The radius of the cone is found by comparing the ratios of heights and radii in a similar triangle. By applying the given values, the radius of the cone is found to be 9 cm.
Explanation:In mathematics, problems related to finding the dimensions of cones and frustums are common. As the problem mentions, a frustum is created when a smaller, similar cone is removed from a larger cone. To calculate the dimensions of the frustum, we use the properties of similar triangles.
For similar triangles:
Base ratios are equal to height ratios.
Therefore:
Height ratio (height of frustum/height of cone) = Base ratio (radius of frustum/radius of cone)
Applying the given values, it becomes:
3/9 = 3/Radius of cone
Hence, the radius of the cone is 9 cm.
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Final answer:
The volume of a frustum can be calculated using the formula [tex]\(V = \frac{1}{3}\pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})\)[/tex]. Given the dimensions provided for the frustum with the larger cone having a radius of 4.5 cm, the frustum a radius of 3 cm, and height of 3 cm, the volume of the frustum is approximately 134.25π cubic centimeters.
Explanation:
To work out the volume of a frustum that is formed by removing a small cone from a larger, similar cone, you can make use of the formula for the volume of a frustum:
V = [tex]\(\frac{1}{3}\pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})\),[/tex]
where:
h is the height of the frustum,
r₁ is the radius of the larger base (radius of the original cone),
r₂ is the radius of the smaller base (radius of frustum).
Given:
Radius of the large cone (r₁) = 4.5 cm,
Radius of the frustum (r₂) = 3 cm,
Height of the frustum (h) = 3 cm.
The height of the original cone (9 cm) is not needed to calculate the volume of the frustum. Plugging the given values into the formula:
[tex]\(V = \frac{1}{3}\pi \times 3 \times (4.5^{2} + 3^{2} + 4.5 \times 3)\)[/tex]
[tex]\(V = \pi (20.25 + 9 + 13.5)\)[/tex]
[tex]\(V = \pi (42.75)\)[/tex]
[tex]\(V = 134.25\pi \text{cm}^{3}\)[/tex]
Therefore, the volume of the frustum is approximately 134.25π cubic centimeters.
Melanie bought bags of colored sand that each cost the same. She spent a total of $24. Find three possible costs per bag and the number of bags that she could have purchased
Answer:
Step-by-step explanation:
Factors of 24 are 1,2,3,4,6,8,12,24
Since we are looking for prices as well as quantity, these numbers must go in pairs.
1*24 = option 1
2*12 = option 2
3*8 = option 3
4*6 = option 4
Any of the above 4 pairs can suit the figures. They can even be reversed except option 1 ($1 and 24 bags but not $24 and 1 bag because the question says bags not bag).
Determine the models that could represent a compound interest account that is growing exponentially.
Select all the correct answers.
A(t) = 2,675(1.003)12t
A(t) = 4,170(1.04)t
A(t) = 3,500(0.997)4t
A(t) = 5,750(1.0024)2t
A(t) = 1,500(0.998)12t
A(t) = 2,950(0.999)t
Answer:A(t)= 2,675(1.003)12t
A(t)=4170(1.04)t
A(t)=5750(1.0024)2t
Step-by-step explanation:Exponential growth is also called growth percentage.
It is calculated using 100% of the original amount plus the growth rate . Example if the amount grows by 5% yearly.5%=0.05
It is written thus(1+0.005)=1.05.
It is usually written in decimal.
The formular for compound interest that is growing exponentially is written as
A=P (1 + i)^N
Looking at the 5 A(t) equations,only 3 of it are growing exponentially.
Clovis is standing at the edge of a cliff, which slopes 4 feet downward from him for every 1 horizontal foot. He launches a small model rocket from where he is standing. With the origin of the coordinate system located where he is standing, and the x-axis extending horizontally, the path of the rocket is described by the formula y = −2x² + 160x.
(a) Give a function h = f(x) relating the height h of the rocket above the sloping ground to its x-coordinate.
(b) Find the maximum height of the rocket above the sloping ground. What is its x-coordinate when it is at its maximum height?
(c) Clovis measures its height h of the rocket above the sloping ground while it is going up. Give a function x = g(h) relating the x-coordinate of the rocket to h.
(d) Does this function still work when the rocket is going down? Explain.
Answer:
a) -2x^2 + 164x
b) 3362 feet
c) (82 , -328)
d) yes
Step-by-step explanation:
y = -2x^2 + 160x
Slope = 4 feet downward for every 1 horizontal foot.
a) h(x) = -2x^2 + 160x - (-4x)
= -2x^2 + 160x + 4x
= -2x^2 + 164x
b) The highest point occurs at the vertex of the parabolic equation. x is the same as the number of the axis of symmetry.
x = -b/2a
From the equation, a = -2 , b= 164
x = -164/ 2(-2)
x = -164/-4
x = 41
Put x = 41 into the value of h(x)
h(x) = -2x^2 + 164x
= -2(41^2) + 164(41)
= -2(1681) + 6724
= -3362 + 6724
= 3362 feet.
