AP CAL AB!!!! HELP. A ladder 10 feet long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall, and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when θ=π/3 ?
Answers
Answer:
[tex]\displaystyle \frac{dx}{d\theta}=5\text{ ft/rad}[/tex]
Step-by-step explanation:
Please refer to the diagram below.
We can use the sine function. Recall that the sine function is the ratio between the opposite side and the hypotenuse. Hence:
[tex]\displaystyle \sin(\theta)=\frac{x}{10}[/tex]
We want to find how fast x changes with respect to θ. So, we want to find dx/dθ. Therefore, let's take the derivative of both sides with respect to θ:
[tex]\displaystyle \frac{d}{d\theta}[\sin(\theta)]=\frac{d}{d\theta}\left[\frac{1}{10}x\right][/tex]
Differentiate. Since it's with respect to θ, we can differentiate the left-hand side like normal. On the right, we will implicitly differentiate. This yields:
[tex]\displaystyle \cos(\theta)=\frac{1}{10}\frac{dx}{d\theta}[/tex]
Multiply both sides by 10. So, dx/dθ is:
[tex]\displaystyle \frac{dx}{d\theta}=10\cos(\theta)[/tex]
We want to find dx/dθ when θ is π/3. Thus, substitute π/3 for θ:
[tex]\displaystyle \frac{dx}{d\theta}=10\cos\left(\frac{\pi}{3}\right)[/tex]
Evaluate:
[tex]\displaystyle \frac{dx}{d\theta}=10\left(\frac{1}{2}\right)=5\text{ ft/rad}[/tex]
So, x is changing with respect to θ at a rate of 5 feet per radian.
Related Questions
A car has a 16-gallon fuel tank. When driven on a highway, it has a gas mileage of 30 miles per gallon. The gas mileage (also called "fuel efficiency") tells us the number of miles the car can travel for a particular amount of fuel (one gallon of gasoline, in this case). After filling the gas tank, the driver got on a highway and drove for a while. How many miles has the car traveled if it has 10 gallons of gas left in the tank?
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Answer:
30
Step-by-step explanation:
A parent purchased a dictionary for $13.50 and a pack of printer paper for $4.50. The sales tax rate was 4%. What was the total bill, including the sales tax?
Answers
Answer:
C $18.72
Step-by-step explanation:
1. $13.50+$4.50= $18.00
2. $18.00* 4%=$0.72 sale tax
3. $18.00+$0.72= $18.72 your answer
Answer:
its c!
Step-by-step explanation:
$18.72
8 + 6 + 12 = __ + 12 + 6
Answers
Answer:
The number 8 goes in the blank,
Step-by-step explanation:
Its basic commutative property.
Hope this helps!
PLEASE HELP!!!!!!!!!!!!!!!
Answers
Select all ordered pairs that satisfy the function
(-3, -1)
(-2, -1)
0 (
33)
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Amber is traveling to he mom's house. The scale on the map is 2 inches=10 miles. She knows her mom lives 40 miles away. How far would that be on the map?
Answers
8 inches on the map. To approach this problem, just set up a proportion:
2in =10 mi x in =40 mi.
If we use that to set up an equation we get
80 =10x.
Solve that, you get 8.
Hope this helps, let me know if you need more help.
If 4 is added to double the number n, the result is the same as
if n were quadrupled. What is the value of n?
(A) 2
(B) 4
(C) 6
(D) 8
(E) 10
Answers
Answer:
A) 2
Step-by-step explanation:
For this problem, you can create an equation.
2n+4=4n
Then, divide by 2.
n+2=2n
Finally, subtract n from both sides.
2=n
What is the midpoint of (5,5) and (-3,-5)t
Answers
Answer: (1,0)
Step-by-step explanation:
Which property is illustrated by the following statement?
9a (bc) = (9a)bc
O A. Commutative Property of Addition
O B. Inverse Property of Multiplication
O C. Commutative Property of Multiplication
O D. Associative Property of Multiplication
Answers
Answer:
Associative Property
Step-by-step explanation:
The associative property states that it doesn't matter where the parenthesis are put, the answer will always be the same
What is the pythagorean thereom? And how is it used to solve triangles
Answers
Answer:
Pythagorean Thereom is the method in which finds the lengths of each side of a triangle. The formula is [tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex]
Each letter in the formula defines a certain side of a triangle.
The picture shown shows the sides of each triangle and how they correspond with the formula.
This helps solve triangles to find its sum, as well as, to figure out a side depending on what is given.
