The dosage of Jack and Jill is obtained using geometric series formula which is 500 mg and 225 mg respectively.
What is geometric series?
A geometric series is a collection of terms in mathematics that have an unlimited number and a fixed ratio between each term. A geometric series is typically expressed as a + ar + ar² + ar³ + ..., where r is the common ratio between neighbouring terms and a is the coefficient of each term.
Let's start by finding the amount of medication that Jack takes in total.
Each hour, Jack takes a dosage that is 80% of the previous dosage (since it drops by 20%).
So his dosages form a geometric sequence with first term 100 and common ratio 0.8 -
100, 80, 64, 51.2, 40.96, ...
The sum of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by -
sum = a / (1 - r)
In this case, a = 100 and r = 0.8, so the sum is -
sum = 100 / (1 - 0.8) = 500
Therefore, Jack takes a total of 500 mg of medication.
Now let's find the original dosage for Jill.
If Jill takes half the amount of medication that Jack eventually took, then she takes -
500 / 2 = 250 mg
Let's call Jill's original dosage x.
Then her dosages form a geometric sequence with first term x and common ratio 0.7 (since it drops by 30%).
The sum of this sequence must be 250.
x + 0.7x + 0.49x + 0.343x + ... = 250
This is an infinite geometric series with first term x and common ratio 0.7, so we can use the formula for the sum of an infinite geometric series -
sum = x / (1 - 0.7) = x / 0.3
Simplifying the equation from before -
x / 0.3 × (1 + 0.7 + 0.49 + 0.343 + ...) = 250
x / 0.3 × (1 / (1 - 0.7)) = 250
x / 0.3 × 3.333... = 250
x = 250 / 1.111... = 225
Therefore, Jill's original dosage was 225 mg.
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Edgar and his older sisters, Anne and Maya, are going to visit their grandmother. Edgar is too young to drive, so Anne and Maya each drive the same amount of time to complete the 5-hour trip.
How long does each of Edgar's sisters drive?
Write your answer as a proper fraction or mixed number.
Anne and Maya drive for 2 1/2 hours.
What is mixed fraction?A mixed fraction is a combination of a whole number and a proper fraction. For example, 2 1/3 is a mixed fraction, where the whole number is 2 and the proper fraction is 1/3.
To convert an improper fraction into a mixed fraction, Divide the numerator by the denominator and write down the whole number result as the whole number part of the mixed fraction.
Write down the remainder as the numerator of the fractional part of the mixed fraction.
Here we have
Anne and Maya each drive the same amount of time to complete the 5-hour trip.
Let both of them be derived for 'x' hours
=> x + x = 5 hours
=> 2x = 5 hours
=> x = 5/2
Hence, Anne and Maya drive for 5/2 hours
Here 5/2 can be converted as 2 1/2
Therefore,
Anne and Maya drive for 2 1/2 hours.
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Negative forty four plus 22 times 16 divided by 100
Answer:
[tex]\boxed{R:-40.48}[/tex]
Step-by-step explanation:
Rewriting the statement we have:
[tex]-44+22 \times\frac{ 16}{100}[/tex]
According to the order of operations, we solve:
First, exponents and squaresThen, multiplication and divisionFinally, addition and subtractionsimplify:
[tex]-44+22 \times\frac{ 4}{25}\\=-44 +\frac{88}{25} \\=\frac{-44(25)+88}{25} = \frac{-1012}{25} \approx -40.48[/tex]
solve the absolute value equation or indicate that the equation
has no solution.
|2x-1|=9
The equation has two solutions:[tex]x=5[/tex] and [tex]x=-4[/tex].
To solve an absolute value equation, we need to consider both the positive and negative solutions. We can do this by setting up two separate equations and solving each one individually.
The first equation is:
[tex]2x-1 = 9[/tex]
We can solve for x by isolating the variable on one side of the equation:
[tex]2x = 10[/tex]
[tex]x = 5[/tex]
The second equation is:
[tex]2x-1 = -9[/tex]
We can also solve for [tex]x[/tex] in this equation:
[tex]2x = -8[/tex]
[tex]x = -4[/tex]
Therefore, the solution to the absolute value equation[tex]|2x-1|=9[/tex] is [tex]x=5[/tex] or [tex]x=-4[/tex].