The maximum height occurs at 41 feet out from the top of the sloping ground at a height of 3362ft about the top edge of the cliff.
c) h(x) = -2x^2 + 164x
2x^2 - 164x + h = 0 when 0 ≤ x ≤ 41
Solve the equation using the formula (-b+/-√b^2 - 4ac) / 2a
a = 2, b= -164 , c = h
= [-(-164) +/- √(-164)^2 - 4(2)(h) ] / 2(2)
= (164 +/- √26896 - 8h)/ 4
This gives the value of -328 ≤ h ≤ 3362 is used because the rocket hits the sloping ground of (82 , -328)
d) the function still works when it is going down
We first find the function h=f(x) for the rocket's height above the ground. The maximum height is found using the vertex formula with x=(-b)/(2a). We can also determine the function x=g(h) for when the rocket is going up, but this does not work for when the rocket is going down due to needing negative roots.
Explanation:(a) Because the ground slopes 4 feet downward for every 1 horizontal foot, the ground line equation is y=-4x. So, to find the height of the rocket 'h' above the ground, we subtract the equation of the rocket's path from this equation for height of the ground, which gives h=f(x)=y-(-4x)=-2x²+160x+4x=-2x²+164x.
(b) To find the maximum height (vertex) of the parabolic path the rocket takes, we use x=(-b)/(2a) from the standard quadratic equation format (ax²+bx+c=0). Here, a=-2 and b=164. So, x=(-164)/(2*-2)=41. Therefore, the maximum height is when x=41, and we substitute x=41 into the equation h=f(x) to find the maximum height.
(c) To obtain x as a function of h, we can rearrange our equation for h=f(x) to make x=g(h). This will be a square root function because of the x² in the equation. However, since we only want the going up part, we just consider the positive root.
(d) This function doesn't work for when the rocket is going down as it would require considering the negative root, which isn't included in our function g(h) as it involves square roots.
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Factor the expression. x2 – x – 42 (x – 7)(x – 6) (x – 7)(x + 6) (x + 7)(x – 6) (x + 7)(x + 6)
Answer:
(x - 7)x + 6).
Step-by-step explanation:
x^2 – x – 42
6 * -7 = 42 and 6 - 7 = -1 so the factors are:
(x - 7)x + 6).
The factor form of the expression x² - x - 42 is (x - 7)(x + 6).
To factor the expression x² - x - 42, we need to find two binomial factors that, when multiplied together, give us the original expression.
We can start by looking for two numbers that multiply to -42 and add up to -1, which is the coefficient of the x term in the expression.
The pair of numbers that satisfy these conditions are -7 and 6.
If we multiply these two numbers, we get -42, and if we add them, we get -1.
Therefore, we can write the expression as:
x² - x - 42
= (x - 7)(x + 6)
This means that the original expression can be factored as the product of two binomials: (x - 7) and (x + 6).
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The 68-95-99.7 rule tells us how to find the middle 68%, 95% or 99.7% of a normal distribution. suppose we wanted to find numbers a and b so that the middle 80% of a standard normal distribution lies between a and b where a is less than
b. one of the answers below are not true of a and
b. mark the answer that is not true.
Answer:
The values of a and b are -1.28 and 1.28 respectively.
Step-by-step explanation:
It is provided that the area of the standard normal distribution between a and b is 80%.
Also it is provided that a < b.
Let us suppose that a = -z and b = z.
Then the probability statement is
[tex]P (a<Z<b)=0.80\\P(-z<Z<z)=0.80[/tex]
Simplify the probability statement as follows:
[tex]P(-z<Z<z)=0.80\\P(Z<z)-P(Z<-z)=0.80\\P(Z<z)-[1-P(Z<z)]=0.80\\2P(Z<z)-1=0.80\\P(Z<z) = \frac{1.80}{2}\\P(Z<z) =0.90[/tex]
Use the standard normal distribution table to determine the value of z.
Then the value of z for probability 0.90 is 1.28.
Thus, the value of a and b are:
[tex]a = -z = - 1.28\\b = z = 1.28[/tex]
Thus, [tex]P(-1.28<Z<1.28)=0.80[/tex].
Chen has 17CDs. She gives 2 to her brother and buys 4 more. Her brother gives her 1 and she gives 3 to her best friend. How many CDs does Chen have now?
Answer: she has 17 CDs now.
Step-by-step explanation:
The total number of CDs that Chen had initially is 17.
She gives 2 to her brother. This means that she would be having
17 - 2 = 15
She buys 4 more. It means that she would be having
15 + 4 = 19
Her brother gives her 1. So the number that she has is
19 + 1 = 20
she gives 3 to her best friend. Therefore, the number of CDs that Chen has now is
20 - 3 = 17
Cecilia bought a new car the total amount she needs to borrow is 29542 she plans to take out 4 years loan at an APR of 6.3/ what is the monthly payment?
