A right triangle has base x meters and height h meters, where h is constant and x changes with respect to time t, measured in seconds. The angle θ, measured in radians, is defined by tanθ=hx. Which of the following best describes the relationship between dθdt, the rate of change of θ with respect to time, and dxdt, the rate of change of x with respect to time?
Answers
The relationship between dθ/dt, the rate of change of θ with respect to time, and dx/dt, the rate of change of x with respect to time is \(\frac{d\theta}{dt} = \frac{h}{x\sqrt{x^2+y^2} }\frac{dx}{dt}\)
Given the following parameters:
Adjacent side = xOpposite side = hThe angle of elevation = θAccording to SOH CAH TOA identity;
\(tan \theta =\frac{Opposite}{Adjacent}\)
Substitute the given values into the formula
\(tan \theta =\frac{h}{x}\)
\(\theta= tan^{-1}\frac{h}{x}\)
Differentiate with respect to x
\(\frac{d \theta}{dx} = \frac{h}{x\sqrt{x^2+y^2} }\)
According to the chain rule,
\(\frac{d\theta}{dt} =\frac{d\theta}{dx} \times \frac{dx}{dt}\\\frac{d\theta}{dt} = \frac{h}{x\sqrt{x^2+y^2} }\frac{dx}{dt}\)
Hence the relationship between dθ/dt, the rate of change of θ with respect to time, and dx/dt, the rate of change of x with respect to time is \(\frac{d\theta}{dt} = \frac{h}{x\sqrt{x^2+y^2} }\frac{dx}{dt}\)
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Related Questions
a stainless steel patio heaters a square pyramid. The length of one side of the base is 24.6 in the slant height of the pyramid is 91.8 inches what is the height of the pyramid?
Answers
For a square pyramid, its height, slant height, and base form a right triangle:
Where
a is the length of one side of the base
r is the inner radius of the base (and is equal to half the length of a side of the base)
h is the height
s is the slant height
To determine the height (h) you can apply the Pythagorean theorem that states that the square of the hypothenuse is equal to the sum of the squares of the sides of the right triangle:
\(s^2=h^2+r^2\)Write the expression for the height:
\(\begin{gathered} h^2=s^2-r^2 \\ h=\sqrt[]{s^2-r^2} \end{gathered}\)For
s= 91.8inch
r=a/2=24.6/2=12.3in
Replace these values in the expression obtained for h
\(\begin{gathered} h=\sqrt[]{s^2-r^2} \\ h=\sqrt[]{(91.8^2)-(12.3^2)} \\ h=\sqrt[]{8275.95} \\ h=90.97 \end{gathered}\)The height of the pyramid is 90.97inches.
In the accompanying diagram, what is sin E?
Please see image below (math)
Answers
Answer:
\(\sin E=\dfrac{4}{5}\)
Step-by-step explanation:
To find the value of sin E we can use the sine trigonometric ratio.
\(\boxed{\begin{minipage}{9 cm}\underline{Sine trigonometric ratio} \\\\$\sf \sin(\theta)=\dfrac{O}{H}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}\)
From inspection of the given right triangle:
The angle is E, so θ = E.The side opposite angle E is FG, so O = 4.The hypotenuse of the triangle is EF, so H = 5.Substitute these values into the sine ratio:
\(\sin E=\dfrac{4}{5}\)
Together, Sierra and Gavin earned $5,404. Sierra earned four-sevenths
of the money
Answers
5/7 of $5,404 is $3088.
Sierra earned $3088 and Gavin earned $2316
If the perimeter of a rectangle picture frame must be less than 200 in and The width is 36 in what Must the height H of the frame be
Answers
Answer:
So since the formula for perimeter is 2(w+h)
200 = 2(w+h)
that means that
100 = w+h
100 = 36+x
x=
64 UNITS
Step-by-step explanation:
John flew a kite. He started flying the cat at 2:20 p.m. and ended at 3:14 p.m. How many minutes did John fly a kite?
