The cable supporting a ski lift rises 2 feet for each 5 feet of horizontal length. The top of the cable is fastened 1320 feet above the cable’s lowest point. Find the lengths b and c, and find the measure of the angle theta.
Answers
To find the lengths b and c of the cable supporting the ski lift, use the Pythagorean theorem. To find the measure of angle theta, use the tangent function.
Explanation:To find the lengths b and c, we can use the Pythagorean theorem. Since the vertical rise of the cable is 2 feet for every 5 feet of horizontal length, we can create a right triangle where the vertical leg is 2x and the horizontal leg is 5x. The hypotenuse of this triangle, which represents the length of the cable, is 1320 feet. Using the Pythagorean theorem, we can solve for the value of x and then find the lengths b and c.
b = 5x = 5 * (1320 / √(25 + 4))
c = 2x = 2 * (1320 / √(25 + 4))
Next, to find the measure of angle theta, we can use the tangent function. The tangent of an angle is equal to the ratio of the length of the opposite side (2x) to the length of the adjacent side (5x). Therefore, tan(theta) = 2x / 5x = 2x / (5 * (1320 / √(25 + 4))). Taking the arctangent of both sides of the equation will give us the measure of theta.
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The lengths b and c are 3300 feet and 1320 feet respectively, and the angle theta is approximately 21.8°.
To solve this problem, let's denote:
- b is horizontal length from the lowest point to where the cable is fastened.
- c is vertical rise from the lowest point to where the cable is fastened.
- theta is angle that the cable makes with the horizontal.
From the problem statement:
- The cable rises 2 feet for each 5 feet of horizontal length. This gives us the slope of the cable:
[tex]\[ \frac{c}{b} = \frac{2}{5} \][/tex]
- The cable is fastened 1320 feet above the lowest point, which means:
[tex]\[ c = 1320 \text{ feet} \][/tex]
Now, solve for b:
[tex]\[ \frac{c}{b} = \frac{2}{5} \][/tex]
[tex]\[ \frac{1320}{b} = \frac{2}{5} \][/tex]
Cross-multiplying to solve for b:
[tex]\[ 1320 \times 5 = 2 \times b \][/tex]
[tex]\[ 6600 = 2b \][/tex]
[tex]\[ b = \frac{6600}{2} = 3300 \text{ feet} \][/tex]
Now, we have:
[tex]\[ b = 3300 \text{ feet} \][/tex]
[tex]\[ c = 1320 \text{ feet} \][/tex]
To find [tex]\( \theta \)[/tex], use the tangent function:
[tex]\[ \tan(\theta) = \frac{c}{b} = \frac{1320}{3300} = \frac{2}{5} \][/tex]
Therefore, the measure of angle [tex]\( \theta \)[/tex] is:
[tex]\[ \theta = \tan^{-1}\left(\frac{2}{5}\right) \][/tex]
Using a calculator to find [tex]\( \theta \):[/tex]
[tex]\[ \theta \approx \tan^{-1}(0.4) \][/tex]
[tex]\[ \theta \approx 21.8^\circ \][/tex]
The complete question is
The cable supporting a ski lift rises 2 feet for each 5 feet of horizontal length. The top of the cable is fastened 1320 feet above the cable’s lowest point. Find the lengths b and c, and find the measure of the angle theta.
Related Questions
Lucas is applying to both Stanford and Cal-poly. The probability that he gets accepted to Stanford is 0.4. The probability that he gets accepted to Cal-Poly is 0.3. The probability that he is accepted to both schools is 0.15.
a) What is the probability that he is accepted to one of the two schools?
b) What is the probability that he does not get accepted at either school?
Answers
The probability that Lucas is accepted to at least one of the two schools is 55%, while the probability that he does not get into either school is 45%.
Explanation:The subject of the question is probability theory, which is a branch of mathematics. Lucas has a 0.4 chance of getting into Stanford, a 0.3 chance of getting into Cal-Poly, and a 0.15 chance of getting into both. These are independent probabilities because they do not affect each other.
a) To calculate the probability that he gets accepted to at least one of the schools, we add the probabilities of him getting into each school, and then subtract the probability of him getting into both (because we're double counting that scenario). So, that's 0.3 (for Cal-Poly) + 0.4 (for Stanford) - 0.15 (for both) = 0.55, or 55%.
b) In probability theory, the total probability of all possibilities is always 1. So, to find the probability that he doesn't get into EITHER school, we subtract the probability that he gets into at least one school from 1. So, 1 - 0.55 (the answer from part a) = 0.45, or 45%. So, there's a 45% chance that Lucas doesn't get accepted into either school.
