Answer:
[tex]\displaystyle \frac{dy}{dx} = \frac{-\pi}{4} \sin \bigg( \frac{\pi t}{4} \bigg)[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = \cos \bigg( \frac{\pi t}{4} \bigg)[/tex]
Step 2: Differentiate
Trigonometric Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle y' = -\sin \bigg( \frac{\pi t}{4} \bigg) \cdot \frac{d}{dt} \bigg[ \frac{\pi t}{4} \bigg][/tex]Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle y' = -\sin \bigg( \frac{\pi t}{4} \bigg) \cdot \frac{\pi}{4} \frac{d}{dt}[t][/tex]Basic Power Rule: [tex]\displaystyle y' = \frac{-\pi}{4}\sin \bigg( \frac{\pi t}{4} \bigg)[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Manny has 48 feet of wood. He wants to use all of it to create a border around a garden. The equation 2l + 2w = 48 can be used to find the length and width of the garden, where l is the length and w is the width of the garden. If Manny makes the garden 15 feet long, how wide should the garden be? 9 feet 18 feet 30 feet 33 feet
Answer:
You can find a width of 9 feet.
Step-by-step explanation:
The width of the rectangular shaped garden should be 9 feet
What is the Perimeter of a Rectangle?The perimeter P of a rectangle is given by the formula, P=2 ( L + W) , where L is the length and W is the width of the rectangle.
Perimeter P of rectangle = 2 ( Length + Width )
Given data ,
Let the perimeter of the rectangle be P = 48 feet
We can use the equation given to solve for the width w of the garden:
2l + 2w = 48
Substituting l = 15, we get:
2(15) + 2w = 48
30 + 2w = 48
Subtracting 30 from both sides, we get:
2w = 18
Dividing both sides by 2, we get:
w = 9 feet
Hence , the width of the garden should be 9 feet
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A business purchased for $650,000 in 1994 is sold in 1997 for $850,000. What is the annual rate of return for this investment?
solve using elimination 5x-9y=-43 3x+8y=68
Find the greatest common factor of the following monomial.
37x^3y^6 x^6y^2
Jorge's hourly salary is $7.65. last week he worked 23 hour week how much did he earn
what is the coefficient of the term 13a in the expression 13a+6b
Answer:
The coefficient of the term 13a is, 13
Step-by-step explanation:
Coefficient term are the numbers that multiply the variables or letters
In the given expression: [tex]13a+6b[/tex]
13a means 13 times a , and a is the variable,
so 13 is the coefficient .
6b means 6 times b , and b is the variable ,
so 6 is the coefficient .
therefore, the coefficient of the term 13a in the given expression is, 13.
What is 3/4 of 16 in fractions?
how many 6 1/4 -pieces of fabric can be cut from a 75-inch roll
A distribution of numbers has the following five-number summary:
10.0, 15.0, 30.9, 50.0, 63.7
True or False? These numbers can be used to calculate the standard deviation of the distribution. ...?
Answer:
False
Step-by-step explanation:
A five number summary consists of these five statistics: the minimum value, the first quartile, the median, the third quartile, and the maximum value of a set of numbers
A standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
Std deviation is the square root of variance
Variance is the average of squares of deviations of each entry from the mean.
Hence from five point summary, we cannot calculated standard deviation of the distribution
4 ALGEBRA 1B QUESTIONS - 70PTS
HOW MANY SOLUTIONS DOES THE SYSTEM OF EQUATIONS HAVE?
1. 3x = -12y + 15 and x+4y=5
2. y= 4x+ 3 and 2y - 8x = 3
3.x + 4y =12 and 5x - 20y =60
4. y -5x = -6 and 3y - 15x = -12
1. Infinite many solution.
2. No solution
3. One solution
4. No solution
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given:
Depending upon the situation we have three case
[tex]a_1/ a_2 = b_1/ b_2 = c_1 / c_2[/tex] then line is Coincident and have infinite many solution.
and, [tex]a_1/ a_2 \neq b_1/ b_2[/tex] then line is Intersecting and have one solution.
and, [tex]a_1/a_2 = b_1/ b_2 \neq c_1/ c_2[/tex] then line is Parallel and have No solution.