Answer:
697.87
Step-by-step explanation:
You are looking for the monthly payment (PMT) on a loan (borrow = loan).
This is the formula you would use for installment loans (loan payment)
PVA = PMT [(1 - (1+APR/n)^-nY)/APR/n]
NOTE:
PMT = regular payment amount = ?
PVA = starting loan principal (amount borrowed) = 29542
APR = annual percentage rate (as a decimal) = 0.063
n = number of payment periods per year (they told you that it is monthly, so n =12)
Y = loan term in years (can be a fraction) = 4
NOTE: a helpful tip is so start with the original formula and rearrange it to make what you are looking for the subject of the formula.
We're solving for the monthly payment. So rearrange the formula:
PVA = PMT [(1 - (1+APR/n)^-nY)/APR/n]
PMT = [PVA (APR/n)]/(1 - (1+APR/n)^-nY)
PMT = [29542 (0.063/12)] / (1 - (1+ 0.063/12)^-12x4)
∴ PMT = 697.8653...
Round off the answer to as many decimal places as instructed by your lecturer/teacher.
Here we have rounded off to 2 decimal places:
∴ PMT = 697.87
Based on the information given the monthly payment is $697.87 .
Given:
PMT = ?
PVA =29542
APR = 6.3% or 0.063
n =12×4=48
Hence:
PMT = [29542 (0.063/12)] / (1 - (1+ 0.063/12)^-12x4)
PMT = [29542 (0.063/12)] / (1 - (1+ 0.00525)^-12x4)
PMT = [29542 (0.00525)] / (1 - (1.00525)^-48)
PMT = [29542 (0.00525)] / (1-0.777757)
PMT = 155.0955/0.22224274
PMT = 697.865316
PMT= 697.87 (Approximately)
Inconclusion the monthly payment is $697.87 .
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If the last digit of weight measurement is equally likely to be any of the digits 0 through 9. Round your answers to one decimal place (e.g. 98.7). What is the probability that the last digit is 0?
Answer:Probability that the last digit is 0=0.1
Step-by-step explanation:
The likely digits for the last digits runs from 0 through 9 giving a total of 10 digits
The fore P(last digit to be 0) = 1/10 = 0.1
If the last digit of a weight measurement rounded to the nearest tenth place is equally likely to be any of the digits from 0 through 9, then the probability that the last digit is 0 is 0.1 or 10%.
Explanation:The question is asking about the probability that the last digit in a weight measurement, rounded to one decimal place, is 0. Given that the last digit is equally likely to be any digits from 0 through 9, this is a basic probability problem with each outcome being equally likely.
Since there are 10 possible results (the digits 0 through 9), and we are interested in only 1 of these results (the digit 0), the probability can be calculated as 1 divided by 10. Therefore, the probability that the last digit is 0 is 0.1 or 10%.
This is a standard concept in an introductory probability course and is a fundamental idea that will be used in more complex probability problems.
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Finally, the arena decides to offer advertising space on the jerseys of the arena’s own amateur volley ball team. The arena wants the probability of being shortlisted to be 0.14. What is this as a percentage and a fraction? What is the probability of not being shortlisted?
Give your answer as a decimal. Those shortlisted are entered into a final game of chance. There are six balls in a bag (2 blue balls, 2 green balls and 2 golden balls). To win, a company needs to take out two golden balls. The first ball is not replaced.
What is the probability of any company winning advertising space on their volley ball team jerseys?
Answer: 7/50
14%
1/30
Step-by-step explanation:0.14 to fraction =0.14/100=14/100=7/50
0.14 to %= 0.14 ×100=14%
Total number of balls=6
Blue balls=2
Golden balls=2
Green balls=2
Probability of picking the first ball=1/6
Probability of picking the second ball= 1/5
P(winning wit 2 golden balls)=1/6×1/5=1/30
The probability of any company winning advertising space on their volleyball team jerseys is approximately 0.0093, or 0.93%.
Probability of Being Shortlisted
The probability of being shortlisted is given as 0.14.
As a Percentage:[tex]\[ 0.14 \times 100 = 14\% \][/tex]
As a Fraction:[tex]\[ 0.14 = \frac{14}{100} = \frac{7}{50} \][/tex]
Probability of Not Being Shortlisted:The probability of not being shortlisted is:
[tex]\[ 1 - 0.14 = 0.86 \][/tex]
Probability of Winning Advertising Space
To win the advertising space, a company needs to draw two golden balls consecutively without replacement from a bag containing 6 balls (2 blue, 2 green, and 2 golden).
Total Balls:There are 6 balls in total.
