)
ii. How much money have you paid to the loan company so far?
iii. How much interest have you paid so far?
iv. How much equity do you have in your home (equity is value minus remaining debt)
c. Since interest rates have dropped, you consider refinancing your mortgage at a lower 6% rate.
i. If you took out a new 30-year mortgage at 6% for your remaining loan balance, what would your new monthly payments be?
ii. How much interest will you pay over the life of the new loan?
d. Notice that if you refinance, you are going to be making payments on your home for another 30 years. In addition to the 10 years you've already been paying, that's a total of 40 years.
i. How much will you save each month because of the lower monthly payment?
ii. How much total interest will you be paying
Answers
a.
i. 10% of $110,000 = $11,000 down payment
ii. Loan (mortgage) = $110,000 - $11,000 = $99,000
iii. Monthly payment = P * r * (1 + r)n / ((1 + r)n - 1), where P represents the principal amount, r represents the monthly interest rate, and n represents the number of monthly installments.
First, we must determine the quantity of monthly payments. Because it is a 30-year loan, the monthly installments are 30 * 12 = 360.
The monthly interest rate of 9% divided by 12 equals 0.0075.
When we plug in the values, we get:
$793.07 monthly payment
iv. Loan total interest paid = (monthly payment * number of instalments) - principal amount = ($793.07 * 360) - $99,000 = $184,465.20
How much money have you paid to the loan company so far?b
i. Reduction in loan balance = $99,000 - $88,536 = $10,464
ii. To date, the total amount paid to the loan firm is (monthly payment * a number of payments) = ($793.07 * 120) = $95,168.40.
We multiplied by 120 (rather than 10*12) because we're computing total payments made so far, not the total payments made over the life of the loan.
iii. Total interest paid thus far = total money given to the loan business minus principal amount = $95,168.40 - $99,000 = -$3,831.60
The negative figure shows that the interest paid thus far has been less than the principal amount. This is because, in the early phases of the loan, the majority of the monthly payment goes towards interest, with only a little portion going toward principal reduction.
iv. Home equity = value of the home - remaining debt
= $150,000 - $88,536
= $61,464
c.
i. New monthly payment = P * r * (1 + r)n / ((1 + r)n - 1), where P represents the remaining loan value, r represents the new monthly interest rate (6% / 12 = 0.005), and n represents the number of monthly payments remaining (360 - 120 = 240).
When we plug in the values, we get:
$63.63 is the new monthly payment.
Total interest paid on the new loan during the life of the loan = (new monthly payment * number of payments) - remaining loan balance = ($63.63 * 240) - $88,536 = $32,028.40
d.
Monthly savings = old monthly payment minus new monthly payment = $793.07 - $63.63= $729.44
Total interest paid on the new loan during the life of the loan = (new monthly payment * number of payments) - remaining loan balance = ($63.63 * 360) - $88,536 = $37,847.80
Because the new loan is being paid off over a longer period of time, the total interest paid during the life of the new loan is more than the remaining interest on the previous loan.
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