A. 47 minutes
B. 50 minutes
C. 54 minutes
D. 55 minutes
Answers
Answer:
54 minutes
Step-by-step explanation:
3:14 = 3 × 60 = 180 + 14
= 194
2:20 = 2 × 60 = 120 + 20
= 140
194 - 140
= 54
PLS HELP I WILL GIVE U BRAILIEST I need help
Copy and convert into numbers.
1.) 2.75 x 10-2 =
2.) 39,000,000,000 =
3.) .000002 =
4.) 7.368 x 104 =
Answers
Answer:
1)0.0275
2)3.9×10^10
3)2×10^-6
4)did u mean 10^4?
if so then it would be 73,680
explanation
1)2.75×10^-2
2.75×.01=0.0275
2)39,000,000,000.00
in scientific notations the first number has to be lower than 10 and higher than 1>
3.9 works so i would take the dot/period and bring it between 3 and 9, the dot would travel 10 places therefore you raise the 10 to the 10th power
3.9×10^10
4) and since 10 is raised to the 4th power you would just move the dot 4 places right and you get
7.368>73,680
The attendance over a weekly period of time at a movie theater is normally distributed with a mean of 10,000 and a standard deviation of 1000 persons. Find the percent of attendance figures that differs from the mean by 1500 persons or more.
Answers
The percent of attendance figures that differs from the mean by 1500 persons or more is 6.68%.
From the question above, Mean μ = 10,000
Standard Deviation σ = 1,000
The formula for z-score is :
z = (x-μ) / σ
Where, x = observation
z = z-score
Mean μ = 10,000
Standard Deviation σ = 1,000
From the above formula, let's calculate z-score for x = 11,500
z = (x-μ) / σ
z = (11,500 - 10,000) / 1000
z = 1.5
Now, find the probability of attendance figures that differs from the mean by 1500 persons or more.
P(z ≥ 1.5) = 0.0668
To find the percentage, we need to multiply the above value by 100.
P(z ≥ 1.5) × 100 = 0.0668 × 100 = 6.68%
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Consider a line process with 3 processing stages. The production requires each unit to go through Stage A through Stage C in sequence. The characteristics of the Stages are given below: Stage A B C Unit processing time(minutes) 1 2 3 Number of machines 1 1 2 Machine availability 90% 100% 100% Process yield at stage 100% 100% 100% Determine the system capacity. Which stage is the bottleneck? What is the utilization of Stage 3.
Answers
The system capacity is 2 units per minute, the bottleneck stage is Stage A, and the utilization of Stage 3 is 100%.
A line process has three processing stages with the characteristics given below:
Stage A B C Unit processing time(minutes) 1 2 3 Number of machines 1 1 2 Machine availability 90% 100% 100% Process yield at stage 100% 100% 100%
To determine the system capacity and the bottleneck stage and utilization of Stage 3:
The system capacity is calculated by the product of the processing capacity of each stage:
1 x 1 x 2 = 2 units per minute
The bottleneck stage is the stage with the lowest capacity and it is Stage A. Therefore, Stage A has the lowest capacity and determines the system capacity.The utilization of Stage 3 can be calculated as the processing time per unit divided by the available time per unit:
Process time per unit = 1 + 2 + 3 = 6 minutes per unit
Available time per unit = 90% x 100% x 100% = 0.9 x 1 x 1 = 0.9 minutes per unit
The utilization of Stage 3 is, therefore, (6/0.9) x 100% = 666.67%.
However, utilization cannot be greater than 100%, so the actual utilization of Stage 3 is 100%.
Hence, the system capacity is 2 units per minute, the bottleneck stage is Stage A, and the utilization of Stage 3 is 100%.
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What is the truth value for the following conditional statement?
p: true
q: true
∼p → q
A] F T → T
b} T T → F
c} F T → F
D} T F → T
Answers
Answer:
i think the answer is B??
Find the value of z.