First Draw:The probability of drawing a golden ball first:
[tex]\[ \frac{2}{6} = \frac{1}{3} \][/tex]
Second Draw:After drawing one golden ball, there are 5 balls left, including 1 golden ball:
The probability of drawing a golden ball second:
[tex]\[ \frac{1}{5} \][/tex]
Combined Probability:The probability of drawing two golden balls consecutively is the product of the individual probabilities:
[tex]\[ \frac{1}{3} \times \frac{1}{5} = \frac{1}{15} \][/tex]
Final Probability of Winning Advertising Space
Since the company needs to be shortlisted first and then draw the two golden balls to win the advertising space, the combined probability is:[tex]\[0.14 \times \frac{1}{15}\][/tex]
Convert 0.14 to a fraction:[tex]\[0.14 = \frac{7}{50}\][/tex]
Multiply the probabilities:[tex]\[\frac{7}{50} \times \frac{1}{15} = \frac{7}{750}\][/tex]
Convert to a decimal:[tex]\[\frac{7}{750} \approx 0.0093\][/tex]
Barry has 4 wooden identically shaped and sized blocks. 2 are blue, 1 is red and 1 is green. How many distinct ways can barry arrange the 4 blocks in a row? Barry's friend Billie is colour-blind and cannot distinguish between red and green. How many of Barry's distinct arrangements would Billie see different?
Answer:
Step-by-step explanation:
Distinct ways in which Barry can arrange the wooden shaped blocks is calculated from the permutation expression
4 permutation 3 =
P(n,r)=P(4,3) =4! ÷ (4−3)! = 24
Billie's distinct ways of seeing the arrangement would be 4 permutation 2
P(n,r)=P(4,2) =4! ÷ (4−2)! = 12
Answer:
The distinct arrangement Billie would see is P(n,r)=P(4,2) =4! ÷ (4−2)! = 12
Step-by-step explanation:
From the question, we recall the following:
Blue = 2, red =1 green =1
The way this can be solved for which Barry can arrange the wooden shaped blocks is applying the method called permutation
So,
4 permutation 3 = P(n,r)=P(4,3) =4! ÷ (4−3)! = 24
The ways Billie's would see the permutation arrangement is 4 permutation 2
With the expression given as
P(n,r)=P(4,2) =4! ÷ (4−2)! = 12
Melanie is baking breakfast rolls for a band camp fundraiser. She bakes 15 dozen breakfast rolls in 3 hours. After 8 hours, she has baked 40 dozen breakfast rolls. At what rate does Melanie bake breakfast rolls each hour?
Answer:
She bakes rolls at a rate of 60 rolls per hour!
Answer: The rate is 60 per hour
Step-by-step explanation:
A concrete mixer is in volume proportions of 1 part cement, 2 parts water, 2 parts aggregate, and 3 parts sand. How many cubic feet of each ingredient are needed to make 54cu ft of concrete?
Answer:
Step-by-step explanation:
The total volume of cement in cubic feet to be made is 54 cu ft.
A concrete mixer is in volume proportions of 1 part cement, 2 parts water, 2 parts aggregate, and 3 parts sand. This means that the ratio of the ingredients is
1 : 2 : 2 : 3
Total ratio = 1 + 2 + 2 + 3 = 8
Therefore,
Volume of cement needed would be
1/8 × 54 = 6.75 cubic feet
Volume of water needed would be
2/8 × 54 = 13.5 cubic feet
Volume of aggregate needed would be
2/8 × 54 = 13.5 cubic feet
Volume of sand needed would be
3/8 × 54 = 20.25 cubic feet
Jacob brought some tickets to see his favorite singer. He brought some adult tickets and some children tickets for a total of 9 tickets. The adult tickets cost $10 per ticket and the children tickets cost $8 per ticket if he spent a total of $76 then how much are adult and children tickets. Did he buy?
Answer: he bought 2 adult tickets and 7 children tickets.
Step-by-step explanation:
Let x represent the number of adult tickets that he bought.
Let y represent the number of children tickets that he bought.
He brought some adult tickets and some children tickets for a total of 9 tickets. This means that
x + y = 9
The adult tickets cost $10 per ticket and the children tickets cost $8 per ticket if he spent a total of $76, it means that
10x + 8y = 76 - - - - - - - - - - - -1
Substituting x = 9 - y into equation 1, it becomes
10(9 - y) + 8y = 76
90 - 10y + 8y = 76
- 10y + 8y = 76 - 90
- 2y = - 14
y = - 14/ -2
y = 7
x = 9 - y = 9 - 7
x = 2
Phoebe runs at 12km/h and walks at 5km/h. One afternoon she ran and walked a total of 17km. If she ran for the same length of time as she walked for how long did she run
If correct, it should be one hour. Maybe try and solve it yourself to see if this makes sense
Step-by-step explanation:
Assume the total trip that afternoon took t hours
12(t/2) + 5(t/2) = 17 => t = 2
So she ran for 1 hour.
Two hikers are 33 miles apart and walking towards each other. They meet in 10 hours. Find the rate of each Hiker if one joker walks 1.1 mph fast than the other
Answer:
Step-by-step explanation:
Let x represent the rate of the first hiker.
if one hiker walks 1.1 mph fast than the other, it means that the rate of the second hiker would be x + 1.1
Two hikers are 33 miles apart and walking towards each other. They meet in 10 hours. This means that in 10 hours, both hikers travelled a total distance of 33 miles.
Distance = speed × time
Distance covered by the first hiker in 10 hours would be
x × 10 = 10x
Distance covered by the second hiker in 10 hours would be
10(x + 1.1) = 10x + 11
Since the total distance covered by both hikers is 33 miles, then
10x + 10x + 11 = 33
20x + 11 = 33
20x = 33 - 11 = 22
x = 22/20 = 1.1 miles per hour
The rate of the second hiker would be
1.1 + 1.1 = 2.2 miles per hour.
At a corner gas station, the revenue R varies directly with the number g of gallons of gasoline sold. If the revenue is $56.40 when the number of gallons sold is 12, find a linear equation that relates revenue R to the number g of gallons of gasoline. Then find the revenue R when the number of gallons of gasoline sold is 7.5.
Answer:
(i) R = 4.70g
(ii) R = $35.25
Step-by-step explanation:
(i) R ∞ g
Removing the proportionality symbol, we have
R = kg, where k is the constant of proportion
56.40 = k(12)
Divide both sides by 12
56.40/12 = k(12)/12
$4.70 = k
k = $4.70
So, R = 4.70g (which is the linear equation relating Revenue, R to number of gallons, g)
(ii) When g = 7.5,
R = 4.70 * 7.5 = 35.25
R = $35.25
The revenue R at a gas station varies directly with the number of gallons of gasoline sold g. The linear equation relating R to g is R = 4.7g. The revenue when the number of gallons sold is 7.5 is $35.25.
Explanation:
In this particular scenario, we're dealing with a problem of direct variation. In a direct variation, as one quantity increases, the other increases proportionally. This can be represented by a linear equation of the form y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation (the ratio of y to x).
Here, the Revenue (R) varies directly with the number of gallons of gasoline sold (g). We can calculate the constant of variation (k) by dividing the given Revenue (R) by the given number of gallons (g): k = 56.4 ÷ 12 = 4.7. So, the linear equation relating R to g is: R = 4.7g.
To find the revenue R when the number of gallons of gasoline sold is 7.5, substitute g = 7.5 into the equation: R = 4.7 * 7.5 = $35.25
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SHOW YOUR WORK!! Identify the simplest polynomial function having integer coefficients with the given zeros: 3i, −1, 2
Answer:
[tex]p(x)=x^4-x^3+7x^2-9x-18[/tex]
Step-by-step explanation:
The given polynomial has roots 3i, −1, 2
Since [tex]3i[/tex] is a root [tex]-3i[/tex] is also a root.
The factored form of this polynomial is [tex]P(x)=(x-3i)(x+3i)(x+1)(x-2)[/tex]
We need to expand to get:
[tex]p(x)=(x^2-(3i)^2)(x^2-x-2)[/tex]
This becomes [tex]p(x)=(x^2+9)(x^2-x-2)[/tex]
We expand further to get:
[tex]p(x)=x^4-x^3+7x^2-9x-18[/tex]
The polynomial function is [tex]p (x) = x^4 - x^3 + 7x ^2 - 9x - 18[/tex]
The calculation is as follows;The factored form of the given polynomial should be
[tex]P(x) = (x - 3i) (x + 3i) (x + 1) (x - 2)[/tex]
Now we have to expand it
[tex]p(x) = (x^2 - (3i)^2) (x^2 - x - 2)\\\\= (x^2 + 9) (x^2 - x - 2)[/tex]
[tex]p (x) = x^4 - x^3 + 7x ^2 - 9x - 18[/tex]
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For the staff breakfast on Friday Mr. Taylor purchased 5 cartons of eggs (a carton contains a dozen eggs). Je used 2 and 11/12 cartons for scrambled eggs and 1 and 1 and a 3rd cartons for breakfast burritos. How many eggs did he have left?
Answer: he has 9 eggs left.
Step-by-step explanation:
Mr. Taylor purchased 5 cartons of eggs and a carton contains a dozen eggs. A dozen of eggs is 12 eggs. It means that 5 cartons of eggs would contain
5 × 12 = 60 eggs
He used 2 and 11/12 cartons for scrambled eggs. Converting 2 11/12 into improper fraction, it becomes
35/12 cartons .
He used 1 and 1 and a 3rd cartons for breakfast burritos. Converting
1 1/3 into improper fraction, it becomes 4/3 cartons
Total number of cartons that he used would be
35/12 + 4/3 = (35 + 16)/12 = 51/12
The number of cartons left would be
5 - 51/12 = (60 - 51)/12 = 9/12
Since a carton has 12 eggs,
9/12 carton will have 9/12 × 12 = 9 eggs
Follow the steps above, and find c, the total of the payments related to financing, and the monthly payment. A customer buys an automobile from you, the salesman. The price of the car, which includes taxes and license, amounts to $5,955.00. The customer wants to finance the car over 48 months after making a $500 down payment. You inform him that the true annual interest rate is 18%.
the monthly payment is approximately $163.06 and the total payments related to financing are approximately $7834.88.