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Solve using quadratic formula. X^2+9x-13=0
Answers
When the polynomial in P(x) is divided by (x + a), the remainder equals P(a).
A. True
B. False
Answers
According to Remainder Theorem for the polynomials, for every polynomial P(x) there exist such polynomials G(x) and R(x), that [tex] P(x)=(x\pm a)G(x)+R(x) [/tex].
When the polynomial in P(x) is divided by (x + a), then there exist such polynomials G(x) and R(x), that P(x)=(x+a)G(x)+R(x). Note that for x=-a:
This means that P(-a) is the remainder, not P(a).
Answer: correct choice is B.
Answer:
false
Step-by-step explanation:
2614/7=6(47/6b+11/6)
Answers
7 6b 6
multiply everything by the lcm of the denominator, 42b
15684 b = 6b( 6( 47 + 11 )
6b 6
15684 b = 6b( 282 + 66 )
6b 6
15684 b = 284 + 66 b
15618 b = 284
b = 142 = 0.01818
7809
Use the distributive property to simplify the expression 6(s-9
Answers
Answer:
6s-54
Step by step: Put the 6 in front of the s and then multiply 6*9. so the answer is 6s-54
The distributive property is used in mathematics to multiply a number by multiple numbers within parentheses. Using it for expression 6(s - 9) gives us 6s - 54.
Explanation:The expression you need to simplify using the distributive property is 6(s - 9). The distributive property is a fundamental property in mathematics that allows you to multiply a singular number by multiple numbers within parentheses.
In case of the expression 6(s - 9), you'll simply need to distribute or multiply the number 6 to each term inside the parentheses.
6 x s and 6 x -9. Performing these will give us 6s - 54.
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Given that t is the median of the numbers 32, 64, 22, t, 41, 22, and 78, what is the lowest possible value of t?
Answers
Final answer:
The lowest possible value for t as the median of the numbers 32, 64, 22, t, 41, 22, and 78 is 32. This value maintains the order when the numbers are arranged from lowest to highest for t to be the median.
Explanation:
To find the lowest possible value for t given that it is the median of the numbers 32, 64, 22, t, 41, 22, and 78, we first need to arrange the numbers in ascending order.
Since there are seven numbers, the median would be the fourth number when sorted.
To ensure t is the lowest possible median, we sort the remaining numbers excluding t. Thus, we have:
22, 22, 32, 41, 64, 78
The third value is 32, and the fifth value is 41, so for t to be the median, t must be greater than or equal to 32 but less than or equal to 41.
However, since we are looking for the lowest possible value of t, t would be 32, as this is the lowest it can be while still being the median of the sorted list.
The sum of two numbers is 56, and their difference is 10. What are the numbers?
Answers
a + b = 56 and a - b = 10.
if you do (a+b) + (a-b) then you get:
(a+b) + (a-b) = 66 = a + b + a - b
The b's cancel, so you get 2a = 66, which means a = 33. You can use this to solve for b in one of the above equations by plugging in a = 33 and figuring out what b has to be. Then you will have your two numbers.
x - y = 10 so x = y + 10
y + 10 + y = 56
2y +10 = 56
2y = 56 -10
2y = 46
y = 46/2
y = 23
x = y + 10 = 23 + 10 =33
Answer: The numbers are 23 and 33
Proof:
sum: 23 + 33 = 56
difference: 33 -23 = 10
(2x - 3y)(4x - y) ...?
Answers
We will use Foil method to multiply binomials:
(2x - 3y)(4x - y) = 2x * 4x - 2x * y - 3y * 4x + 3y * y =
= 8x² - 2xy - 12xy + 3y²
= 8x² - 14xy + 3y²
3/5 of the village speaks one of the 8 major languages. What fraction of the world speaks a language other than one of the 8 major languages?
Answers
0.00000768 is the correct answer, depending how you do it.
h(x)=(g o f)(x)=1/(x+3)^2 which of the fallowing could be a possible decomposition of h(x)? A. f(x)=1/x^2;g(x)=x+3 B. f(x)=x+3; g(x)=1/x^2 C.f(x)=x^2; g(x)=x+3 D. f(x)=1/x;g(x)=x+3
...?
Answers
If you take B f(x)=x+3 g(x)=1/x^2 plug the f(x) into the g(x) formula (in other words, f(x) becomes the x for g) g(x)=1/x^2 g(x)=1/(x+3)^2
A linear model projects that the number of bachelor's degrees awarded in the United State will increase by 19,500 each year. In 2001, 1.28 million bachelor's degrees were awarded. Use the inverse function to predict the number of years after 2001 that 1.7 million bachelor's degrees will be awarded.
Answers
The linear model predicts that approximately 22 years after 2001 (around the year 2023), 1.7 million bachelor's degrees will be awarded annually in the U.S.
Explanation:The problem is related to the mathematical concept of linear modeling. This model is represented by the function f(x) = 19500x + 1.28 million, where x is the number of years after 2001 and f(x) is the number of bachelor's degrees awarded in that year. The objective is to calculate 'x' when f(x) is 1.7 million. To do this, we use the inverse function.
The inverse function of f(x) = 19500x + 1.28 million is the function x = (f(x) - 1.28 million) / 19500. Substituting the known value f(x) = 1.7 million into the equation, we get: x = (1.7 million - 1.28 million) / 19500 ≈ 21.54. Since we cannot have a fraction of a year, we round this up to the next whole number.
Therefore, according to the linear model, it will be approximately 22 years after 2001, i.e., in the year 2023, when the number of bachelor's degrees awarded is expected to reach 1.7 million.
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Using the linear model, it is predicted that 1.7 million bachelor's degrees will be awarded approximately 22 years after 2001, which corresponds to the year 2023.
Explanation:To predict the number of years after 2001 that 1.7 million bachelor's degrees will be awarded using a linear model, we start with the initial number of degrees awarded in 2001, which is 1.28 million. According to the model, the number of bachelor's degrees increases by 19,500 each year. We set up the equation: 1.28 million + 19,500x = 1.7 million. Here, x represents the number of years after 2001.
Solving for x gives us:
1.7 million - 1.28 million = 19,500x
420,000 = 19,500x
x = 420,000 / 19,500
x ≈ 21.54
Thus, it will take approximately 22 years after 2001 for 1.7 million bachelor's degrees to be awarded, which means this is expected to occur in the year 2023 (since 2001 + 22 = 2023).
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Summer clothes are on sale starting September 1st. A swimsuit is on sale for 40% off the regular selling price of $55.00. What is the markdown to the nearest cent?
Answers
0.55 x 40 = 22 which is 40%
55 - 22 = 33
The answer is $33
solve x2+10x+21=0 ...?
Answers
/\ = 10²-4*1*21
/\ = 100 - 84
/\ = 16
x = (-10+/-\/16)/2
x = (-10+/-4)/2
x' = (-10+4)/2 = -6/2 = -3
x" = (-10-4)/2 = -14/2 = -7
S: {-3; -7}
What decimal is equivalent to 5/6?
Answers
Name an angle adjacent to ∠DGE
Choices:
∠FGI
∠EGH
∠HGJ
Answers
m (DGE)=m (IGJ)
they're interior contrary angle
Answer:
The answer is ∠EGH.
Step-by-step explanation:
In order to determine the correct option, we have to know about adjacent angle.
Two angles are adjacent when they have a common side and vertex (corner point). So in this case, the adjacent angle must have GE or DG common side and the G vertex.
According to the options, the adjacent angle to ∠DGE is:
∠EGH
The other options are not adjacent angles because they do not have a common side.
Finally the correct option is ∠EGH.
what is the quotient when -3^3 +5x+14 is divided by x-2? pleas hurry
Answers
The quotient is -3x² + 3x + 8.
To find the quotient when [tex]\(-3x^3 + 5x + 14\)[/tex] is divided by x - 2, we can use polynomial long division.
The dividend is [tex]\(-3x^3 + 0x^2 + 5x + 14\)[/tex] (since there is no [tex]\(x^2\)[/tex] term) and the divisor is x - 2.
1. Divide the first term of the dividend by the first term of the divisor: [tex]\(-3x^3 \div x = -3x^2\)[/tex].
2. Multiply the divisor by the result of step 1: [tex]\((-3x^2)(x - 2) = -3x^3 + 6x^2\).[/tex]
3. Subtract the result of step 2 from the dividend: [tex]\((-3x^3 + 0x^2 + 5x + 14) - (-3x^3 + 6x^2) = -6x^2 + 5x + 14\).[/tex]
4. Repeat the process with the new dividend [tex]\(-6x^2 + 5x + 14\)[/tex].
Continuing the division process, we find the quotient:
[tex]\[ \frac{-3x^3 + 5x + 14}{x - 2} = -3x^2 + 3x + 8 \][/tex]
So, the quotient is -3x² + 3x + 8.
Select the number property that will justify the following expression.
If cba represents three numbers multiplied together, what property allows you to rearrange the factors to read abc?
a. transitive
b. idenity addition
c. communitive addition
d. associative - addition
Answers
I believe this may be what your looking for ;-)
Select the number property that will justify the following expression.
If cba represents three numbers multiplied together, what property allows you to rearrange the factors to read abc?
commutative - multiplication used twice
The Commutative Property of multiplication justifies the rearrangement of factors in a multiplication problem, such as changing cba to abc.
If cba represents three numbers multiplied together, the c. Commutative Property of multiplication allows you to rearrange the factors to read abc. This property states that the order of factors can be switched without changing the product, meaning ab = ba. The options provided in the query (transitive, identity addition, and associative - addition) are not related to multiplication or the rearranging of factors.
according to poll for an upcoming school board election is 40% of voters are likely to vote for the incumbent. The polls show a margin of error of +-3 percent points.
Answers
The original price is 50. the discount is 15%. what is the new price
Answers
15% off of 50
notice that 50 is 1/2 of hundred
15% of 100 is 15
therefor
15% of 50 is 15/2=7.5
that is discount
50-7.5=42.5
new price=$42.5
Twice the difference of two numbers is 14 and their sum is 3
find the numbers.
Answers
You have to solve the system of equations
2(a-b)=14
a+b=3
2(x-y) = 14
Substitute value from eq (1) in eq (2),
2(3-y-y) = 14
2(3-2y) = 14
6-4y = 14
-4y = 8
y = -2
Put it into eq (1),
x-2 = 3
x = 5
So, x = 5 & y = -2
Hope this helps!
What is the value of the function y=2x−3 when x=−1 ?
Answers
When x=-1
y=2×(-1)-3=-2-3= -5
Suzanne is going to rent a car while she is out of town. One car rental company offers a flat rate of $35 per day plus $0.10 per mile. Another car rental company offers the same car for $25 per day plus $0.25 per mile. She will need the car for 5 days. How many miles would she need to drive for the first rental company to be a better deal?
Answers
r1 will be a better deal when when r2 > r1r2 > r1 when35*5 + .10*m > 25*5 + .25m5(35-25) > (.25 - .10)m5*10 > (.15)m50/.15 > mm > 50/.15 = 33 1/3
Ans. When 33 1/3 miles have been driven, rental car company r1 will be a better deal.
Answer:
She need to drive more than 2000 miles for the first rental company to be a better deal .
Step-by-step explanation:
Case 1) One car rental company offers a flat rate of $35 per day plus $0.10 per mile
Let m be the no. of miles
She will need the car for 5 days.
So, total cost = [tex]35 \times 5 +0.10m=175+0.10m[/tex]
Case 2) Another car rental company offers the same car for $25 per day plus $0.25 per mile.
Let m be the no. of miles
She will need the car for 5 days.
So, total cost = [tex]25 \times 5 +0.25m=125 +0.25m[/tex]
Now we are supposed to find How many miles would she need to drive for the first rental company to be a better deal?
So, [tex]175+0.10m <125 +0.25m [/tex]
[tex]175+125 <125 +0.25m-0.10m [/tex]
[tex]300 < 0.15m [/tex]
[tex]\frac{300}{0.15} < m [/tex]
[tex]2000 < m [/tex]
So, she need to drive more than 2000 miles for the first rental company to be a better deal
Find the area of a square whose side is (3x + 2).
...?
Answers
A square has sides of equal length
Therefore
Area= (3x + 2) x (3x + 2)
= (3x + 2)²
what is 1/4 divided by 5?
Answers
1/4 divided by 5 results into '0.05'
Given that we need to find 1/4 divided by 5
Thus let the required answer be denoted as "A"
Thus using basic definitions of divisions and fractions we can find out the resultant value
[tex]A=\frac{\frac{1}{4} }{5} \\A=\frac{1}{4*5} \\A=\frac{1}{20}[/tex]
Thus '1/20' is the required answer
Now the resultant value can also be simplified to get a decimal value
[tex]A=\frac{1}{20} \\A=\frac{1}{2*10} \\A=\frac{1}{2} *\frac{1}{10} \\A=0.5*0.1\\A=0.05[/tex]
Thus '0.05' is the final answer
Find the first,fourth,and tenth terms of the arithmetic sequence described by the given rule. A(n) = 12+(n-1)(3)
Answers
The first, fourth, and tenth terms of the arithmetic sequence A(n) = 12+(n-1)(3) are 12, 21, and 39 respectively.
Explanation:The arithmetic sequence A(n) given by the rule A(n) = 12+(n-1)(3) represents the value of the nth term in the sequence. The first term of the sequence (n=1) can be found by substitifying n=1 into the equation, which will yield [tex]A(1)=12+(1-1)(3)=12.[/tex] The fourth term A(4) can be found similarly by substituting n=4 into equation resulting in [tex]A(4)=12+(4-1)(3)=21.[/tex]The tenth term A(10) can also be calculated in the similar manner giving [tex]A(10)=12+(10-1)(3)=39.[/tex] So, the first, fourth, and tenth terms of the arithmetic sequence are 12,21, and 39 respectively.
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The table shows the heights in inches of trees after they have been planted. Determine the equation which relates x and y.
Height In Pot, x | Height Without Pot, y
30 | 18
36 | 24
42 | 30
48 | 36
A. y = x - 6
B. y = x - 10
C. y = x - 12
D. y = x + 6
Answers
Let's check all choices.
A. y = x - 6
18 = 30 - 6
18 = 24
INCORRECT
B. y = x - 10
18 = 30 - 10
18 = 20
INCORRECT
C. y = x - 12
18 = 30 - 12
18 = 18
CORRECT
24 = 36 - 12
24 = 24
CORRECT
...
D. y = x + 6
18 = 30 + 6
18 = 36
INCORRECT
1) what is the volume of a cylinder with a radius of 2 cm and a height of 10 cm? ( Express your answer using symbol pie)
A) 4(symbol pie) cm^3
B) 10(symbol pie) cm ^3
C) 20(symbol pie) cm ^3
D) 40(symbol pie) cm ^3
2) a cylinder has a base with a radius of 3 cm and a height of 12 cm. what is the volume of the cylinder? ( use symbol pie=3.14)
A) 113.04 cm^3
B) 339.12 cm^3
C) 452.16 cm^3
D) 1356.48 cm^3
3) A cylinder has a base with a radius of 3 cm and a height of 12 cm. what is the volume of the cylinder? ( use symbol pie=3.14)
A) 113.04 cm^3
B) 339.12 cm^3
C) 452.16 cm^3
D) 1356.48 cm^3
4) A cylinder and a rectangular prism have the same volume and same height. The base of the prism is a square with length 7 cm.
what is the approximate radius of the cylinder?
A) 4.0 cm
B) 3.5 cm
C) 2.2 cm
D) 1.5 cm
Answers
Answer:
1 a 2 d 3b 4 c
Step-by-step explanation:
sorry it took me 6 years to get back to you
let Ax=b be any consistent system of linear equations, and let x1 be a fixed solution. show that every solution to the system can be written in form x=x1+x2, where x0 is a solution to Ax=0. show that every matrix of this form is a solution.
...?
Answers
so if Ax = 0 can be expressed as x1 - x2
any solution of Ax=b can be expressed as x1 + x2
hope this helps
What is another way to write the expression 82p
Answers
To convert 82 pints to quarts, divide 82 by the conversion factor of 2, which provides the result of 41 quarts.
Explanation:To rewrite the expression 82p where p represents pints into quarts, first understand that pints are smaller than quarts. Because there are 2 pints in a quart, you use division for the conversion. Since the conversion factor is 2, you would divide the number of pints by 2 to find out how many quarts that is.
To convert 82p, divide 82 by 2:
82 ÷ 2 = 41
Therefore, 82 pints is equal to 41 quarts.
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holly can make 3 necklaces with 6 diamonds. if holly has 30 diamonds, how many necklaces can she make?
Answers
Edit: I read the problem wrong, check the other comment.
If Holly has 30 diamonds and she needs 6 to make 3 necklaces, you do 30 ÷ 6 = 5. You then take 5 × 3 to give you the amount of necklaces she can make, which is 15.