Answers
2z = 15
z = 7.5
algebra 2-2
solve \(\sqrt{x}+3=-6\)
√x+3=-6
Answers
The solution of equation is x=-3
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
Given
The equation;√x+3=-6
Now,
√x=-6-3
√x=-9
x=-3
Therefore the answer of the given equation will be -3
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simplifique o seguinte produto notável: (2x-6)²
Answers
Answer:
Multiply first and add the negative sign and multiply all of it 2 times
Step-by-step explanation:
Answer:
4x +36
Step-by-step explanation:
= (2x - 6) * (2x - 6)
= 4x -12x -12x +36
= 4x +36
suppose your score on the gre (graduate records exam) was at the 90th percentile. what does that mean?
Answers
If your score on the GRE (Graduate Records Exam) is at the 90th percentile, it means that you have performed better than or equal to 90% of the test takers who took the exam. In other words, your score is higher than or equal to the scores of 90% of the individuals who participated in the test.
Being at the 90th percentile indicates that you have achieved a relatively high score compared to the majority of test takers. It demonstrates that you have performed well and are among the top performers on the GRE. This percentile rank is often used to compare and assess individuals' performance in standardized tests, helping to provide a reference point for evaluating their relative standing in the test-taking population.
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HELLPPPPPP please it’s due today
Answers
Answer:
i think its b but im not sure
Step-by-step explanation:
If a+b+c = 9 and ab+bc+ca = 40, find a^2 + b^2 + c^2
Answers
Explanation
By definition.
\(\begin{gathered} (a+b+c)^2=a^2+b^2+c^2+2((ab)+(bc)+(ac)) \\ \end{gathered}\)so
Let
\(\begin{gathered} (a+b+c)=9 \\ ab+bc+ac=40 \\ \text{now, replace} \end{gathered}\)\(\begin{gathered} (a+b+c)^2=a^2+b^2+c^2+2((ab)+(bc)+(ac)) \\ 9^2=a^2+b^2+c^2+2(40) \\ 81=a^2+b^2+c^2+80 \\ \text{subtract 80 in both sides} \\ 81-80=a^2+b^2+c^2+80-80 \\ 1=a^2+b^2+c^2 \end{gathered}\)hence
\(a^2+b^2+c^2=1\)I hope this helps you
Chi-square distributions that are positively skewed have a research hypothesis that is?
Answers
Answer:
A one tailed test
Step-by-step explanation:
Chi-Square Distributions That Are Positively Skewed Have A Research Hypothesis That Is A One-Tailed Test.
Chi-Square distributions are positively skewed, with the degree of skew decreasing with increasing degrees of freedom.
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One-Tailed Test
In probability theory and statistics, the chi-square distribution with k degrees of freedom is the distribution of the sum of squares of k independent standard normal random variables. The chi-square distribution is a continuous probability distribution. The shape of the chi-square distribution depends on its degrees of freedom k. It is used to describe the distribution of the sum of squares of random variables. The chi-square distribution is positively skewed, with decreasing skewness as the degrees of freedom increase. The chi-squre distribution approaches the normal distribution on increase of degree of freedom.Chi-Square distributions that are positively skewed have a research hypothesis that is a One-Tailed Test.
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Given the definition of a sequence fn whose domain is the set of all non-negative integers:
f0 = 4
fn = fn - 1 + 5n for n≥1
Find f2 and f3 .
Answers
f2 = 19 and f3 = 34 for the given sequence. the definition of a sequence fn whose domain is the set of all non-negative integers:
To find the values of f2 and f3 for the given sequence, we can use the recursive definition of the sequence.
Given:
f0 = 4
fn = fn-1 + 5n for n ≥ 1
To find f2, we substitute n = 2 into the recursive formula:
f2 = f2-1 + 5(2)
= f1 + 10
Next, we find f1 by substituting n = 1 into the recursive formula:
f1 = f1-1 + 5(1)
= f0 + 5
Now, we can substitute the value of f1 into the expression for f2:
f2 = (f0 + 5) + 10
To find f3, we repeat the process:
f3 = f3-1 + 5(3)
= f2 + 15
Substituting the value of f2 into the expression for f3:
f3 = (f0 + 5 + 10) + 15
Let's evaluate the expressions to find the values of f2 and f3:
f1 = f0 + 5 = 4 + 5 = 9
Substituting f1 into the expression for f2:
f2 = (9 + 10) = 19
Substituting f2 into the expression for f3:
f3 = (19 + 15) = 34
Therefore, f2 = 19 and f3 = 34 for the given sequence.
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1. If a square-thread screw having a major diameter of 19 mm and six threads per 25. 4mm is used to lift a load of 17. 80 KN, compute the efficiency of the power screw in raising the load. Use a coefficient of friction of 0. 20
Answers
The efficiency of the power screw in raising the load is approximately 6976%.
The efficiency of a power screw can be calculated using the formula: Efficiency = (Ideal Mechanical Advantage / Actual Mechanical Advantage) x 100% To find the Ideal Mechanical Advantage (IMA) of the screw, we can use the formula:
IMA = (π / tan(α)) x (π / 2) x (D / P)
Where: π is the mathematical constant pi (approximately 3.14159) tan(α) is the coefficient of friction (given as 0.20) D is the major diameter of the screw (given as 19 mm) P is the pitch of the screw, which can be calculated as (25.4 mm / number of threads per inch) First, let's calculate the pitch: Number of threads per inch = 6
Pitch = 25.4 mm / 6 = 4.23 mm (approximately)
Now, we can substitute the values into the formula to find IMA:
IMA = (π / tan(α)) x (π / 2) x (19 mm / 4.23 mm)
Using the given coefficient of friction, we have:
IMA = (π / 0.20) x (π / 2) x (19 mm / 4.23 mm)
Simplifying the equation, we get:
IMA = 9.87 x 1.57 x 4.49
Calculating IMA, we find:
IMA ≈ 69.76
Now, we need to find the Actual Mechanical Advantage (AMA), which can be calculated using the formula:
AMA = Load / Effort
Where:
Load is the weight being lifted (given as 17.80 kN)
Effort is the force applied to turn the screw
Since the screw is being used to lift the load, the Effort is equal to the load.
AMA = 17.80 kin / 17.80 kin
AMA = 1
Now, we can calculate the efficiency using the formula:
Efficiency = (IMA / AMA) x 100%
Efficiency = (69.76 / 1) x 100%
Efficiency ≈ 6976%
The efficiency of the power screw in raising the load is approximately 6976%.
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The oven that is used To make the pizza is majors temperature in Fahrenheit the baking temperature for the pizza is that buy a new employer to 310° F is this temperature correct if he had been instructed to set the temperature is 180°C use the formula °C=(°F-32°)x1,8]
Answers
Answer:
°F = 356
Step-by-step explanation:
°C = (°F - 32) ÷ 1.8
180 = (°F - 32) ÷ 1.8
180×1.8 = °F - 32
324 = °F - 32
°F = 324 + 32
°F = 356
Kayla drives 15 miles in 20 minutes. If she drove one hour in total at the same rate,
how far did she go?
Answers
Answer:
45 miles
Step-by-step explanation:
first divide miles/minutes: 15 / 20 = 0.75
then multiply: 0.75 x 60 = 45
Describe a pattern in the sequence of numbers. Predict the next number.
1.) 256, 64, 16, 4, ... ________.
2.) 2, 6, 18, 54, ... _______
Answers
\({ \qquad\qquad\huge\underline{{\sf Answer}}} \)
In the first sequence,
let's take ratio of the next term with its preceding term, so we will get the common ratio ~
\(\qquad \sf \dashrightarrow \: \cfrac{4}{16} = \dfrac{1}{4} \)
So, we can imply that The next term of the sequence is 1/4 times the previous term.
hence, the unknown term will be 1/4 times of previous term.
That is : 1/4 × 4 = 1Therefore, the next term here is 1
In the second sequence,
Do the same procedure as above, we will get common ratio as :
\(\qquad \sf \dashrightarrow \: \cfrac{18}{6} = 3\)
So, the next term is 3 times the preceding term, that is :
3 × 54 = 162Therefore, the next term is 162
pattern: divide by 4
next term: 1
2.
pattern: multiply by 3
next term: 162
Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of the given equation. (4,-3); y=-5x+8
Answers
Answer:
Step-by-step explanation:
The slope of the line is m = -5
4 is your x1
-3 is your y1
Plug into the equation y - y1 = m(x - x1)
y + 3 = -5(x - 4)
y + 3 = -5x + 20
-3 -3
y = -5x + 17
Beatrice makes a jar of dry soup mix. The ingredients are:
1 3/4 = 7/4cups uncooked rice
1/2 cup dry lentils
1 2/3 = 5/3 cups uncooked pasta
1/3 cup beef bouillon
1/4 cup dried onion flakes
1/2cup dried peas
How much soup mix does this recipe make?
Answer the questions to find out.
The sum 7/4+ 1/2+5/3 +1/3 +1/4 +1/2 gives the amount of mix.
1. Which property could you use to change the grouping of the addends?
Answers
The soup mix that this recipe makes is 5.
The property could you use to change the grouping of the addends is the associative property of addition.
How to calculate the value of the mix?From the information, the following can be deduced:
7/4cups uncooked rice
1/2 cup dry lentils
1 2/3 = 5/3 cups uncooked pasta
1/3 cup beef bouillon
1/4 cup dried onion flakes
1/2cup dried peas
The soup mix that this recipe make will be the addition of all the values. This will be:
= 7/4+ 1/2+5/3 +1/3 +1/4 +1/2
= 5
The associative property of addition can change the grouping.
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Taylor bought a watch onsale for 60% off the originalprice, and another 10% off thediscounted price. If the watchoriginally cost $82, what wasthe final sale price?
Answers
Original Cost - $82
First part: Apply the 60% off discount
Convert first the 60% into decimal form
60% ÷ 100% = 0.6
Multiply it to the original cost to determine the discount
$82 ˣ 0.6 = $49.2
Subtract the discount to the original price, to determine the discounted price.
$82 - $49.2 = $32.8
Second part:
The discounted price is now $32.8 for which we will apply another 10% discount.
Again, convert 10% into decimal
10% ÷ 100% = 0.1
Multiply it to the discounted price, to determine the second discount.
$32.8 ˣ 0.1 = $3.28
Subtract the discount to the already discounted price.
$32.8 - $3.28 = $29.52
Therefore, the final sale price of the watch is at $29.52.
In the parallelogram shown in the picture solve for Z
Answers
In a parallelogram, opposite angles are congruent.
Since angles z and 124° are opposite angles in the parallelogram, they are congruent angles.
Therefore we have:
\(z=124°\)Hey y'all, do you think you can answer this question? (oh, and P.S. sorry about the already filled in parts. Ignore them)
Answers
The surface area is increased by the square of the factor used to increase both radius and height of the cylinder.
Factor of 2: Area factor of 4, Factor of 3: Area factor of 9, Factor of 5: Area factor of 25, Factor of 10: Area factor of 100, Factor of 20: Area factor of 400
How to analyze the change in surface area of a cylinder by changing radius and height by same factorIn this problem we must analyze the change in the surface area of the cylinder, when radius and height are changed by same factor. The situation is described by following formula:
A = 2π · (k · r)² + 2π · (k · r) · (k · h)
A = k² · (2π · r² + 2π · r · h)
A = k² · A'
Where:
k - Enlargement factorr - Radiush - HeightA' - Initial areaThus, the resulting area factor is:
Factor of 2: Area factor of 4
Factor of 3: Area factor of 9
Factor of 5: Area factor of 25
Factor of 10: Area factor of 100
When both radius and height of the cylinder are multiplied by same factor, then the surface area is increased by the square of the former factor.