1. 3x = -12y + 15 and x+4y=5
3x + 12y - 15= 0 and x+ 4y- 5 = 0
So, [tex]a_1/ a_2 = b_1/ b_2 = c_1 / c_2[/tex] then line is Coincident and have infinite many solution.
2. 4x - y +3 = 0 and -8x + 2y -3 = 0
So, [tex]a_1/a_2 = b_1/ b_2 \neq c_1/ c_2[/tex] then line is Parallel and have No solution.
3. x+ 4y - 12= 0 and 5x - 20y -60 = 0
So, [tex]a_1/ a_2 \neq b_1/ b_2[/tex] then line is Intersecting and have one solution.
4. -5x + y +6 = 0 and -15x + 3y +12 = 0
So, [tex]a_1/a_2 = b_1/ b_2 \neq c_1/ c_2[/tex] then line is Parallel and have No solution.
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Prove:
lim x^3 = 8.
x approaches 2
What is the relationship between the two quantities?
The age of a car and the value of the car.
a. Positive Correlation
b. Negative Correlation
c. No Correlation ...?
Hannah brought 3 / 4 yards of felt for a project. she use 1/8 yard.what fraction of a yard of fill did she have left over?
Find tan θ if sec θ = square root of thirty seven divided by six and sin θ < 0.
tan(θ) = -1/6.
Given that sec(θ) = √37/6 and sin(θ) < 0, we can find tan(θ) using the trigonometric identity:
sec²(θ) - tan²(θ) = 1
First, we need to find the value of tan(θ) using the given information:
Since sec(θ) = √37/6, we know that sec(θ) = 1/cos(θ), so cos(θ) = 6/√37.
Since sin²(θ) + cos²(θ) = 1, we can find sin(θ):
sin²(θ) + (6/√37)² = 1
sin²(θ) + 36/37 = 1
sin²(θ) = 1 - 36/37
sin²(θ) = (37-36)/37
sin²(θ) = 1/37
Given that sin(θ) < 0, sin(θ) is negative. Therefore, sin(θ) = -1/√37.
Now that we have sin(θ) and cos(θ), we can find tan(θ):
tan(θ) = sin(θ)/cos(θ)
tan(θ) = (-1/√37) / (6/√37)
tan(θ) = -1/6
Therefore, tan(θ) = -1/6.
Find the solution set for y = 2x + 1, given the replacement set {(-2, -3), (-1, -1), (0, 2), (1, 3), (2, 6)} ...?
Angle θ lies in the second quadrant, and sin θ = 3/5.
cos θ =
tan θ =
Answer:
[tex]cos\Theta =\frac{4}{5}[/tex]
[tex]tan\Theta =\frac{3}{4}[/tex]
Step-by-step explanation:
Given : sin θ = 3/5
To Find : value of cos θ and tan θ
Solution :
use the identity:
[tex]sin^{2}\Theta +cos^{2}\Theta =1[/tex]
putting value of sin θ
⇒ [tex](\frac{3}{5})^{2} + cos^{2}\Theta =1[/tex]
⇒[tex]\frac{9}{25} +cos^{2}\Theta =1[/tex]
⇒[tex]cos^{2}\Theta = 1-\frac{9}{25}[/tex]
⇒[tex]cos^{2}\Theta = \frac{16}{25}[/tex]
⇒[tex]cos\Theta = \sqrt{\frac{16}{25}}[/tex]
⇒[tex]cos\Theta =\frac{4}{5}[/tex]
Thus , [tex]cos\Theta =\frac{4}{5}[/tex]
Now to find value of tan θ
Since we know that
⇒ [tex]tan\Theta =\frac{sin\Theta }{cos\Theta }[/tex] (identity)
⇒[tex]tan\Theta =\frac{\frac{3}{5} }{\frac{4}{5} }[/tex]
⇒[tex]tan\Theta =\frac{3}{5}\div \frac{4}{5}[/tex]
⇒[tex]tan\Theta =\frac{3}{5}\times \frac{5}{4}[/tex]
⇒[tex]tan\Theta =\frac{3}{4}[/tex]
Thus , the value of
[tex]tan\Theta =\frac{3}{4}[/tex]
[tex]cos\Theta =\frac{4}{5}[/tex]
Final answer:
This detailed answer provides the values of cos θ and tan θ for an angle θ in the second quadrant given sin θ = 3/5. It explains the process using the Pythagorean identity and the characteristics of the second quadrant.
