If x is increasing at a
rate of 2 units per second,
find the rate of change of theta
at the instant when x= 12 units.
Answers
Answer:
[tex]\frac{d\theta}{dt}=-\frac{2}{5}[/tex] at [tex]x=12[/tex]
Step-by-step explanation:
[tex]\frac{dx}{dt}=2[/tex]
[tex]\frac{d\theta}{dt}=?[/tex]
[tex]x=12[/tex]
[tex]cos(\theta)=\frac{adjacent}{hypotenuse}[/tex]
[tex]cos(\theta)=\frac{x}{13}[/tex]
[tex]\frac{d}{dt}cos(\theta)=\frac{d}{dt}\frac{x}{13}[/tex]
[tex]-sin(\theta)\frac{d\theta}{dt}=\frac{1}{13}\frac{dx}{dt}[/tex]
[tex]-sin(\theta)\frac{d\theta}{dt}=\frac{1}{13}(2)[/tex]
[tex]-sin(\theta)\frac{d\theta}{dt}=\frac{2}{13}[/tex]
[tex]\frac{d\theta}{dt}=\frac{2}{-13sin(\theta)}[/tex]
[tex]cos(\theta)=\frac{x}{13}[/tex]
[tex]cos(\theta)=\frac{12}{13}[/tex]
[tex]\theta=cos^{-1}(\frac{12}{13})[/tex]
[tex]\frac{d\theta}{dt}=\frac{2}{-13sin(\theta)}[/tex]
[tex]\frac{d\theta}{dt}=\frac{2}{-13sin(cos^{-1}(\frac{12}{13}))}[/tex]
[tex]\frac{d\theta}{dt}=\frac{2}{-13(\frac{5}{13})}[/tex]
[tex]\frac{d\theta}{dt}=\frac{2}{-5}[/tex]
[tex]\frac{d\theta}{dt}=-\frac{2}{5}[/tex]
The value of θ is given by the inverse sine function, from which the rate of
change of θ with respect to x can be derived.
The rate of change of θ at the instant when x = 12 units is -0.4 rad/sReasons:
[tex]\displaystyle \frac{dx}{dt} = \mathbf{2 \ units \ per \ second}[/tex]
[tex]\displaystyle cos(\theta) = \frac{x}{13}[/tex]
[tex]\displaystyle \theta = arccos \left(\frac{x}{13}\right)[/tex]
[tex]\displaystyle \frac{d}{dx} \theta = \frac{d\left(arccos \left(\frac{x}{13}\right)\right)}{dx} = \mathbf{\frac{\sqrt{169-x^2} }{x^2-169}}[/tex]
Using chain rule of differentiation, we have;
[tex]\displaystyle \frac{d\theta}{dt} = \mathbf{ \frac{d\theta}{dx} \times \frac{dx}{dt}}[/tex]
Therefore;
[tex]\displaystyle \frac{d\theta}{dt} =\frac{\sqrt{169-x^2} }{x^2-169}\times \frac{dx}{dt} = \mathbf{\frac{\sqrt{169-x^2} }{x^2-169}\times 2}[/tex]
When x = 12, we get;
[tex]\displaystyle \frac{d\theta}{dt} =\frac{\sqrt{169-12^2} }{12^2-169}\times 2 = -\frac{2}{5} = -0.4[/tex]
The rate of change of the angle, θ, with time at the instant when x = 12 is -0.4 rad/s
Learn more about rate of change here:
https://brainly.com/question/24686935
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Answers
Answer:
7.5 gallons
Step-by-step explanation:
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Answer:
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Step-by-step explanation:
-2 - 19 x -1
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Answer:
17 is the answer.
Step-by-step explanation:
Given:
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=> -2 - 19t
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Therefore, 17 is the answer.
Hoped this helped.
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Answers
Answer:
4 * 4 ÷ 4 = 4
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Step-by-step explanation:
[tex]4 * 4 \div4\\\rule{150}{0.5}\\\text{Use PEMDAS}\\\\4 * 4 \div4\\\\4 * 1\\\\\boxed{4}[/tex]
==============================
[tex]8 \div (2*2)\\\rule{150}{0.5}\\\text{Use PEMDAS}\\\\8 \div (2*2)\\\\8 \div 4\\\\\boxed{2}[/tex]
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Answers
Answer:
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Answers
Step-by-step explanation:
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Answers
Answer:
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Answer:
your answer will be 14/45
Step-by-step explanation:
2/5 × 7/9
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Answers
Check the picture below.