Factor of 20: Area factor of 400
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create a graphic organizer under the topic of differential equations. the graphic organizer will show important relationships among the subtopics you learned about in this unit. you will submit this as a portfolio item at the end of this lesson. all the following key terms, along with a description of each topic, should be included in your graphic organizer. differential equation initial value problem general solution exact differential equation slope field indefinite integrals substitution methods for integration separable differential equations select the link to review the graphic organizer portfolio rubric to understand how you will be graded.
Answers
Here is a possible graphic organizer on the topic of differential equations.The following descriptions correspond to each subtopic included in the graphic organizer.
Differential Equation: an equation that involves an unknown function and its derivatives or differentials.Initial Value Problem: a type of differential equation that requires the solution to satisfy a given condition at a specific point.General Solution: a solution that contains arbitrary constants and includes all possible solutions of a differential equation.Exact Differential Equation.
a differential equation that can be solved by using an integrating factor that makes the left-hand side of the equation exact.
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What does 5 less than N mean?.
Answers
5 less than N means we subtract 5 to the given variable N. So the expression will be N - 5.
What is Algebra ?Algebra shows the numbers by alphabets in a equation including mathematical operations. In algebra we used alphabet which shows a variable unknown quantity, unknown variables which helps in changing a mathematical statement into an equation.
In the given question a mathematical statement is given and we have to change the mathematical equation into an algebraic expression by using the information.
What does 5 less than N means ?The statement 5 less than N means 5 is being subtracted from N. The mathematical expression for 5 less than N is equal to N - 5.
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price of two games in ratio a:b. If both prices decreased by £10, ratio = 3:5. If both prices increase by £5, ratio = 3:4. Work out ratio a:b.
Answers
The ratio of a: b is = 5:7
The price of the two games are in the ratio a:bWhen the prices are both decreased by £10, the ratio becomes 3:5When the prices are both increased by £5, the ratio becomes 3:4
Here the ratio is a:b
When the prices are both decreased by £10, the new ratio
= a-10 : b-10
When the prices are both increased by £5, the new ratio
= a+5 : b+5
By the given conditions,
a-10 : b-10 = 3:5
or, \(\frac{a-10}{b-10\\}\) =\(\frac{3}{5}\)
or, 5(a-10) = 3(b-10)
or, 5a -50 = 3b-30
or, 5a -3b = 20 .... ...(1)
and a+5 : b+5 =3 : 4
or,\(\frac{a+5}{b+5}\) = \(\frac{3}{4}\)
or, 4(a+5) = 3(b+5)
or, 4a +20 = 3b+15
or, 4a-3b= -5 .... ...(2)
subtract the equation(2) from equation(1)
5a -3b - (4a -3b) =20 - (-5)
or, 5a-3b-4a+3b = 20+5
or, 5a-4a = 25
Therefore, a =25
Putting a=25 in equation (1), we get
125-3b=20
or, 3b= 105
therefore, b=35
a :b =25:35
= 5:7
Hence, the ratio of a: b = 5:7
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Stereo Inc. sells a stereo system for $400 down and monthly payments of $90 for the next 4 years. If the interest rate is 2.75% per month, find:
a) The cost of the stereo.
Answer = $
b) The total amount of interest paid.
Answer = $
Answers
a) The cost of the stereo system is $4,760.
b) The total amount of interest paid is $1,760.
To find the cost of the stereo system, we need to calculate the sum of the down payment and the total of monthly payments over 4 years. The down payment is $400, and the monthly payment is $90 for 48 months (4 years). Thus, the total cost of the stereo system is $400 + ($90 × 48) = $4,760.
To calculate the total amount of interest paid, we need to subtract the initial principal amount (down payment) from the total cost of the stereo system. The initial principal amount is $400, and the total cost is $4,760. Therefore, the total interest paid is $4,760 - $400 = $1,760.
In summary, the cost of the stereo system is $4,760, and the total amount of interest paid is $1,760.
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