In conclusion, the equation has two solutions: [tex]x=5[/tex] and [tex]x=-4[/tex].
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You need a home loan of $55,000 after your down payment. How much will your monthly house payment be if the bank charges 5.25% APR for a loan of 20 years? (Simplify your answer completely. Round your answer to the nearest cent.)
#2
A farmer purchased 265 acres of land for $4,300/acre. He paid 25% down and obtained a loan for the balance at 6.75% APR over a 20-year period. How much is the annual payment? (Simplify your answer completely. Round your answer to the nearest cent.)
#3
Larry purchased a new combine that cost $240,500, minus a rebate of $4,500, a trade-in of $9,500, and a down payment of $5,000. He takes out a loan for the balance at 8% APR over 4 years. Find the annual payment. (Simplify your answer completely. Round your answer to the nearest cent.)
The monthly house payment will be $375.84. The annual payment for A farmer that purchased 265 acres of land for $4,300/acre will be $75,436.20. The annual payment of Larry purchased a new combine that cost $240,500will be $64,514.88.
To calculate the monthly house payment for a home loan of $55,000 at 5.25% APR for 20 years, we can use the formula:
M = P * (r / (1 - (1 + r)^(-n)))
Where M is the monthly payment, P is the principal amount, r is the monthly interest rate, and n is the number of monthly payments.
In this case, P = $55,000, r = 5.25% / 12 = 0.004375, and n = 20 * 12 = 240.
Plugging in these values, we get:
M = $55,000 * (0.004375 / (1 - (1 + 0.004375)^(-240)))
M = $375.84
Therefore, the monthly house payment will be $375.84.
#2
To calculate the annual payment for a loan of 265 acres of land at $4,300/acre with a 25% down payment and 6.75% APR over 20 years, we can use the same formula as above, but with different values for P, r, and n.
In this case, P = 265 * $4,300 * (1 - 0.25) = $845,625, r = 6.75% / 12 = 0.005625, and n = 20 * 12 = 240.
Plugging in these values, we get:
M = $845,625 * (0.005625 / (1 - (1 + 0.005625)^(-240)))
M = $6,286.35
Since this is the monthly payment, we can multiply it by 12 to get the annual payment:
A = $6,286.35 * 12 = $75,436.20
Therefore, the annual payment will be $75,436.20.
#3
To calculate the annual payment for a loan of $240,500 for a new combine with a rebate of $4,500, a trade-in of $9,500, and a down payment of $5,000 at 8% APR over 4 years, we can use the same formula as above, but with different values for P, r, and n.
In this case, P = $240,500 - $4,500 - $9,500 - $5,000 = $221,500, r = 8% / 12 = 0.006667, and n = 4 * 12 = 48.
Plugging in these values, we get:
M = $221,500 * (0.006667 / (1 - (1 + 0.006667)^(-48)))
M = $5,376.24
Since this is the monthly payment, we can multiply it by 12 to get the annual payment:
A = $5,376.24 * 12 = $64,514.88
Therefore, the annual payment will be $64,514.88.
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answer quick please show work!! plss
the total cost of the clarinet, including shipping and handling, is $600.
What is percentage?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.
Given, Jordan bought a clarinet priced at $517. Shipping and handling cost an additional 16% of the price.
The cost of shipping and handling is 16% of the price of the clarinet, which is:
0.16 * $517 = $82.72
the total cost of the clarinet including shipping and handling, we need to add the cost of the clarinet and the cost of shipping and handling:
Total cost = $517 + $82.72 = $599.72
Rounding this to the nearest dollar, we get:
Total cost = $600
Therefore, the total cost of the clarinet, including shipping and handling, is $600.
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Use the Scores data set. Dr. Z is interested in discovering if there is a difference in depression scores between those who do not watch or read the news and those who continue with therapy as normal. She divides her clients with depression into 2 groups. She asks Group 1 not to watch or read any news for two weeks while in therapy and asks Group 2 to continue with therapy as normal. The Scores data set is a record of the results of the measure, administered after 2 weeks
1. Independent Variable: Group (No News or watch news)
Dependent Variable: Depression Scores
2. Null Hypothesis: μ = 0
Alternative Hypothesis: μ < 0
3. Yes, you can reject the null hypothesis at a = 0.05 because there is a considerable difference in the depression ratings of the two groups
1. According to the information provided, the score for depression depends on whether a group watches the news or not.
Group is an independent variable (No News or watch news).
Depression scores are a dependent variable.
2. Null Hypothesis: The two groups do not differ in their depression levels, which is the null hypothesis.
Alternate Hypothesis: Those who don't read or watch the news will score less depressed than people who receive therapy as usual.
μ = 0 is the null hypothesis.
Between the two groups, there is no difference in depression scores.
Differential Hypothesis: μ < 0
Those who don't read or watch the news will score less depressed than people who receive therapy as usual.
3. Due to the p-value of (0.024) < α(0.05) at α = 0.05, we can rule out the null hypothesis.
The rejection zone of the t-distribution with 12 degrees of freedom has the t-score of -2.58.
As a result, it is clear that there is a considerable difference in the depression ratings of the two groups, with the group that does not watch or read the news scoring significantly lower on the depression scale than the group that continues to get therapy as usual.
The proper response is "Yes".
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The complete question is:
Dr. Z is interested in discovering if there is a difference in depression scores between those who do not watch or read the news and those who continue with therapy as normal. She divides her clients with depression into 2 groups. She asks Group 1 not to watch or read any news for two weeks while in therapy and asks Group 2 to continue with therapy as normal. The dataset score is a record of the results of the measure, administered after 2 weeks.
Independent Samples T-Test
t df p
Score -2.580 12 0.024
1. Identify IV and DV.
2. State the null hypothesis and the directional (one-tailed) alternative hypothesis
3. Can you reject the null hypothesis at a = 0.05?
Can someone please help ?
Answer:
Height = 8
Volume = 48
Step-by-step explanation:
(4x4)x3 = 48
From the top of a cliff, the angle of depression of a boat on the sea is 60°. If the top of the cliff is 25m above the sea level, calculate the horizontal distance from the bottom of the cliff to the boat.
I need help on this asap!
Answer:
Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
the lines p and q intersect at point O.
what is the balue of x?
those are opposite angles therefore they are equal
3x - 3 = 2x + 13
solve for x
x = 16
Answer:
See below.
Step-by-step explanation:
We are asked to find the value of x.
We should know that the 2 angles given are Vertical Angles.
What are Vertical Angles?Vertical Angles are angles that are opposite of each other. These angles are equal to each other if 2 or more lines are straight.
Let's set the angles equal to each other.
[tex]3x-3=2x+13[/tex]
Simplify:
[tex]x=16[/tex]
The value of x is 16.
Set up (do not solve ) the partial fraction decomposition for (4x-3)/(x^(2)(x+3)(x^(2)+2x+9)(x^(2)+1)^(3)).
To find the values of these constants, we need to multiply both sides of the equation by the denominator of the original fraction and then equate the numerators. This will give us a system of equations that we can solve for the constants.
The partial fraction decomposition for the given fraction is:
(4x-3)/(x^(2)(x+3)(x^(2)+2x+9)(x^(2)+1)^(3)) = A/x + B/x^(2) + C/(x+3) + D/(x^(2)+2x+9) + E/(x^(2)+1) + F/(x^(2)+1)^(2) + G/(x^(2)+1)^(3)
Where A, B, C, D, E, F, and G are constants that need to be determined.
To find the values of these constants, we need to multiply both sides of the equation by the denominator of the original fraction and then equate the numerators. This will give us a system of equations that we can solve for the constants.
However, since the question only asks to set up the partial fraction decomposition, we do not need to solve for the constants at this time.
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Which is a y-intercept of the continuous function in the table?
The y-intercept of the continuous function in the table is (0, –6) which can be determined by looking at the first point, (0, –6).
What is a continuous function?A continuous function is a function that is defined for all values of its independent variable, and whose graph is a continuous line without breaks or gaps. It typically means that a function's value for an input can be calculated without having to consider the values of the function for other inputs.
The y-intercept of a continuous function is the point at which the graph of the function crosses the y-axis. It is the point where the x-coordinate is zero. In other words, it is the point where the graph of the function intersects the y-axis.
In the given table, the y-intercept can be determined by looking at the first point, (0, –6). This means that the y-coordinate is –6 when the x-coordinate is 0. Therefore, the y-intercept of the continuous function in the table is (0, –6).
The y-intercept also helps us to determine the slope of the function. The slope is the rate at which the function changes as we move along the x-axis. The slope can be calculated by looking at the change in the y-coordinate for a given change in the x-coordinate.
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The y-intercept οf the cοntinuοus functiοn in the table is (0, –6) which can be determined by lοοking at the first pοint, (0, –6).
What is a cοntinuοus functiοn?A cοntinuοus functiοn is a functiοn that is defined fοr all values οf its independent variable, and whοse graph is a cοntinuοus line withοut breaks οr gaps. It typically means that a functiοn's value fοr an input can be calculated withοut having tο cοnsider the values οf the functiοn fοr οther inputs.
The y-intercept οf a cοntinuοus functiοn is the pοint at which the graph οf the functiοn crοsses the y-axis. It is the pοint where the x-cοοrdinate is zerο. In οther wοrds, it is the pοint where the graph οf the functiοn intersects the y-axis.
In the given table, the y-intercept can be determined by lοοking at the first pοint, (0, –6). This means that the y-cοοrdinate is –6 when the x-cοοrdinate is 0. Therefοre, the y-intercept οf the cοntinuοus functiοn in the table is (0, –6).
The y-intercept alsο helps us tο determine the slοpe οf the functiοn. The slοpe is the rate at which the functiοn changes as we mοve alοng the x-axis. The slοpe can be calculated by lοοking at the change in the y-cοοrdinate fοr a given change in the x-cοοrdinate.
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Complete question:
Which is a y-intercept of the continuous function in the table?
(0, –6) (–2, 0) (–6, 0) (0, –2)Work out the value of r when 2^(r)=(1)/(32) Give your answer as an integer or as a fraction in its simplest form.
The value of r when 2^(r)=(1)/(32) is -5. This can be derived by taking the logarithm of both sides of the equation. Since log2 (2^r) = r and log2 (1/32) = -5, then r = -5.
In order to work out the value of r when 2^(r) = 1/32, we need to use logarithms. Logarithms are a way of expressing a number as the power of another number. Specifically, the logarithm of a number x is the exponent y to which a given base must be raised to equal x.
In this case, the base is 2, so log2 (2^r) = r. Therefore, to work out the value of r, we need to find the logarithm of 1/32. This can be done by dividing 1 by 32 and taking the logarithm of the result. We get log2 (1/32) = -5. Therefore, r = -5.
Since the question specifies that we should give the answer as an integer or as a fraction in its simplest form, we must conclude that the answer is -5. It is impossible to reduce this fraction any further, so it is already in its simplest form.
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Find the geometric mean for 64 and 25
Answer:
40
Step-by-step explanation:
The geometric mean will equal the square root of the product of the two numbers.
√64 * √25
8 * 5 = 40
√25+∛27-2(-3)°= calculate without using a calculator
Answer: 12 no caculator needed
Step-by-step explanation:
(1 point) Find the angleαin radians between the vectors[−12] and [6−1]α=
The angle between the vectors "[−12] and [6−1]α" is approximately 2.03 radians.
To find the angle between the vectors, we can use the formula:
cos(α) = ([tex]V{1}[/tex] • [tex]V{2}[/tex]) / (||[tex]V{1}[/tex]|| ||[tex]V{2}[/tex]||)
Where v1 and v2 are the two vectors, and ||[tex]V{1}[/tex]|| and ||[tex]V{2}[/tex]|| are the magnitudes of the vectors.
For the given vectors:
[tex]V{1}[/tex] = [−1 2] and [tex]V{2}[/tex] = [6 −1]
[tex]V{1}[/tex]• v2 = (−1)(6) + (2)(−1) = −6 + −2 = −8
||[tex]V{1}[/tex]|| = √((-1)^2 + 2^2) = √(5)
||[tex]V{2}[/tex]|| = √(6^2 + (-1)^2) = √(37)
cos(α) = (−8) / (√(5) √(37)) = −8 / √(185)
α = cos^-1(−8 / √(185))
α ≈ 2.03 radians
Therefore, the angle between the vectors is approximately 2.03 radians.
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Its a screen shot
right down there its for math
Answer:
Step-by-step explanation:
2-Bs: 4 x 3.8 = 15.2 2 x 15.2 = 30.4 m²
2-Cs: 11.5 x 3.8 = 43.7 2 x 43.7 = 87.4 m²
Total S.A. = 92 + 30.4 + 87.4 = 209.8 m²
Answer: 2.98m squared i think im not sure im just guessing im still not fully awake yet bro.
Step-by-step explanation:
so confusinggggggggggggggggg
Convert the expression as follows:
[tex]\cfrac{y+2}{y+7} =\cfrac{k}{1} \ \ fraction\ form\ of\ the\ expression[/tex][tex]y+2=k(y+7) \ cross-miultiply[/tex][tex]y+2=ky+7k\ \ distribute[/tex][tex]y-ky=7k-2\ \ collect\ terms\ with\ \ variable\ y[/tex][tex]y(1-k)=7k-2\ \ factor\ out\ y[/tex][tex]y=\cfrac{7k-2}{1-k}\ \ divide \ both\ sides\ by\ 1-k[/tex]To Show:-
y = ( 7k - 2 )/( 1 - k)Answer:-
The ratio given to us is ,
[tex]\implies (y + 2) : ( y + 7) = k : 1 \\[/tex]
In fraction form we can write it as ,
[tex]\implies \dfrac{y+2}{y+7}=\dfrac{k}{1} \\[/tex]
Now solve for y , by cross multiplying,
[tex]\implies 1( y + 2 ) = k( y + 7) \\[/tex]
Simplify the brackets,
[tex]\implies y + 2 = ky + 7k \\[/tex]
Subtract ky on both sides ,
[tex]\implies y - ky + 2 = 7k \\[/tex]
Subtract 2 on both sides,
[tex]\implies y - ky = 7k - 2 \\[/tex]
Take out y as common from LHS ,
[tex]\implies y ( 1 - k ) = 7k - 2 \\[/tex]
Divide both the sides by (1-k) ,
[tex]\implies \underline{\underline{ \green{y =\dfrac{7k-2}{1-k}}}} \\[/tex]
Hence Proved !
and we are done!
Find the effective interest rate for the specified account. nominal yield, 9%; compounded twice a year 9.00% 9.31% 9.38% 9.20% Use the formula for future value, A - P(1 + rt), to find the missing quantity. A=$5580; P=$4500;r=6% A.t= 4 years B. t= 5 years C. t= 3 years D. t= 6 years
Option a) t= 4 years. The effective interest rate for the specified account is 9.31%. This is because the effective interest rate is the rate that actually applies to the balance in the account, taking into account the compounding frequency. The formula for the effective interest rate is:
Effective interest rate = (1 + nominal yield / compounding frequency) ^ compounding frequency - 1
In this case, the nominal yield is 9% and the compounding frequency is 2 (since it is compounded twice a year). Plugging these values into the formula, we get:
Effective interest rate = (1 + 0.09 / 2) ^ 2 - 1
Effective interest rate = (1.045) ^ 2 - 1
Effective interest rate = 1.092025 - 1
Effective interest rate = 0.092025
Converting this to a percentage, we get:
Effective interest rate = 9.31%
Therefore, the correct answer is 9.31%.
For the second part of the question, we can use the formula for future value to find the missing quantity. The formula is:
A = P(1 + rt)
Plugging in the given values, we get:
5580 = 4500(1 + 0.06t)
Solving for t, we get:
5580 = 4500 + 270t
1080 = 270t
t = 4
Therefore, the correct answer is t = 4 years, or choice A.
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3 = Problem 3. Given the vectors ū and ✓ find the vector w = 4ữ - 5 (in coordinates), and its magnitude | W |. Use exact values; assume mile as the length unit. (-1) 2 Problem 4. Let O be an angle in standard position, such that the point P(-15, 8) is on its terminal side. Find the exact values of all the six trigonometric functions of 0.
Given vectors ū = (u1, u2) and ✓ = (v1, v2), we can find vector w = 4ữ - 5✓ by using the formula:
w = (4u1 - 5v1, 4u2 - 5v2)
To find the magnitude of vector w, we can use the formula:
|w| = √((4u1 - 5v1)² + (4u2 - 5v2)²)
Since we do not have the exact values of u1, u2, v1, and v2, we cannot compute the exact values of w and |w|.
w = (4u1 - 5v1, 4u2 - 5v2), |w| = √((4u1 - 5v1)² + (4u2 - 5v2)²)
Problem 4:
Given the point P(-15, 8) on the terminal side of angle O in standard position, we can find the values of the six trigonometric functions of O using the following formulas:
sin O = y/r
cos O = x/r
tan O = y/x
csc O = r/y
sec O = r/x
cot O = x/y
Where x = -15, y = 8, and r = √(x² + y²) = √((-15)² + 8²) = √(225 + 64) = √289 = 17
Therefore, the exact values of the six trigonometric functions of O are:
sin O = 8/17
cos O = -15/17
tan O = -8/15
csc O = 17/8
sec O = -17/15
cot O = -15/8
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Solve the following system of equations:
2x-6y+9z=-8
5x+y+2z=10
3x+y-8z = -28
(Please add an explanation for better understanding.)
I've been working on this for a week and I can't figure it out...
Answer:
This
Step-by-step explanation:
( x y z) = (-1,7,4) explanation is big
What is the missing angel
The missing angle as labelled in the task content and required to be determined is; 33°.
What is the measure of the missing angle?It follows from the task content that the missing angle is to be determined from the task content.
By the use of trigonometric ratios;
The missing angle, x can be determined as follows;
tan (x) = 9.09 / 14
x = tan-¹ (9.09/14).
x = 32.995°
The missing angle is therefore; 33°.
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I need help on this asap!!
The amount in the second case is more. Then the better job is the second one.
What is the solution to the equation?In other words, the collection of all feasible values for the parameters that satisfy the specified mathematical equation is the convenient storage of the bunch of equations.
If he can make $2,000 worth of sales per week. Then the total amount in the first case is given as,
⇒ $31,200 + $2,000 x 0.09 x 52
⇒ $40,560
And the total amount in the second case is given as,
⇒ $26,000 + $2,000 x 0.15 x 52
⇒ $41,600
The amount in the second case is more. Then the better job is the second one.
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A
radio transmission tower is 578 feet tall. A guy wire is to be
attached 6 feet from the top and is to make an angle of 20° with
the ground? How many feet long should the guy wire be? Round your
ans
The guy wire's length should be approximately 120.5 feet.
To find out, we can use trigonometry. The guy wire, the tower, and the ground form a right triangle, with the guy wire as the hypotenuse. We know the angle between the guy wire and the ground (20°), and we know the height of the tower (578 feet). We want to find the length of the guy wire. Using the trigonometric function tangent, we can set up the following equation:
tan(20°) = 578 / (guy wire length - 6)
Solving for the guy wire length, we get:
guy wire length = 578 / tan(20°) + 6
guy wire length ≈ 120.5 feet
Therefore, the guy wire should be approximately 120.5 feet long.
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FIRST OT HELP GETS BRAINLIEST! PLEASE SHOW WORK!
The table shows the value of printing equipment for 3 years after it is purchased. The values form a geometric sequence. How much will the equipment be worth after 7 years?
Geometric sequence: [tex]a_{n} = a_{1} r^{n - 1}[/tex]
Year Value $
1................12,000
2...............9,600
3...............7,680
The equipment will be worth approximately $3,145.73 after 7 years.
What is the worth of the equipment after 7 years?
To find the value of the printing equipment after 7 years, we need to first determine the common ratio (r) of the geometric sequence.
We can do this by dividing any term by the preceding term. Let's divide the value in year 2 by the value in year 1:
r = 9,600/12,000 = 0.8
Now we can use the formula for a geometric sequence to find the value of the equipment after 7 years:
a₇ = a₁r⁶
where;
a₁ is the value in year 1 and r is the common ratio we just found.Plugging in the values we have:
a₇ = 12,000 x 0.8⁶
a₇ ≈ $3,145.73
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Consider the following inequality: |x+4|-1>=0 Step 1 of 2 : Solve the absolute value inequality and express the solution in interval notation and algebraic notation.
The solution of the absolute value inequality is (-∞,-5] U [-3,∞) in interval notation and x ≥ -3 or x ≤ -5 in algebraic notation.
Step 1: To solve the absolute value inequality, we need to isolate the absolute value expression on one side of the inequality. We can do this by adding 1 to both sides of the inequality:
|x + 4| ≥ 1
Step 2: Now, we can split the inequality into two separate inequalities:
x + 4 ≥ 1 and x + 4 ≤ -1
Step 3: Solve each inequality for x:
x ≥ -3 and x ≤ -5
Step 4: Express the solution in interval notation and algebraic notation. In interval notation, the solution is (-∞,-5] U [-3,∞). In algebraic notation, the solution is x ≥ -3 or x ≤ -5.
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3. The bottom of a ramp is 15 feet from the entrance of a building. If the angle of the ramp is
about 3.2°, about how high does the ramp rise off the ground to the nearest inch?
The ramp rises about 10.4 inches off the ground to the nearest inch.
What is ramp ?A ramp is an inclined plane that connects two different levels, allowing people or objects to move between them with less force than if they had to be lifted vertically.
We can use trigonometry to solve this problem. Let's call the height of the ramp "h". Then we have:
tan(3.2°) = h/15
To solve for h, we can multiply both sides by 15:
h = 15 tan(3.2°)
Using a calculator, we get:
h ≈ 0.87 feet
To convert this to inches, we can multiply by 12:
h ≈ 10.4 inches
So, the ramp rises about 10.4 inches off the ground to the nearest inch.
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Expand 3(x+4) I tried this question but I cant get it quite right. Could you help me?
Rates and Unit Rates
Use ratios to solve.
3
-
Tyler reads page per minute.
4
What is the ratio of pages to minutes?
How many pages will he read in 16 minutes?
Tyler has to read 24 pages for homework.
How long will it take him to read 24 pages?help me pls
The ratio of pages to minutes is 3:4. If he read 16 minutes, then the number of pages is 12 pages. And Tyler has to read 24 pages for homework. Then the number of minutes is 32 minutes.
What is the ratio?The utilization of two or more additional numbers that compares is known as the ratio.
Tyler reads 3/4 pages per minute. Then the ratio of pages to minutes is given as,
Ratio = (3/4) / 1
Ratio = 3: 4
If he read 16 minutes, then the number of pages is calculated as,
⇒ (3/4) x 16
⇒ 3 x 4
⇒ 12 pages
Tyler has to read 24 pages for homework. Then the number of minutes is given as,
⇒ 24 / (3/4)
⇒ 24 x (4/3)
⇒ 8 x 4
⇒ 32 minutes
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2. [P] Consider the following equation.x2−918−x−312=x+3x(a) Give the restrictions onx. (b) What is the least common denominator? (c) Keeping these restrictions in mind, solve the equation. If there is no solution or infinitely many solutions, then so state.
The given equation is x2−9/18−x−3/12=x+3/x
(a) The restrictions on x are that x cannot equal 0, 6, or -4. This is because if x were to equal any of these values, the denominator of one of the fractions in the equation would equal 0, which is undefined.
(b) The least common denominator of the fractions in the equation is 36. This is because 36 is the smallest number that is divisible by 18, 12, and x.
(c) To solve the equation, we can multiply each term by the least common denominator to eliminate the fractions:
36x2 - 18(9) - 36x - 3(3) = 36x + 3(36)
Simplifying and rearranging the terms gives us:
36x2 - 72x - 207 = 0
Using the quadratic formula, we can find the values of x:
x = (-(-72) ± √((-72)2 - 4(36)(-207)))/(2(36))
Simplifying gives us:
x = (72 ± √12960)/72
So the solutions are x = (72 + √12960)/72 and x = (72 - √12960)/72.
However, we must check these solutions against the restrictions on x. The first solution, (72 + √12960)/72, is approximately 6.8, which does not violate any of the restrictions. The second solution, (72 - √12960)/72, is approximately -0.8, which also does not violate any of the restrictions. Therefore, the solutions to the equation are x = (72 + √12960)/72 and x = (72 - √12960)/72.
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