To find the monthly payment and the total payments related to financing, we need to follow these steps:
1. Calculate the total amount financed.
2. Use the total amount financed to calculate the monthly payment using the formula for monthly payments on a fixed-rate loan.
3. Multiply the monthly payment by the number of months to find the total payments related to financing.
Given:
- Price of the car = $5955.00
- Down payment = $500.00
- Finance period = 48 months
- Annual interest rate [tex]\(= 18\%\)[/tex]
Step 1: Calculate the total amount financed.
The total amount financed is the difference between the price of the car and the down payment.
[tex]\[ \text{Total amount financed} = \text{Price of the car} - \text{Down payment} \][/tex]
[tex]\[ \text{Total amount financed} = \$5955.00 - \$500.00 \][/tex]
[tex]\[ \text{Total amount financed} = \$5455.00 \][/tex]
Step 2: Calculate the monthly payment.
To calculate the monthly payment, we use the formula for the monthly payment on a fixed-rate loan:
[tex]\[ M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} \][/tex]
Where:
- M is the monthly payment
- P is the principal amount (total amount financed)
- r is the monthly interest rate (annual interest rate divided by 12)
- n is the number of payments (finance period in months)
First, we need to convert the annual interest rate to a monthly interest rate:
[tex]\[ r = \frac{18\%}{12} = 0.18 \times \frac{1}{12} = 0.015 \][/tex]
Now, we plug in the values:
[tex]\[ M = \frac{5455 \times 0.015 \times (1 + 0.015)^{48}}{(1 + 0.015)^{48} - 1} \][/tex]
[tex]\[ M ≈ \frac{5455 \times 0.015 \times (1.015)^{48}}{(1.015)^{48} - 1} \][/tex]
Using a calculator, we find that the monthly payment M is approximately $163.06.
Step 3: Calculate the total payments related to financing.
[tex]\[ \text{Total payments} = \text{Monthly payment} \times \text{Number of months} \][/tex]
[tex]\[ \text{Total payments} = \$163.06 \times 48 \][/tex]
[tex]\[ \text{Total payments} ≈ \$7834.88 \][/tex]
So, the monthly payment is approximately $163.06 and the total payments related to financing are approximately $7834.88.
Evan cut a triangular piece of cloth to use in a quilt. The perimeter of the cloth is 934 cm. The base of the triangular cloth is 214cm. The remaining two sides are the same length.Choose Yes or No to tell if each expression models how to find the length of the other two sides of Evan's cloth.934−2s=214 114+2s=934 2s=934−214 2s−214=934
Answer:
934−2s=214; Yes
114+2s=934; No
2s=934−214; Yes
2s−214=934; No
Step-by-step explanation:
The base of the triangular cloth is 214cm. The remaining two sides are the same length.
Let s be the length of other sides.
Perimeter = Sum of all sides of a triangle.
[tex]Perimeter = s+s+214[/tex]
[tex]Perimeter =2s+214[/tex]
It is given that the perimeter of the triangular cloth is 934 cm.
[tex]2s+214=934[/tex] .... (1)
Equation (1) can be rewritten as
[tex]2s=934-214[/tex] and [tex]214=934-2s[/tex]
On solving we get
[tex]2s=720[/tex]
Divide both sides by 2.
[tex]s=360[/tex]
Therefore, the length of the other two sides of Evan's cloth is 360 cm.
Ben decided to volunteer forty hours to community service projects. The garden project took 2/3 of the time. How many hours did the garden project take?
Answer:
26.67 hours
Step-by-step explanation:
Given:
Number of hours of service projects (N) = 40 hours
Time taken to complete the garden project is two-third of the total time.
Therefore, the time taken to complete the garden project can be obtained by multiplying the part to the total time taken.
So, the hours taken for garden project is given as:
[tex]x=\frac{2}{3}\times N\\\\x=\frac{2}{3}\times 40\\\\x=\frac{80}{3}\\\\x=26.67\ hours[/tex]
Therefore, it took 26.67 hours to complete the garden project.
You are adding an addition to your patio. The area ( in square feet) of the addition can be represented by k² - 3k - 10.
a) The area of the patio before the addition was 50 square feet. Find it.
b) Find the area of the addition and the area of the entire patio after the addition.
Answer:
[tex]k^{2}-3k+40[/tex]
Step-by-step explanation:
We suppose;
A= area before addition
B= Area of addition [tex]k^{2}-3k-10[/tex]
a) As area of pation before addition is 50 - it means A= 50
b) Area of addition and the area of entire pation after addition = A+B
= 50 + [tex]k^{2}-3k-10[/tex]
=[tex]k^{2}-3k+40[/tex]
Answer:We suppose;
A= area before addition
B= Area of addition
a) As area of pation before addition is 50 - it means A= 50
b) Area of addition and the area of entire pation after addition = A+B
= 50 +
=
the sum of three consecutive number is 114. what is the smallest of the three numbers?
Answer:
37
Step-by-step explanation:
37+38+39
Answer: the smallest of the three numbers is 37
Step-by-step explanation:
Let x represent the smallest number.
Since the three numbers are consecutive, it means that the next number would be x + 1
Also, the last and also the largest number would be x + 2
If the sum of the three consecutive numbers is 114, it means that
x + x + 1 + x + 2 = 114
3x + 3 = 114
Subtracting 3 from the Left hand side and the right hand side of the equation, it becomes
3x + 3 - 3 = 114 - 3
3x = 111
Dividing the Left hand side and the right hand side of the equation by 3, it becomes
3x/3 = 111/3
x = 37
Rewrite as a combination of multiple logarithms:
log_8 (10xy^3)
Answer:
The answer to your question is letter B. log₈10 + log₈x + 3log₈y
Step-by-step explanation:
Just remember the properties of logarithms
- The logarithm of a product is the sum of logarithms.
- The logarithm of a power is equal to the power times the log.
Then
log₈(10xy³) = log₈ 10 + log₈x + log₈y³
and finally
log₈10 + log₈x + 3log₈y
To find the equation of a line, we need the slope of the line and a point on the line. Since we are requested to find the equation of the tangent line at the point (4, 2), we know that (4, 2) is a point on the line. So we just need to find its slope. The slope of a tangent line to f(x) at x = a can be found using the formula
Answer:
[tex]x-4y+4=0[/tex]
[tex]f(x)=\sqrt x[/tex] and x=4
Step-by-step explanation:
We are given that a curve
[tex]y=\sqrt x[/tex]
We have to find the equation of tangent at point (4,2) on the given curve.
Let y=f(x)
Differentiate w.r.t x
[tex]f'(x)=\frac{dy}{dx}=\frac{1}{2\sqrt x}[/tex]
By using the formula [tex]\frac{d(\sqrt x)}{dx}=\frac{1}{2\sqrt x}[/tex]
Substitute x=4
Slope of tangent
[tex]m=f'(x)=\frac{1}{2\sqrt 4}=\frac{1}{2\times 2}=\frac{1}{4}[/tex]
In given question
[tex]m=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}[/tex]
[tex]\frac{1}{4}=\lim_{x\rightarrow 4}\frac{f(x)-f(4)}{x-4}[/tex]
By comparing we get a=4
Point-slope form
[tex]y-y_1=m(x-x_1)[/tex]
Using the formula
The equation of tangent at point (4,2)
[tex]y-2=\frac{1}{4}(x-4)[/tex]
[tex]4y-8=x-4[/tex]
[tex]x-4y-4+8=0[/tex]
[tex]x-4y+4=0[/tex]
The equation of the tangent line of a function at a particular point can be found by using the formula y - y1 = m(x - x1), where the slope m is the derivative of the function at the specific point. In this case, find the derivative at x = 4 and substitute into the formula along with the point (4,2).
Explanation:To find the equation of the tangent line of a function at a particular point, we can indeed utilise the slope-point form of a straight line equation, which is y - y1 = m (x - x1). In this case the point on the line is (4,2).
However regarding the slope, it is calculated as the derivative of the function f(x) at the point x = a.
Let us assume the function f(x). The derivative f '(x), also known as the slope of the tangent line at any point x, is found by taking the derivative of f(x). So to find the slope at x = 4, you would calculate f '(4).
Substitute the value of the derivative at the point (4,2) which represents our m(slope), x1=4 and y1=2 into the linear equation y - y1 = m(x - x1) to generate the equation of the tangent line.
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What is the slope intercept form of the equation y+18=2(x-1)
Step-by-step explanation:
Given,
The equation y + 18 = 2( x - 1)
To write the given equation in the slope intercept form = ?
∴ The equation y + 18 = 2( x - 1)
⇒ y + 18 = 2x - 2
⇒ y = 2x - 2 - 18
⇒ y = 2x - 20
⇒ y = 2x + ( - 20) ..... (1)
We know that,
The equation of slope intercept form,
y = mx + c
Where, m is the sope and c is the y-intercept
∴ The slope intercept form of the given equation is: y = 2x + ( - 20)
PLEASE HELP ASAP!!! I NEED CORRECT ANSWERS ONLY PLEASE!!! I NEED TO FINISH THESE QUESTIONS BEFORE MIDNIGHT TONIGHT.
Find CD.
Write your answer as an integer or as a decimal rounded to the nearest tenth.