Explanation:
cos θ = -4/5
tan θ = -3/4
Given sin θ = 3/5 and the angle θ lies in the second quadrant, we can determine cos θ using the Pythagorean identity. Since sin θ = 3/5 is positive in the second quadrant, the x-coordinate in the triangle would be negative. Therefore, cos θ = -4/5. Similarly, tan θ can be calculated as tan θ = sin θ / cos θ = (3/5) / (-4/5) = -3/4.
Each triangle in the diagram is a reflection of another triangle across one of the given lines. How can you describe Triangle 3 by using a reflection rule?
A. Rm
(Triangle 2)
B. Rm
(Triangle 1)
C. Rl
(Triangle 1)
D. Rl
(Triangle 2)
Answer:
D
Step-by-step explanation:
its reflected off the L-axis
For the cost function, find the marginal cost at the given production level x. State the units of measurement. (All costs are in dollars.)
C(x) = 15,000 + 70x + 1,000/x ; x = 100
To find the marginal cost at the given production level x, differentiate the cost function C(x) and substitute the value of x. The units of measurement for the marginal cost are dollars per unit (x).
Explanation:To find the marginal cost at the given production level x, we need to find the derivative of the cost function C(x) with respect to x. Let's differentiate the function:
C'(x) = 70 - 1,000/x^2
Now, substitute x = 100 into the derivative:
C'(100) = 70 - 1,000/100^2 = 70 - 1 = 69
The units of measurement for the marginal cost are dollars per unit (x).
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Four runners, Fran, Gloria, Haley, and Imani, compete on a relay team. Haley is the first runner in the relay. The other runners can run in any order. What is the sample space showing the possible orders of the other three runners?
Answer: The sample space showing the possible orders of the other three runners is
[tex]S=\{HFGI,HGFI,HIGF,HFIG,HGIF,HIFG\}[/tex]
Step-by-step explanation:
Since we have given that
Number of runners = 4
Four runners are as follows:
Fran, Gloria, Haley, and Imani.
Since Haley is the first runner in the relay,
So, Sample space showing the possible order of the other three runners.
[tex]S=\{HFGI,HGFI,HIGF,HFIG,HGIF,HIFG\}[/tex]
Hence, the sample space showing the possible orders of the other three runners is
[tex]S=\{HFGI,HGFI,HIGF,HFIG,HGIF,HIFG\}[/tex]
What is the coefficient of the term 53xy ? A. 5 B.1/3 C.5xy D.5/3
Answer:
the answer is D, TRUST ME
Step-by-step explanation:
The coefficient of the term (5/3) x y is 5/3.
What is an algebraic expression?An algebraic expression is consists of variables, numbers with various mathematical operations,
The given expression is,
(5/3) x y
The coefficient of any expression is the numerical term,
Numerical term in the expression is 5/3.
So, the coefficient of the term (5/3) x y is 5/3.
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A map has a scale of 1 cm:12km the distance between the two cities on the map is 6.8cm estimate the actual distance between the cities ...?
Final answer:
The actual distance between two cities on a map with a scale of 1 cm:12 km, where the map distance is 6.8 cm, can be calculated as 6.8 cm × 12 km/cm, resulting in an actual distance of 81.6 kilometers.
Explanation:
Understanding Map Scale Problems
To calculate the actual distance between two cities on a map using the scale given, you would use the following steps:
First, identify the scale of the map provided; in this case, it is 1 cm : 12 km.
Next, measure the distance on the map between the two cities; the distance is 6.8 cm.
Finally, multiply the distance on the map by the scale factor to find the actual distance. That is, 6.8 cm (distance on the map) × 12 km/cm (scale factor) = 81.6 km (actual distance).
Therefore, the estimated actual distance between the two cities is 81.6 kilometers.
If a rectangle has a perimeter of 70, a length of x and a width of x - 9, find the value of the length of the rectangle.
Which property states that adding zero to a quantity does not change the quantity?
A. Associative
B. Additive identity
C. Commutative
D. Reflexive ...?
One pipe fills a storage pool in 3 hours. A second pipe fills the same pool in 3 hours. When a third pipe is added and all three are used to fill the pool, it takes only 1 hour. Find how long it takes the third pipe to do the job alone.
Simplify completely x^2 + 12x +35
________________
3x + 15 ...?
Answer: x + 7/ 3
Assuming the said quantity is divided by 3x+15 as a denominator in fraction form.
After dividing, the simplest form, x + 7/ 3, can be found after dividing both the numerators and the denominators by the smallest common factor.
Proof of validity is shown below.
adam scored x points. imran scored 3 points fewer than adam . shakeel scored twice as many points as imran. together they scored 39 points..... Write down in terms of x an expression for the number of points scored by shakeel.
The value of x is 12 and Shakeel got 18 points.
What is an equation?An equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Given that, Adam scored x points. Imran scored 3 points fewer than Adam. Shakeel scored twice as many points as Imran. together they scored 39 points
Establishing the equation, we get
Adam scored x points.
Imran scored x-3 points.
Shakeel scored 2(x-3) points.
Together they scored 39 points:
x + (x-3) + 2(x-3) = 39
x + x -3 + 2x - 6 = 39
4x = 48
x = 12
∵ Shakeel scored = 2(x-3) points.
Putting x = 12, to get the marks scored by him,
Shakeel's score = 2(12-3) = 18
Hence, The value of x is 12 and Shakeel got 18 points.
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which of the following represents the most accurate estimation of 96-38?
If x+y=15 and x-y=25, then what is the value of 2x^2 - y^2?
A) 425
B) 775
C) 825
D) 1,575
E) 1,625 ...?
Help Me Please???:):D
four years after a hedge maple tree was planted, its height was 9 feet. eight years after it was planted, the hedge maple tree's height was 12 feet. what is the growth rate of the hedge maple? what was its height when it was planted?? ...?
The growth rate of the hedge maple tree is 0.75 feet per year, and the tree's original height when it was planted was 6 feet.
Explanation:We are tasked with finding the growth rate of a hedge maple tree and its original height when it was planted based on the given data. The tree's height was recorded at two different times: Four years after it was planted, its height was 9 feet, and eight years after planting, its height reached 12 feet.
To calculate the growth rate per year, we take the difference in height over the difference in time:
Growth rate = (Height at 8 years - Height at 4 years) / (Time at 8 years - Time at 4 years)
Thus, Growth rate = (12 feet - 9 feet) / (8 years - 4 years) = 3 feet / 4 years = 0.75 feet per year.
To determine the original height when the tree was planted, we can subtract the total growth over the four years from the height at four years after planting:
Original height = Height at 4 years - (Growth rate * 4 years)
Original height = 9 feet - (0.75 feet/year * 4 years) = 6 feet.
The music shop records the number of cds sold for 5 months. what fraction of cds were sold in april? write the fraction in simplest form.
Answer:
yoo hoo
Step-by-step explanation: This was in 2016 but its 1/5