22x^3 y^4+28x^2 y^5+34xy^4
Answers
Answer:
2xy4 • (11x2 + 14xy + 17)
Step-by-step explanation:
(((22•(x3))•(y4))+((28•(x2))•(y5)))+(2•17xy4)
(((22•(x3))•(y4))+((22•7x2)•y5))+(2•17xy4)
(((2•11x3) • y4) + (22•7x2y5)) + (2•17xy4)
5.1 Pull out like factors :
22x3y4 + 28x2y5 + 34xy4 = 2xy4 • (11x2 + 14xy + 17)
Trying to factor a multi variable polynomial :
5.2 Factoring 11x2 + 14xy + 17
Final result :
2xy4 • (11x2 + 14xy + 17)
A store sells a shirt that is discounted by 10%. A coupon is used for an additional 10% off the discounted price.
What is the total percentage discount on the shirt? PLEASEEEE HELPPPP
Answers
Answer:20%
Step-by-step explanation: 10% because the shirt was already discounted and another 10% from the coupon
Gamers charges $8 to rent a video game. The Discount Club charges $4 to rent a video game after a membership fee of $10 is paid. For what number of video game rentals is Gamers less expensive than Discount Club? Write an inequality that represents this situation. (do not solve) How did you determine which inequality symbol to use for the situation?
Answers
Answer
8x>10+4x for x=>3
Just two games.
for the third game you would have paid $24 in total
Step-by-step explanation:
Given that
gamers charge $8
Discount Club charges $4 after a membership of $10
total for the Discount club is given by the expression
let x be the number of games rent
Y=10+4
So at the first rent, you will pay Discount Club
$14 and the second rent you would have spent
14+4= $18
and the third rent you would have spent
18+4= $22
the inequality is given as
for x=>3
8x>10+4x
EXTRA POINTS * how do you graph y=2x + 1
Answers
Step-by-step explanation:
first u need to pick a random coordinate to substitute the X . example 0
y = 2(0) + 1
y =1 , X= 0
X= 1
y= 2(1) + 1
y= 3, X= 1
then you continue this same process and again until u have at least two points to draw a straight line. it kinda depended if u want to fill the entire graph.
What kind of number does distance always represent?
Answers
Answer:
Step-by-step explanation:There are other kinds of numbers that can be graphed on the number line, too. ... which are natural numbers and which are whole numbers is to think of a “hole,” which can be represented by 0. ... Notice that distance is always positive or 0.
20x3² - 4 ) divided by 50
Evaluate the expression.
Answers
Answer:
2/5x^9−2/25
Step-by-step explanation:
I dont quite understand how to do this
Answers
Answer:
10
Step-by-step explanation:
substitute -3 in for x
f(-3)=(-3)²+1=
9+1=10
(You can ignore the g(x) part - it has nothing to do with this question.)
Pls help me with this question I give out 20 points
Answers
Answer:
55 dollars
Step-by-step explanation:
y = total cost, x = number of sessions attended
Non-members equation: y = 5x
Members equation: y = 3x + 25
y = 3(10) + 25
y = 30 + 25
y = 55 dollars
Hope this helps!
=55 dollars
Fill in the blanks 1. The degree of a linear equation is ? 2. The constant term has a degree ? 3. The solution of the equation ax + b = 0 is ? 4. If 2x - 3 = 0 then the value of x is ? 5. The value of x which satisfies the equation (- 8) = 2x + 4 is ? 6. The solution of the equation 10 - (x + 2) = 7x is ? 7. A linear equation in one variable has _________ solution ? 8. The sum of the two consecutive multiples of 10 is 210. The smallest multiple is 9. Convert the statement "adding 15 to 4 times x is 39" into an equation 10. Value of "S" in 1/3 + S = 2/5 is ?
Answers
Answer:
Step-by-step explanation:
A linear equation is of the form ax+b and the highest degree of x is 1. You can see that from the power of x which is 1.
The power of x is zero from the equating. It is not reflecting because anything raise to power if zero gives 1
To get the solution of the equation ax+b = 0
We will make x the subject of the formula
ax = -b
Divide both side's by a
ax/a = -b/a
x = -b/a
If 2x-3 = 0
2x = 0+3
2x = 3
2x/2 = 3/2
x = 3/2
Given -8 = 2x+4
2x = -8-4
2x = -12
x = -12/2
x = -6
A linear equation in one variable is known to have ONE solution. This is as demonstrated in the equation above, we cans see that the only solution is -6(just one solution)
Convert the statement "adding 15 to 4 times x is 39" into an equation 1
The product of 4 and x is 4x
Adding 15 to 4x will give
15+4x
If the resulting expression is equal to 39, the final equation will be;
15+4x = 39
10) given 1/3+S = 2/5, we are to find S
S = 2/5-1/3
S = 6-5/15
S = 1/15
Hence the value of S is 1/15
A truckload of plastic soda bottles was finely shredded at a recycling center. The plastic shreds were placed into 55-liter drums. What it the mass of the plastic shreds inside one of the drums in kg? Assume lowest density within the range.
Answers
Answer:
75,900 g
Step-by-step explanation:
m=d×v
(1.38g/cm^3) × (55,000)
The mass of the drum is 75.9 kg.
What is the mass of the drum?The mass of the drum is the product of the volume and density of the drum.
What are Arithmetic operations?Arithmetic operations can also be specified by the addition, subtract, divide, and multiply built-in functions.
+ Addition operation: Adds values on either side of the operator.
For example 4 + 2 = 6
- Subtraction operation: Subtracts the right-hand operand from the left-hand operand.
for example 4 -2 = 2
* Multiplication operation: Multiplies values on either side of the operator
For example 4*2 = 8
We have been given that the plastic shreds were placed into 55-liter drums.
We have to determine the mass of the plastic shreds inside one of the drums in kg.
Here density of the drum = 1.38g/cm³ and the volume of the drum = 55,000
The mass of the drum = (1.38g/cm³) × (55,000)
The mass of the drum = 75,900 g
The mass of the drum = 75.9 kg
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Write and solve an equation to find your average speed in an airplane
if you flew 800 miles in 2.5 hours.
Answers
Answer:
320 miles per hour
Step-by-step explanation:
Speed=800miles/2.5 hours
320milesperhour
A train travels 540 miles in 12 hours.
Complete the table to represent the situation.
Answers
2: 90
5: 225
10: 450
Stefan spends $6.50 on supplies for a lemonade stand and sells each cup of lemonade for $1.25. What is x, the number of cups of lemonade Stefan must sell to earn a profit of more than $50?
⊝
Stefan has a balance of $6.50 in his savings account and deposits $1.25 each week. What is x, the number of weeks Stefan must deposit $1.25 in order to have a balance of more than $50 in his savings account?
⊝
Stefan earns 1.25% interest on the balance in his checking account and has to pay a monthly charge of $6.50. What is x, the balance that Stefan must have in his checking account in order to have an ending balance greater than $50 after interest and fees?
⊝
Stefan charges $1.25 for gasoline plus $6.50 per hour for mowing lawns. What is x, the number of hours he has to mow lawns to earn more than $50?
Answers
Answer: 40
Step-by-step explanation:50/1.25
Find the rate. Principle= $50, Interest= $10, Time= 2½ years
Answers
Answer: It should be 50% or 0.5
Step-by-step explanation:
y=7 x=6 what is the value of y when x=12?
Answers
Answer:
Y = 14
Step-by-step explanation:
6 times 2 = 12
7 times 2 = 14
is the following graph a function?
A. yes
B. no
Answers
Answer:
Yes
Step-by-step explanation:
You can use the Vertical line test to figure this out, and sure enough none of the points touch the vertical line.
One angle of a triangle measures 40°. The other two angles are in a ratio of 5:9. What are the measures of those two angles?
Answers
Answer:
50 and 90
Step-by-step explanation:
(180-40)/(5 + 9) = 10
=> 50 and 90
help pleaseeeeeeeeeeeeee now
Answers
Answer: per 1 ticket sold it costs 10 dollars
Step-by-step explanation:
1 and ten meet on the graph constantly
I have a F in this class may someone help me..?
Answers
Answer:
x = -10
Step-by-step explanation:
9 x -10 = -90
9x=-90
-90/9= -10
Check:
-10 times 9= -90
A rotating lawn sprinkler sprays water in a circular area of grass, as shown in the picture. The diameter of the circular area of grass is 16 ft.Which measurement is closest to the area in square feet that this sprinkler sprays with water?
Answers
The measurement that is closest to the area in square feet that this sprinkler sprays with water is 201.062 ft².
Area of Circle[tex]\rm{ Area\ of\ circle = \pi \times (radius)^2[/tex]Given to us,Diameter of the circular area of grass, d = 16 ft.,
The radius of the circular area of grass, [tex]r = \dfrac{Diameter}{2} = \dfrac{16}{2} = 8\ ft[/tex],
Area of sprinkler spraysArea in which the sprinkler sprays with water = Area of the circle
[tex]= \pi \times r^2\\= \pi \times 8^2\\=\pi \times 64\\= 201.0619\approx 201.062\ ft^2[/tex]
Hence, the measurement that is closest to the area in square feet that this sprinkler sprays with water is 201.062 ft².
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