CD =
Answer:
Step-by-step explanation:
This is classic right triangle trig. We have the reference angle of 59 degrees, we have the side adjacent to that angle of 4 units, and we are looking for CD which is the hypotenuse of the triangle. That is the cosine ratio:
[tex]cos\theta=\frac{adj}{hyp}[/tex]
Filling in:
[tex]cos(59)=\frac{4}{hyp}[/tex]
Doing some algebraic acrobats there to solve for the hypotenuse gives you:
[tex]hyp=\frac{4}{cos(59)}[/tex]
Use your calculator to solve this in degree mode:
hyp = 7.8 units
Jill found a new fruit punch recipe that calls for orange juice and lemon-lime soda. If orange juice costs $3.60 per bottle and lemon-lime soda costs $1.80 per bottle and the recipe calls for 3 times as many bottles of lemon-lime soda as orange juice, at most how many bottles of orange juice can she buy if she only has $54.00?
Answer:
She can buy at most 6 bottles of orange juice.
Step-by-step explanation:
Consider the provided information.
The recipe calls for 3 times as many bottles of lemon-lime soda as orange juice,
Let she buy x bottles of orange juice.
According to question: Lemon lime soda = 3x
Orange juice costs $3.60 per bottle and lemon-lime soda costs $1.80 per bottle. She only has $54.00
[tex]3.60x+1.80(3x)=54[/tex]
[tex]3.60x+5.4x=54[/tex]
[tex]9x=54[/tex]
[tex]x=6[/tex]
Hence, she can buy at most 6 bottles of orange juice.
Jill can buy at most 6 bottles of orange juice.
Explanation:To find out how many bottles of orange juice Jill can buy, we need to determine the cost of the orange juice and the cost of the lemon-lime soda based on the given prices. Let's assume she can buy 'x' bottles of orange juice. Since the recipe calls for 3 times as many bottles of lemon-lime soda, she can buy 3x bottles of lemon-lime soda. The total cost of the orange juice and the lemon-lime soda must not exceed $54.00.
The cost of the orange juice is $3.60 per bottle, so the cost of 'x' bottles of orange juice is 3.60x dollars. The cost of the lemon-lime soda is $1.80 per bottle, so the cost of 3x bottles of lemon-lime soda is 1.80 * 3x = 5.40x dollars.
To find the maximum number of bottles of orange juice she can buy, we need to solve the inequality:
3.60x + 5.40x ≤ 54.00
Combining like terms, we have:
9.00x ≤ 54.00
Dividing both sides of the inequality by 9.00, we get:
x ≤ 6
Jill can buy at most 6 bottles of orange juice.
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The first four terms of an arithmetic sequence are given.
27, 32, 37, 42, ...
What is the 60th term of the sequence?
Answer:
[tex]a_6_0=322[/tex]
Step-by-step explanation:
we know that
The rule to calculate the an term in an arithmetic sequence is
[tex]a_n=a_1+d(n-1)[/tex]
where
d is the common difference
a_1 is the first term
we have that
[tex]a_1=27\\a_2=32\\a_3=37\\a_4=42[/tex]
[tex]a_2-a_1=32-27=5[/tex]
[tex]a_3-a_2=37-32=5[/tex]
so
The common difference is d=5
[tex]a_4-a_3=42-37=5[/tex]
Find 60th term of the sequence
[tex]a_n=a_1+d(n-1)[/tex]
we have
[tex]a_1=27\\d=5\\n=60[/tex]
substitute
[tex]a_6_0=27+5(60-1)[/tex]
[tex]a_6_0=27+5(59)[/tex]
[tex]a_6_0=322[/tex]
The 60th term of the sequence should be 322 when the first four terms should be given.
Calculation of the 60th term of the sequence:Since
a1 = 27
a2 = 32
a3 = 37
And, a4 = 42
So,
= 27 + 5(60 - 1)
= 27 + 5(59)
= 322
hence, The 60th term of the sequence should be 322 when the first four terms should be given.
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a scale drawing of a rectangle is made by using a scale factor of 5/8. the original and the scale drawing are shown below. which method can be used to find the dimensions of the original rectangle
Answer:
L_original = 28.8 in
H_original = 19.2 in
Step-by-step explanation:
Given:
- Length of scaled rectangle L_scale = 18 in
- width of the scaled rectangle H_scale= 12 in
- Scale factor = (5/8)
Find:
-Which method can be used to find the dimensions of the original rectangle
Solution:
- The best way to determine the original dimensions of the rectangle is by ratios. We have the scale factor as (5/8). so we can express:
L_scale = (5/8)*L_original
L_original = L_scale*(8/5)
L_original = 18*(8/5) = 28.8 in
H_scale = (5/8)*H_original
H_original = H_scale*(8/5)
H_original = 12*(8/5) = 19.2 in
- Hence, the original dimensions are:
L_original = 28.8 in
H_original = 19.2 in
Answer:
B. [tex]18 / \frac{5}{8}= 28\frac{4}{5}[/tex] [tex]inches[/tex] [tex]and[/tex] [tex]12 /\frac{5}{8} = 19\frac{1}{5}[/tex] [tex]inches[/tex]
Step-by-step explanation: