A lighthouse is located on an island 33 miles from the closest point on a straight shoreline. If the lighthouse light rotates clockwise at a constant rate of 99 revolutions per minute, how fast does the beam of light move towards the point on the shore closest to the island when it is 5252 miles from that point
Answers
Question:
A lighthouse is located on an island 3 miles from the closest point on a straight shoreline. If the lighthouse light rotates clockwise at a constant rate of 9 revolutions per minute, how fast does the beam of light move towards the point on the shore closest to the island when it is 52 miles from that point
Answer:
The beam of light moves at [tex]16278\pi[/tex] miles/min
Step-by-step explanation:
This question is illustrated with the attached image
Taking the instructions in the question, one at a time.
A revolution of 9 per minute implies that:
[tex]\frac{d\theta}{dt} = \frac{9 * 2\pi\ rad}{1\ min}[/tex]
[tex]\frac{d\theta}{dt} = 18\pi \frac{rad}{min}[/tex]
Take tan of the angle in the attachment:
[tex]tan(\theta) =\frac{52}{3}[/tex]
Differentiate both sides with respect to time
[tex]\frac{d\ tan(\theta)}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
Rewrite as:
[tex]\frac{d\ tan(\theta)}{d\theta} * \frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
In calculus:
[tex]sec^2(\theta) =\frac{d\ tan(\theta)}{d\theta}[/tex] -- Chain rule
So:
[tex]sec^2(\theta) *\frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
In trigonometry:
[tex]sec^2(\theta) = tan^2(\theta) + 1[/tex]
So:
[tex](tan^2(\theta) + 1)\frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
Recall that:
[tex]tan(\theta) =\frac{x}{3}[/tex]
[tex]((\frac{52}{3})^2 + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{52^2}{9} + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{2704}{9} + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{2704+9}{9})\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{2713}{9})\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
Recall that: [tex]\frac{d\theta}{dt} = 18\pi \frac{rad}{min}[/tex]
[tex](\frac{2713}{9}) * 18\pi =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex]2713 * 2\pi =\frac{1}{3} * \frac{dx}{dt}[/tex]
Multiply both sides by 3
[tex]3 * 2713 * 2\pi =\frac{1}{3} * \frac{dx}{dt} * 3[/tex]
[tex]3 * 2713 * 2\pi =\frac{dx}{dt}[/tex]
[tex]16278\pi =\frac{dx}{dt}[/tex]
[tex]\frac{dx}{dt} = 16278\pi[/tex] miles/min
Hence:
The beam of light moves at [tex]16278\pi[/tex] miles/min
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Step-by-step explanation:
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Step-by-step explanation:
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Step-by-step explanation:
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✧ [tex] \underline{ \underline{ \large{ \tt{R\: E \: M \: E \: M \: B \: E \: R}}} }: [/tex]
If [tex]( {x}^{a} )^{b} [/tex] is an algebraic term then [tex] {(x}^{a} )^{b} = {x}^{a \times b} [/tex] i.e When an algebraic term in the index form is raised to another index , the base is raised to the product of two indices.
⟶ [tex] \large{ \tt{( {7}^{3} ) ^{3} = {7}^{3 \times 3} = {7}^{9}}} [/tex]
☥ [tex] \boxed{ \boxed{ \large \tt{Our \: Final \: Answer : \underline{ \tt{ {7}^{9}}}}}} [/tex]
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⋆ Have a wonderful day / night ! ♪
Let me know if you have any questions regarding my answer! ツ
[tex] \underline{ \underline{ \mathfrak{Carry \: On \: Learning}}}[/tex] !! ✎
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what is 1 times 89what is 1 * 17
Answers
Answer:
1 times 89 = 89 lol 1 x 7= 7
Step-by-step explanation:
hope this helps lol
Simplify
(7x^2 - 11 + 16x) + (5x - 3x^2 - 7)
Answers
Answer:
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Answer:
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Sweet corn of a certain variety is known to produce individual ears of com with a mean weight of 8 ounces. A farmer is testing a new fertilizer designed to produce larger ears of com, as measured by their weight. He selects a random sample of 10 ents of com and determines their weight (in ounces). The data are as follows: 8.25 7359,05 8.00 8.25 8.75 9.00 7.60 8.45 8.55
Do the data provide convincing evidence at the a=0.05 level that the fertilizer had a positive impact on the weight of the com cars?
Answers
Answer:
We can conclude that there is significant evidence that dL fertilizer has a positive impact on larger weight of coms
Step-by-step explanation:
Given the data :
8.25 7.35 9.05 8.00 8.25 8.75 9.00 7.60 8.45 8.55
Using calculator :
Mean, x = 8.325
Standard deviation, s = 0.560
Null hypothesis :
H0: μ = 8
H1 : μ > 8
The test statistic,
(x - μ) / (s/sqrt(n))
(8.325 - 8) / (0.56 / sqrt(10))
0.325 / 0.177
= 1.835
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Pvalue = 0.04985
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Answer:
Step-by-step explanation:
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Answers
Answer:
Step-by-step explanation:
The area of a trapezoid is the average of the parallel bases times the height or distance between the two bases
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Answer:
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Step-by-step explanation:
So this question states that yesterday, 10 preschoolers chose to play with stuffed animals and 10 different preschoolers chose to so other activities.
We can assume that there were 20 preschoolers at school yesterday (10 + 10 = 20).
From this given information, we can conclude that the preschoolers who would chose to play with stuffed animals would be the same amount at the preschoolers who would chose to do other activities.
Using all of the given information, we can also find that:
10/20 or 1/2 of the preschoolers would chose to play with the stuffed animals
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10/20 or 1/2 of the preschoolers would chose to do other activities
We are then given that there are 10 preschoolers at school today. We are asked to find out how many of the preschoolers are expected to chose to play with the stuffed animals.
Previously, we found that 1/2 of the preschoolers yesterday chose to play with the stuffed animals. We can use this to find the expected amount of preschoolers who will chose to play with the stuffed animals today.
We can do this by multiplying 1/2 with the number of preschoolers today, or 10:
10 × 1/2 = 10/2 = 5
So it is expected that 5 preschoolers would chose to play with the stuffed animals today.
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Answers
Answer:
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Step-by-step explanation:
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Answers
Answer: 2(7x/20 + 5y/4 + 3)
Enter an algebraic expression for the word expression.
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Answers
9514 1404 393
Answer:
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Step-by-step explanation:
The wording "the quotient of A and B" is used to signify A/B.
Here, we want the quotient of -5 and y:
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Answer:
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Step-by-step explanation:
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Step-by-step explanation:
Answer:
5/12
Step-by-step explanation:
We are going to change both number so they have the same denominators.
1/4x3/3= 3/12
2/3x4/4=8/12
Once we changed the number we are going to subtract the numbers.
8/12-3/12=5/12
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Step-by-step explanation:
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Step-by-step explanation:
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Answers
The procedure is attached.
Answer:
x=-2
Step-by-step explanation:
[tex]\sqrt{x+3}+4=5[/tex]
subtract 4 from each side
[tex]\sqrt{x+3} +4-4=5-4[/tex]
square both sides
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[tex]x+3=1[/tex]
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Answers
Missing information:
How fast is the temperature experienced by the particle changing in degrees Celsius per meter at the point
[tex]P = (\frac{1}{2}, \frac{\sqrt 3}{2})[/tex]
Answer:
[tex]Rate = 0.935042^\circ /cm[/tex]
Step-by-step explanation:
Given
[tex]P = (\frac{1}{2}, \frac{\sqrt 3}{2})[/tex]
[tex]T(x,y) =x\sin2y[/tex]
[tex]r = 1m[/tex]
[tex]v = 2m/s[/tex]
Express the given point P as a unit tangent vector:
[tex]P = (\frac{1}{2}, \frac{\sqrt 3}{2})[/tex]
[tex]u = \frac{\sqrt 3}{2}i - \frac{1}{2}j[/tex]
Next, find the gradient of P and T using: [tex]\triangle T = \nabla T * u[/tex]
Where
[tex]\nabla T|_{(\frac{1}{2}, \frac{\sqrt 3}{2})} = (sin \sqrt 3)i + (cos \sqrt 3)j[/tex]
So: the gradient becomes:
[tex]\triangle T = \nabla T * u[/tex]
[tex]\triangle T = [(sin \sqrt 3)i + (cos \sqrt 3)j] * [\frac{\sqrt 3}{2}i - \frac{1}{2}j][/tex]
By vector multiplication, we have:
[tex]\triangle T = (sin \sqrt 3)* \frac{\sqrt 3}{2} - (cos \sqrt 3) \frac{1}{2}[/tex]
[tex]\triangle T = 0.9870 * 0.8660 - (-0.1606 * 0.5)[/tex]
[tex]\triangle T = 0.9870 * 0.8660 +0.1606 * 0.5[/tex]
[tex]\triangle T = 0.935042[/tex]
Hence, the rate is:
[tex]Rate = \triangle T = 0.935042^\circ /cm[/tex]
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Answers
Answer:
there is no image
Step-by-step explanation:
solve for 26 - 4s = 9s.
Answers
26-4s=9s
⇔26=9s+4s
⇔26=13s
⇔s=2
Answer:
s = 2
Step-by-step explanation:
26 - 4s = 9s
add 4s to 4s and 9s
that equals 13
26/13 = 13s/13
s = 2
Expand the following expression 8(3x+10y)
Select one:
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24x+10y
8xy(13)
24x+80y
Answers
Answer:
8(3x+10y)
the answer is 24x+80y
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= 2 x 22/7 x 12
= 44 / 7 x 12
= 75.4285714286
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Answer:
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Step-by-step explanation:
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Answer:
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Area: 18u^2
Step-by-step explanation:
perimeter is just adding all the sides together so 6+3+6+3
area is multiplying length by width so 6•3
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Evaluate the expression where x=5 and y =2
Answers
Answer:
6
Step-by-step explanation: