Expanding the desired form, we have
[tex]A \sin(\omega t + \phi) = A \bigg(\sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi)\bigg)[/tex]
and matching it up with the given expression, we see that
[tex]\begin{cases} A \sin(\omega t) \cos(\phi) = 2 \sin(4\pi t) \\ A \cos(\omega t) \sin(\phi) = 5 \cos(4\pi t) \end{cases}[/tex]
A natural choice for one of the symbols is [tex]\omega = 4\pi[/tex]. Then
[tex]\begin{cases} A \cos(\phi) = 2 \\ A \sin(\phi) = 5 \end{cases}[/tex]
Use the Pythagorean identity to eliminate [tex]\phi[/tex].
[tex](A\cos(\phi))^2 + (A\sin(\phi))^2 = A^2 \cos^2(\phi) + A^2 \sin^2(\phi) = A^2 (\cos^2(\phi) + \sin^2(\phi)) = A^2[/tex]
so that
[tex]A^2 = 2^2 + 5^2 = 29 \implies A = \pm\sqrt{29}[/tex]
Use the definition of tangent to eliminate [tex]A[/tex].
[tex]\tan(\phi) = \dfrac{\sin(\phi)}{\cos(\phi)} = \dfrac{A\sin(\phi)}{\cos(\phi)}[/tex]
so that
[tex]\tan(\phi) = \dfrac52 \implies \phi = \tan^{-1}\left(\dfrac52\right)[/tex]
We end up with
[tex]y(t) = 2 \sin(4\pi t) + 5 \cos(4\pi t) = \boxed{\pm\sqrt{29} \sin\left(4\pi t + \tan^{-1}\left(\dfrac52\right)\right)}[/tex]
where
• amplitude:
[tex]|A| = \boxed{\sqrt{29}}[/tex]
• angular frequency:
[tex]\boxed{4\pi}[/tex]
• phase shift:
[tex]4\pi t + \tan^{-1}\left(\dfrac 52\right) = 4\pi \left(t + \boxed{\frac1{4\pi} \tan^{-1}\left(\frac52\right)}\,\right)[/tex]
ASAP help me with this question <3 number 12
Answer:
False
Step-by-step explanation:
Consider an isoscles trapezoid with angles 75° and 105°.
The diameter of a circle is 6 inches find the circumference to the nearest tenth
Answer:
18.8 inches
Step-by-step explanation:
The circumference is [tex]\displaystyle{C = 2\pi r}[/tex]. The diameter is two times of radius which can be expressed as [tex]\displaystyle{d = 2r}[/tex].
We can rewrite the equation of circumference as [tex]\displaystyle{C = \pi d}[/tex]. Thus, substitute d = 6 in:
[tex]\displaystyle{C = 6\pi}\\\\\displaystyle{C = 18.849}[/tex]
Rounding to the nearest tenth, we will get:
[tex]\displaystyle{C = 18.8 \ \sf \ inches}[/tex]
Hence, the circumference is 18.8 inches.
How many numbers are in the list 433, 429, 425, .... -103, -107?
The negative really confuses me
Answer:
135
Step-by-step explanation:
You can add the number which are in negative o
as well
The number is in descending order by 4.
Then add the first and last number i.e.
433+107===> 540
divide 540 by 4
540/4
= 135
Amanda rented a bike from Ted's Bikes.
It costs $10 for the helmet plus $4.25 per hour.
If Amanda paid about $33.38, how many hours did she rent the bike?
a) Let h = the number of hours she rented the bike. Write the equation you would use to solve this problem.
Answer:
5.5 hours
Step-by-step explanation:
I am going to assume that a helmet is necessary prerequisite for renting the bike.
I also am assuming that the hourly rate can be turned into a sub hourly rate if the full hour isn’t used.
10+4.25h=33.38
4.25h=23.38
h=23.38/4.25
h=5.5
12. The base of a triangle with an area of 36 squared inches is 4.2 inches. What is the area of a similar
triangle whose base measures 5.6 inches?
The 4.2 inches base length and 36 in.² area of the given triangle and the 5.6 inches base length of the similar triangle gives the area of the similar triangle as 64 square inches
Which method can be used to find the area of the similar triangle given the dimensions?Area of a triangle = (Base length × Height)/2
Area of the given triangle = 36 in.²
Base length of the given triangle = 4.2 inches
Base length of the similar triangle = 5.6 inches
Therefore;
Area of the given triangle = (Base length × Height)/2
Which gives;
36 = (4.2 × h)/2
Where;
h = Height of the given triangle
36 × 2 = 4.2 × h
[tex]h = \mathbf{\frac{36 \times 2}{4.2}} = 17 \frac{1}{7} [/tex]
Height of the given triangle, h = 17+ 1/7
The ratio of corresponding sides of similar triangles are the same, which gives;
[tex] \frac{5.6}{4.2} = \frac{h'}{17 \frac{1}{7}} [/tex]
Where;
h' = The height of the similar triangle
Which gives;
[tex] h' = \frac{5.6}{4.2} \times 17 \frac{1}{7} = 22 \frac{6}{7} [/tex]
The area, A', of the similar triangle is therefore;
[tex] A' = \frac{1}{2} \times 5.6 \times 22 \frac{6}{7} = 64 [/tex]
The area of the similar triangle A' = 64 in.²The area can also be obtained using the scale factor of area as follows;
(4.2/5.6)² = 36/A'Which gives;
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You are currently evaluating your business and trying to decide how much you need to sell to make a profit. Choose one of the following options for your cost and revenue functions. The variable, x, represents the number of units sold.
c(x)=300+260x
r(x)=300x-xsquared
For the option you chose, find the value(s) of x (the number of units sold) to break-even. Show all your work by typing it in or uploading a picture of your handwritten work. What is your profit function, P(x)? What is your profit when you sell 10 more than a break-even point? Is that what you expected? Show all your work
From the given functions, of the cost, c(x) = 300 + 260•x, and revenue, r(x) = 300•x - x², we have;
First part;
The values of x to break-even are;
x = 30, or x = 10
Second part;
The profit function, P(x) is presented as follows;
P(x) = x•(40 - x) - 300
Third part;
The profit (loss) when 10 more units is sold than the break-even point, x = 30 is -($300) unexpected The profit when 10 more units is sold than the break-even point, x = 10 is $100How can the given functions be used to find the profit made?The cost is c(x) = 300 + 260•x
Revenue is r(x) = 300•x - x²
First part;
At the break even point, we have;
c(x) = r(x)Which gives;
300 + 260•x = 300•x - x²
x² + 260•x - 300•x + 300 = 0
x² - 40•x + 300 = 0Factoring the above quadratic equation gives;
x² - 40•x + 300 = (x - 30)•(x - 10) = 0
At the break even point, x = 30, or x = 10
The values of x at the break even point are;
x = 30 units soldx = 10 units soldSecond part;
Profit = Revenue - Cost
The profit function, P(x), is therefore;
P(x) = r(x) - c(x)
Which gives;
P(x) = (300•x - x²) - (300 + 260•x)
P(x) = 300•x - x² - 300 - 260•x
P(x) = 300•x - 260•x - x² - 300
P(x) = 40•x - x² - 300
The profit function is therefore;
P(x) = x•(40 - x) - 300Third part;
When 10 more units are sold than the break even point, we have;
x = 30 + 10 = 40 or x = 10 + 10 = 20
The profit at x = 40 or x = 20 are;
P(40) = 40•(40 - 40) - 300 = -300
P(40) = -($300)When the number of units sold, x = 40, the profit is, P(40) = -($300) unexpected loss
The profit (loss) when the number of units sold increases to 40, of -($300) is unexpected.At x = 20, we have;
P(20) = 20•(40 - 20) - 300 = 100
P(20) = $100When the number of units sold, x = 20, the profit is, P(20) = $100Learn more about functions here:
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If the lengths of a triangle are 8, 12, and 18 can it be a right triangle?
Hello,
The triangle be right only if : 18² = 8² + 12²
We have : 18² = 18 × 18 = 324
and 8² + 12² = 8 × 8 + 12 × 12 = 208
324 ≠ 208 → the triangle can not be a right triangle
The value of your stock investment in a certain company decreased 10 % over the last year. If the value of your stock investment is $7000
today, then what was the value of your stock investment a year ago
The value of the stock a year ago is $7778
How to determine the value of the stock a year ago?The given parameters are:
Depreciation rate, r = 10%
Value of investment today, V(t) = $7000
The value of the stock a year ago is then calculated as:
Value of investment today = Value of investment last year * (1 - Depreciation rate)
Substitute the known values in the above equation
7000 = Last year * (1 - 10%)
Evaluate the difference of 1 and 10%
7000 = Last year * (90%)
Divide both sides by 90%
Last year = 7778
Hence, the value of the stock a year ago is $7778
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23. What would be the location of the vertex (3, 4) if it were reflected across
the y-axis?
A. (3,-4)
B. (-3, 4)
C. (4, -3)
D. (-4, 3)
A teacher records the number of students present in her 1st period class each day. This count is a ___________ random variable.
A. Discrete
B. Countable
C. Continuous
D. Finite
This count is a discrete random variable (option A).
What is a discrete random variable?A discrete random variable is a variable that contains integers that can only be a limited number of possible values. A discrete random variable is can contain only a finite set of numbers .
An example of discrete random variable is the number of students in the first period class. It is impossible for the number of students in the class to go on indefinitely.
Discrete random variable has the following properties:
It is finiteIt is numericIt is countableIt contains non-negative integers.A continous random variable is a variable that has an infinite number.
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SUPER URGENT PLEASE ANSWER ASAP
Answer:
shown below
Step-by-step explanation:
Look at the white squares, count them. 1, 2, 3. the next will be 4 and 5. draw 4 white squares, then house it in black squares like the others. to know if u did it right, count the black squares, there should be 8, then 9.
30% of 70 is what percent of 2000?
Answer:
1.05
Step-by-step explanation:
Because 30% of 70 is [tex]30\% \times 70 = \frac{30}{100} \times 70 = 0.3 \times 70 = 21[/tex], so the question becomes "21 is what percent of 2000?"
To answer this, we can solve for [tex]x[/tex] in the equation [tex]21 = x\% \times 2000[/tex], which is:
[tex]21 = x\% \times 2000\\21 = \frac{x}{100} \times 2000\\21 = 20x\\x=\frac{21}{20}=1.05[/tex]
So the answer is 1.05 percent.
George paid a total of $985 to rent a storage unit.
He initially paid $290 plus an extra amount for
each month that he rented the storage unit. Write
an equation to find the extra amount George paid
each month if he had the storage unit for 5 months.
[?] + 5m= [ ]
Answer:
290 + 5m = 985
Step-by-step explanation:
Given information:
Initial payment = $290Total payment = $985Extra amount each month = mLength of rental = 5 monthsFrom the given information, we can create the following equation:
Initial payment plus extra amount each month for 5 months equals total payment.
⇒ 290 + 5m = 985
To solve the equation:
Subtract 290 from both sides:
⇒ 290 + 5m - 290 = 985 - 290
⇒ 5m = 695
Divide both sides by 5:
⇒ 5m ÷ 5 = 695 ÷ 5
⇒ m = 139
Therefore, the extra amount George paid each month was $139.
Answer: [tex]\Large\boxed{290+5m=985}[/tex]
Step-by-step explanation:
Given information
Initial cost = $290
Total cost = $985
Number of months = 5 months
Cost each month = ?
Set variable
Let [ m ] be the cost each month
The equation in verble sense
Initial paid + Extra each month = Total cost
Convert verbal equation to mathematical sense
$290 + 5m = $290
Therefore, to find the extra amount paid each month, the equation is [tex]\Large\boxed{290+5m=985}[/tex]
---------------------------------------------------------------------------------------------------------
EXTRA The following will be solving the derived equation, please ignore this part if it is useless for you.
Given equation
290 + 5m = 985
Subtract 290 on both sides
290 + 5m - 290 = 985 - 290
5m = 695
Divide 5 on both sides
5m / 5 = 695 / 5
m = $139
Hope this helps!! :)
Please let me know if you have any questions
Find the experimental probability that only 2
of 4 children in a family are boys.
The problem has been simulated by tossing 4
coins (one to represent each child). Let "heads"
represent a boy and "tails" represent a girl. A
sample of 20 coin tosses is shown.
HTHH HTTH TTTT THTT HTHT
HHTT HHHT THHT
TTHH
HTTT HTHT TTHH THTH HTHH
TTHT HTTT HTHT HHHT HHHH
HTTH
Experimental Probability = [?]%
The experimental probability that only 2 of 4 children in a family are boys is 50%.
What is experimental probability?Experimental probability refers to the chance of an expected success being achieved in a series of experiments conducted.
Experimental probability is the number of times that the expected success occurs as a fraction of the total number of times the experiment was conducted.
Like all probabilities, the experimental probability is based on the likelihood that what the experimenter expects is achieved.
Expected number of boys = 2
The number of children in the family = 4
Experimental probability = 50% (2/4 x 100)
Thus, we can conclude, based on the experimental probability, that 50% (or 2) of the 4 children in the family are boys.
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Answer:
15%
Step-by-step explanation:
Step 1: Count samples with one H
Why; Only 1 of 4 children in families are boys. The problem is simulated with samples.
Step 2: Count samples with only one H.
Why; The H represents the one boy in the family.
Step 3: Add the samples you counted with one H.
Answer: Your answer should be 3/20 after you change it into a percentage your final answer is 15%
the first term of a geometric sequence is 400 and the common ratio is 0.5. what is the 5th term of the sequence.
Answer:
25
Step-by-step explanation:
Geometric sequence
General form of a geometric sequence:
[tex]a_n=ar^{n-1}[/tex]
where:
a = first termr = common ratio[tex]a_n[/tex] = nth termGiven values:
a = 400r = 0.5n = 5Substitute the given values into the formula and solve:
[tex]\implies a_5=400(0.5)^{5-1}[/tex]
[tex]\implies a_5=400(0.5)^{4}[/tex]
[tex]\implies a_5=400(0.0625)[/tex]
[tex]\implies a_5=25[/tex]
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Identify the theorem or postulate that is related to the measures of the angles in the pair, and find the unknown angle measure.
m∠1=(50x−50)∘, m∠2=(20x+70)∘
Alt. Ext. ∠s Thm.
m∠1 = 130°, m∠2 = 130°
Same-Side Int. ∠s Thm.
m∠1 = 130°, m∠2 = 130°
Alt. Int. ∠s Thm.
m∠1 = 150°, m∠2 = 150°
Corr. ∠s Post.
m∠1 = 150°, m∠2 = 150°
The theorem used and the angle measures are:
D. corresponding angles postulate
m∠1 = 150°, m∠2 = 150°
What is the Corresponding Angles Postulate?According to the corresponding angles postulate, if two angles lie on same relative corner along a transversal that cuts two parallel lines (corresponding angles), then their measures are equal.
Thus, ∠1 and ∠2 are corresponding angles, therefore:
m∠1 = m∠2 [corresponding angles postulate]
Given the following:
m∠1 = (50x − 50)∘,
m∠2 = (20x + 70)∘
Therefore, we would have:
50x − 50 = 20x + 70
Subtract 20x from both sides
50x − 50 - 20x = 20x + 70 - 20x
30x − 50 = 70
Add 50 to both sides of the equation
30x − 50 + 50 = 70 + 50
30x = 120
Divide both sides by 30
30x/30 = 120/30
x = 4
Plug in the value of x
m∠1 = (50x − 50) = 50(4) − 50 = 150°
m∠2 = (20x + 70)∘ = 20(4) + 70 = 150°
Thus, the answer is: D. corresponding angles postulate
m∠1 = 150°, m∠2 = 150°
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Briana received a 10-year subsidized student loan of $26,000 at an annual interest rate of 4.125%. Determine her monthly payment (in dollars) on the loan after she graduates in 2 years. (Round your answer to the nearest cent.)
Answer:
$264.78
Step-by-step explanation:
The government pays the interest on Briana's loan while she is in school. Her monthly payments will be for a loan of $26,000. The amount of the payment can be found from the amortization formula, or using a calculator or spreadsheet.
Payment amountFor a loan of principal amount P at annual rate r for t years, the monthly payment is ...
A = P(r/12)/(1 -(1 +r/12)^(-12t)) = $26000(0.04125/12)/(1 -(1 +0.04125/12)^(-120)) = $264.78
Briana's monthly payment after she graduates is $264.78.
Use the coordinate grid to determine the coordinates of point A:
Coordinate grid shown from negative 2 to positive 2 on x axis and negative 2 to positive 2 on y axis. There are increments of 1 over 4 for each grid line on each of the two axes. Only the whole numbers are labeled on either side of the axis. A point A is shown at the intersection of 5 grid lines to the left of the y-axis and 3 grid lines above the x-axis.
What are the coordinates of point A?
( fraction negative 1 and 1 over 4, fraction 3 over 4)
( fraction 1 and 1 over 4, fraction negative 3 over 4)
( fraction negative 3 over 4, fraction 1 and 1 over 4)
( fraction 3 over 4, fraction negative 1 and 1 over 4)
Based on the given coordinate grid, and the increment for each grid line, the coordinates of point A is ( fraction negative 1 and 1 over 4, fraction 3 over 4) or (-1¹/₄, ³/₄).
What are point A's coordinates?
The x-value of point A is said to be 5 grid lines to the left of y-axis. This means that the x value will be negative.
The increments are 1/4 so the x-value is:
= -1/4 x 5
= -1¹/₄
The y-value is given by the 3 grid lines above the x-axis which means that this value is positive:
= 1/4 x 3
= 3 /4
The coordinates of A are:
(-1¹/₄, ³/₄)
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If 7 and 9 are the remainders when 98 and 165
respectively, are divided by a positive integer a, then a =
Answer:
Step-by-step explanation:
98-7=91
165-9=156
(91/156)/13=7/12
Answer = 13
It is divided by a positive integer a, which will equal the number 13.
What is an arithmetic operation?The four basic mathematical operations are the addition, subtraction, multiplication, and division of two or even more integers. Among them is the examination of integers, particularly the order of actions, which is crucial for all other mathematical topics, including algebra, data organization, and geometry.
As per the data provided by the question,
The number are 98 and 165.
The remainders are 7 and 9.
So, the actual number is,
98 - 7 = 91 and,
165 - 9 = 156
They are divided by the number 13, so the value of a will be equal to 13.
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one of the angles of a triangle is 106° and the other two angles are equal find each of the equal angles
Answer:
37°
Step-by-step explanation:
• recall that the sum of the interior angles of a triangle
is equal to 180°
Then
The measure of the two other angles :
= 180 - 106
= 74°
The measure of each of the equal angles :
= 74 ÷ 2
= 37°
Answer:
the 2 angles r 37
Step-by-step explanation:
the angles of triangle sums up to 180, and lets call the unknown angle x, since they're both equal, x+x=2x
so when 3 angles of triangle adds up, we get 180
106+2x=180
2x=180-106
2x=74, divide both sides by 2 to get x
x=74/2
x=37
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If one angle in a parallelogram is a right angle prove the other are the same.
Answer: If one angle of a parallelogram is a right angle, then all other angles would also be right angles and the parallelogram would be a rectangle.
1) La suma de dos números es 58. Si uno de los números es 12 más que el otro
número,
cuáles son los números?
Answer:
Los números son 23 y 35.
Step-by-step explanation:
Q: The sum of two numbers is 58. If one of the numbers is 12 more than the other number, what are the numbers?
Set the numbers as x and y. Using the given information, we have two equations:
x+y=58
x=y+12
Solving using substitution, we have:
y+y+12=58 -> y=23
Substituting, x=23+12=35
The numbers are 23 and 35.
A ball is thrown from an initial height of 2 meters with an initial upward velocity of 25 m/s . The ball's height h (in meters) after t seconds is given by the following.
The value of values of t for which the ball's height is 7 meters then is 4.79secs.
What is velocity?Velocity can be regarded as a vector measurement of the rate as well as direction of motion.
It is the the speed at which something moves in one direction and with the given velocity, the time as well as the distance can be calculated as :
Given:
h=2+25t-5t^2
But h=7
Then 2+25t-5t^2=7
-5t^2++25t +2-7=0
-5t^2+25t -5=0
we can then factorize as :
-5(t^2+5t -1)=0
Solving the equation using quadratic formula, the the roots of equations are: 4.79 and -5.0
The value of values of t for which the ball's height is 7 meters then is 4.79secs.
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COMPLETE QUESTION :
A ball is thrown from an initial height of 2 meters with an initial upward velocity of 25 m/s . The ball's height h (in meters) after t seconds is given by the following. h=2+25t-5t^2
Find all values of t for which the ball's height is 7 meters.
Round your answer(s) to the nearest hundredth.
He wants to earn 481$ By mowing lawns and he charge $22 for each long what is the maximum number of lungs that he needs to mow to reach his goal of $481
[tex]\stackrel{\stackrel{total~amount}{goal}}{481}~~ = ~~22(lawn)\implies \cfrac{481}{22}=lawn\implies \stackrel{rounded}{22}\approx lawn[/tex]
Knowing the first day Anna and Tamara stop for sightseeing and lost 2 hours of travel time and the second day, they gain 1 hour because they did not stop for lunch, write an algebraic expression that represents how many hours Anna and Tamara traveled the first two days. Let x be the number of hours driven.
The algebraic expression that represents the number of hours Anna and Tamara traveled the first two days is (2x - 1) hours.
How to write Algebraic Word Problems?
We are told that;
On the first day, Anna and Tamara stop for sightseeing and lost 2 hours of travel time.
On the second day, they did not stop for lunch and gained 1 hour.
Now, we are told that x is the number of hours driven per day. Thus, for two days, number of hours driven is expressed as 2x.
Now, since they lost 2 hours on the first day, then total hours travelled is;
2x - 2 hours
On the second day, they gained 1 hour. Thus;
Final total number of hours traveled = 2x - 2 + 1
Final total number of hours traveled = (2x - 1) hours
Thus, we can conclude that the algebraic expression that represents how many hours Anna and Tamara traveled the first two days is (2x - 1) hours.
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create a pattern of 6 numbers by multiplying a number by 5 to get the next number.starts with 4
Answer:
45 405 415 425 435 445
Step-by-step explanation:
this are the pattern of 6 numbers multiplying by number 5 to get the number starts with 4
Answer:
4,20,100,500,1500,8500.
Step-by-step explanation:
The distance from TJ's house to school and back is 0.4 km. In one week TJ travelled 2 km. How many times did TJ go to school?
$$
Answer: 5 times
Step-by-step explanation: The distance from his house to school and back is 0.4 km. Since he traveled 2km in a week, we have to divide 2 by0.4. 0.4x5 = 2 so he went to school 5 times a week.
Please help, type your answers in order .
Step-by-step explanation:
we know from the laws of motion that in the equation
h = 20t - 1.86t²
the gravitational acceleration is in the factor of the t² term.
h = v0t + 1/2 × g × t²
v0 being the initial velocity (20 m/s).
and we can therefore see, that the gravitational acceleration "a" on Mars is 2×1.86 = 3.72 m/s²
(a)
the velocity v after 2 seconds is (first law of motion)
v = v0 + at = 20 - 3.72×2 = 12.56 m/s
gravity pulling down, so negative acceleration.
(b)
first we need the time when the rock is at 25 m.
25 = 20t - 1.86t²
0 = -1.86t² + 20t - 25
the solution to a quadratic equation is always
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = t
a = -1.86
b = 20
c = -25
t = (-20 ± sqrt(400 - 4×-1.86×-25))/(2×-1.86) =
= (-20 ± sqrt(214))/-3.72
t1 = (-20 + 14.62873884...)/-3.72 =
= 1.443887409... s ≈ 1.44 s
t2 = (-20 - 14.62873884...)/-3.72 =
= 9.308800763... s ≈ 9.31 s
that means, on its way up the rock reached 25m after
1.44 s.
on its way down the rock reached 25 m after
9.31 s.
velocity 0 (the rock came to a stop before falling back down) was after
0 = 20 - 3.72t
3.72t = 20
t = 20/3.72 = 5.376344086... s
that means the rock was falling after this point back to the ground and was gaining speed again.
that accelerating phase until the rock was again at 25 m was
9.308800763... - 5.376344086... = 3.932456677... s
long
the velocity at these points was
v-up = 20 - 3.72×1.443887409... = 14.62873884... m/s ≈
≈ 14.63 m/s
v-down = 0 + 3.72×3.932456677... = 14.62873884... m/s ≈
≈ 14.63 m/s
as expected : the rock did a perfectly symmetric flight path between the two 25 m marks.
so time and speed on both sides had to be identical.
but we have proven it.
Answer:
a) 12.56 m/s
b) up: 14.63 m/s (2 d.p.)
down: -14.63 m/s (2 d.p.)
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{9 cm}\underline{The Constant Acceleration Equations (SUVAT)}\\\\s = displacement in m (meters)\\u = initial velocity in m s$^{-1}$ (meters per second)\\v = final velocity in m s$^{-1}$ (meters per second)\\a = acceleration in m s$^{-2}$ (meters per second per second)\\t = time in s (seconds)\\\\When using SUVAT, assume the object is modeled\\ as a particle and that acceleration is constant.\end{minipage}}[/tex]
The average gravitational acceleration on Mars is 3.721 m/s² (about 38% of that of Earth). The fact that the rock is thrown from the surface of Mars rather than the surface of Earth is of no consequence when using the equations of constant acceleration (SUVAT equations) as long as acceleration is not used in the calculations.
Part (a)Given:
[tex]s=20t-1.86t^2, \quad u=20, \quad t=2[/tex]
[tex]\begin{aligned}\textsf{Using }\: s&=\dfrac{1}{2}(u+v)t\\\\\implies 20(2)-1.86(2)^2 & = \dfrac{1}{2}(20+v)(2)\\\\32.56 & = 20+v\\\\v & = 32.56-20\\\\v & = 12.56\:\: \sf m/s\end{aligned}[/tex]
Therefore, the velocity of the rock after 2 s is 12.56 m/s.
Part (b)Find the time when the height of the rock is 25 m:
[tex]\begin{aligned}20t-1.86t^2 & = s\\\implies 20t-1.86t^2 & = 25\\1.86t^2-20t+25 & = 0\end{aligned}[/tex]
Quadratic Formula
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]
Therefore:
[tex]a=1.86, \quad b=-20, \quad c=25[/tex]
[tex]\implies t=\dfrac{-(-20) \pm \sqrt{(-20)^2-4(1.86)(25)} }{2(1.86)}[/tex]
[tex]\implies t=\dfrac{20 \pm \sqrt{214}}{3.72}[/tex]
[tex]\implies t=9.308800763..., 1.443887409...[/tex]
Therefore, the height of the rock is 25 m when:
t = 9.31 s (2 d.p.)t = 1.44 s (2 d.p.)To find the velocity (v) of the rock at these times, substitute the found values of t into the equation, along with s = 25 and u 20 m/s:
[tex]\begin{aligned}\textsf{Using }\: s&=\dfrac{1}{2}(u+v)t\\\\\implies v & = \dfrac{2s}{t}-u\\\\v & = \dfrac{2(25)}{t}-20\\\\v & = \dfrac{50}{t}-20\\\\\end{aligned}[/tex]
When t = 1.443887409...s
[tex]\implies v=\dfrac{50}{1.443887409...}-20=14.63\:\: \sf m/s \: \:(2\:d.p.)[/tex]
When t = 9.308800763...s
[tex]\implies v=\dfrac{50}{9.308800763...}-20=-14.63\:\: \sf m/s \: \:(2\:d.p.)[/tex]
Therefore, the velocity of the rock when its height is 25 m is:
up: 14.63 m/s (2 d.p.)down: -14.63 m/s (2 d.p.)Learn more about constant acceleration equations here:
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is there anybody to solve this need step by step solution details
The circumference of the circles from biggest to smallest are; 30π, 15π and 7.5π
How to find the area of a circle?The formula for area of a circle is;
A = πr²
In geometry, the area enclosed by a circle of radius r is πr². Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
The area of a circle formula is useful for measuring the region occupied by a circular field or a plot. Suppose, if you have a circular table, then the area formula will help us to know how much cloth is needed to cover it completely.
The area formula will also help us to know the boundary length i.e., the circumference of the circle. Does a circle have volume? No, a circle doesn't have a volume
Area of entire circle is 706 cm².
Thus;
πr² = 706
r² = 706/π
r = √(706/π)
r ≈ 15
Circumference of biggest circle = 2πr
Circumference of biggest circle = 2π * 15
Circumference of biggest circle = 30π
Circumference of second biggest circle = 2π * (15/2)
Circumference of second biggest circle = 15π
Circumference of smallest circle = 2π * (15/4)
Circumference of smallest circle = 7.5π
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There is a rectangular prism with a length of 8 m, a width of 5 m, and a height of 7 m.
Find the surface area of this rectangular prism.
PLEASE HELP ME :)