Answer:
ΔABEStep-by-step explanation:
In ΔBCD,
Two sides are equal, i.e,
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In ΔBDE,
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Answers
Step-by-step explanation:
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Answers
Answer:
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Step-by-step explanation:
To figure out the tip, you need to find 20% of $52.60.
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You should leave $10.52 for the waiter if you want to leave him exactly 20%.
To figure the tax, you need to find 8% of $52.60.
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Next, add them up:
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The total bill, including tax and tip, will be $67.33.
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Answer:
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Answers
Answer:
Step-by-step explanation:
Find the centre first by reoriganisation it into the form (x - cx)^2 + (y -cy)^2 = r^2
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Given :
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Equation of a Line -- 2x - y + a = 0
The following steps can be used in order to determine the value of 'a':
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[tex]x^2-2x+y^2-4y-4=0[/tex]
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[tex](x-1)^2+(y-2)^2 = 3^2[/tex]
Step 2 - From the above step, it can be concluded that the center of the circle is (1,2).
Step 3 - If (1,2) is the center of the circle then it also satisfies the equation of diameter (2x - y + a = 0).
2x - y + a = 0
2(1) - 2 + a = 0
a = 0
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https://brainly.com/question/23528207
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Step-by-step explanation:
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Answer:
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Step-by-step explanation:
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Answers
Answer:
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Step-by-step explanation:
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Answer:
50%
explanation
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Answers
Answer:
The graph is attached below.
Step-by-step explanation:
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Answers
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Substituting the given parameters into the formula, we have;
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Answers
Answer:
See belowStep-by-step explanation:
1. c increased by 5 is 5 ⇒ h. c + 5 = 5 2. 4 more than a number is 12 ⇒ b. n + 4 = 12 3. The product of 4 and n is 12 ⇒ i. 4n = 12 4. The difference of 27 and k is 15 ⇒ a. 27 - k = 155. 8 less than 3 times x is 10 ⇒ f. 3x - 8 = 10 6. 3 times x increased by 8 is 20 ⇒ d. 3x + 8 = 20 7. A number minus 8 is 3 ⇒ e. k - 8 = 3 8. Three times a number is 9 ⇒ j. 3n = 99. Twice a number plus 1 is 11 ⇒ g. 2x + 1 = 11 10. 6 less than a number is 5 ⇒ c. c - 6 = 15how many seconds are there in 3 minute By the process
Answers
Answer:
Step-by-step explanation:
You can easily convert 3 minutes into seconds using each unit definition:
Minutes
minute = 60 s = 60 s
Seconds
1 s
= 1 sec / 60 = 0.01667 min
With this information, you can calculate the quantity of seconds 3 minutes is equal to.
In 3 min there are 180 s.
Which is the same to say that 3 minutes is 180 seconds.
find the perimeter of the triangle A(4,-2) B(9,-2) C(9,10)
Answers
Answer:
30 units
Step-by-step explanation:
A graph can help you see the lengths of the sides. This is a right triangle, so the length of the hypotenuse is found using the Pythagorean theorem:
5² +12² = b² = 169
b = √169 = 13
The perimeter is 5+12+13 = 30 units.
Work out the area of the shaded shape.
Answers
Answer:
Step-by-step explanation:
A = 8(14) - 9(8 - 4) = 76 m²
(a) $21,000.00 is to be divided in the ratio 5:7:9 respectively for three siblings Antony, Terry
and Jack. Calculate how much money each sibling will get.
Answers
Answer:
See below for answer
Step-by-step explanation:
Set up the equation 5x+7x+9x=2100 and solve for x:
5x+7x+9x=2100
21x=2100
x=100
Now multiply the proportion by 100:
5:7:9 * 100 = 500:700:900
This means that Antony will get $500, Terry will get $700, and Jack will get $900.
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Answers
Answer:
Step-by-step explanation:
A translation 1 unit left and 2 units down :)
1 unit to the left and 2 units down.
