Answer:
16 feet
Step-by-step explanation:
The relationships between displacement (position), velocity and acceleration are:
[tex]\boxed{\boxed{\begin{array}{c}\textbf{DISPLACEMENT (s)}\\\\\text{Differentiate} \downarrow\qquad\uparrow\text{Integrate}\\\\\textbf{VELOCITY (v)}\\\\\text{Differentiate}\downarrow\qquad\uparrow \text{Integrate}\\\\\textbf{ACCELERATION (a)}\end{array}}}[/tex]
To find the distance a particle travels in its first 4 seconds of travel, we first need to determine if the particle changes direction at any point during this time.
The instant(s) when the particle changes direction is when its velocity is zero. Therefore, set the velocity function v(t) to zero and solve for t:
[tex]\begin{aligned}v(t)&=0\\-t^2+4&=0\\-t^2&=-4\\t^2&=4\\t&=\pm 2 \end{aligned}[/tex]
So, the particle changes direction at t = 2 seconds in its first 4 seconds of travel. This means that to find the distance the particle travels in its first 4 seconds of travel, we need to integrate the velocity function over the two intervals [0, 2] and [2, 4] seconds to find the particle's displacement over these intervals.
The displacement of the particle in its first 2 seconds of travel is:
[tex]\begin{aligned}\displaystyle \int^2_0 (-t^2+4)\; \text{d}t&=\left[\dfrac{-t^{2+1}}{2+1}+4t\right]^2_0\\\\&=\left[-\dfrac{t^{3}}{3}+4t\right]^2_0\\\\&=\left(-\dfrac{(2)^{3}}{3}+4(2)\right)-\left(-\dfrac{(0)^{3}}{3}+4(0)\right)\\\\&=-\dfrac{8}{3}+8+0-0\\\\&=\dfrac{16}{3}\end{aligned}[/tex]
The displacement of the particle in its next 2 seconds of travel is:
[tex]\begin{aligned}\displaystyle \int^4_2 (-t^2+4)\; \text{d}t&=\left[\dfrac{-t^{2+1}}{2+1}+4t\right]^4_2\\\\&=\left[-\dfrac{t^{3}}{3}+4t\right]^4_2\\\\&=\left(-\dfrac{(4)^{3}}{3}+4(4)\right)-\left(-\dfrac{(2)^{3}}{3}+4(2)\right)\\\\&=-\dfrac{64}{3}+16+\dfrac{8}{3}-8\\\\&=-\dfrac{32}{3}\end{aligned}[/tex]
The negative value means that the particle is travelling in the opposite direction.
So, the particle travels 16/3 feet in its first 2 seconds of travel, changes direction at t = 2 seconds, and travels 32/3 feet in the opposite direction in the next 2 seconds of travel.
Therefore, the total distance the particle travelled is the sum of the absolute values of the two displacements:
[tex]\textsf{Distance}=\dfrac{16}{3}+\dfrac{32}{3}=\dfrac{48}{3}=16\;\sf feet[/tex]
So, the particle travels 16 feet in its first 4 seconds of travel.
The distance a particle travels in the first 4 seconds, moving according to the velocity equation v(t)= -t² + 4, is found by integrating the velocity function over that interval, which yields 5.33 feet.
Explanation:To find the distance a particle travels given its velocity equation v(t)= -t² + 4 (in feet/sec), we need to integrate the velocity function over the desired time interval.
Since velocity is the rate of change of position with respect to time, the integral of the velocity function from 0 to 4 seconds will give us the displacement of the particle in that time interval.
The integral of v(t) from 0 to 4 seconds can be computed as follows:
∫ v(t) dt from t=0 to t=4= ∫ (-t² + 4) dt from t=0 to t=4= [-t³/3 + 4t] from t=0 to t=4= [(-4³/3 + 4 × 4) - (0³/3 + 4 × 0)]= [(-64/3 + 16) - 0]= (-64 + 48)/3= -16/3= -5.33 feetHowever, the absolute value of -5.33 feet is needed, because distance must be positive. Hence, the particle travels 5.33 feet in the first 4 seconds.
The law of detachment can be used to draw a conclusion from the following statements: "It will be sunny on Tuesday." "I will always go outside if the weather is nice."
Final answer:
Using the law of detachment, if it is sunny on Tuesday and there is a habit of going outside in nice weather, one can conclude that they will go outside on Tuesday.
Explanation:
The law of detachment is a concept in logic used to form conclusions given a conditional statement and its hypothesis being true. For instance, in the scenario provided:
If it is sunny on Tuesday (hypothesis).
I will always go outside if the weather is nice (conditional statement).
By applying the law of detachment, we can conclude that:
I will go outside on Tuesday (conclusion).
This logical reasoning assumes that a 'nice' weather includes a sunny day. Hence, if it is indeed sunny on Tuesday, the condition for going outside is met.
Please help me
True or false. In order for a disjunction statement to be false, both of its component statements must be false. Explain.
The answer is true. This was just on my quiz and I know its correct.
3y + 2z = 12 and y – z = 9.
given f(x)=x^3 and g(x)=1-5x^2, find (fog) (x) and its domain
Answer:
answer is b
Step-by-step explanation:
Please help me this ...
Introduction to addition or subtraction of fractions with different denominator 3/5+1/3
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I need help making an inequality.The problem is down below: The student council wants to rent a ballroom for the junior prom. The ballroom's rental plan is $1500 for 3 h and $125 for each...
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What is the sum of the first eight terms -800, -200, -50, -12.5... etc divide by 4 each time i could do this but im too lazy :p
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Mathematics 5 pts 5 minutes ago
The product of two consecutive positive integers is 1,332. Explain how you can write and solve a quadratic equation to find the value of the larger integer.
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Mathematics 5 pts 5 minutes ago
Introduction to addition or subtraction of fractions with different denominator 3/5+1/3
Amy, who is 17 years old, is 5 years older than twice Moe's age.Write an equation that you could solve to find Moe's age. Solve the equation and describe your steps.
What is the greatest common factor of 60x4y7, 45x5y5, and 75x3y? 5xy 15x3y 45x3y5 75x5y7
Answer: 15x3y
Step-by-step explanation:
We rewrite the expressions by using factorisation method of their primes. This will give:
60x4y7= 2×2×3×5×x×x3×y×y6
45x5y5= 3×3×5×x×x4×y×y4
75x3y= 3×5×5×x×x2×y
Common factors= 3 × 5 × x3 × y
= 15x3y
Answer:
its B
Step-by-step explanation:
In 2010, the national debt of the United States was about 14 trillion dollars. In 2003 it was about 7 x 10 to the 12 dollars. About how many times larger was the national debt in 2010 than 2003? THIS IS SCIENTIFIC NOTATION
Answer: About 2 times.
Step-by-step explanation:
Given : In 2010, the national debt of the United States was about 14 trillion dollars.
We know that 1 trillion [tex]=1,000,000,000,000= 10^{12}[/tex]
then, In 2010, the national debt of the United States was about [tex]14\times10^{12}[/tex]
dollars.
In 2003 it was about [tex]7\times10^{12}[/tex] dollars.
Now, the number of times the national debt was larger in 2010 than 2003 :-
[tex]\dfrac{ 14\times10^{12}}{7\times10^{12}}\\\\=\dfrac{14}{7}\times10^{12-12}\\\\=2\times10^0=2[/tex]
Hence, national debt was about 2 times larger in 2010 than 2003 .
The height of a triangle is 4 cm more than twice the length of the base. the area of the triangle is 63 cm2. find the height of the triangle.
Answer:
THE ANSWER COULD BE 14
A garden table and a bench cost $799 combined. the garden table costs $49 more than the bench. what is the cost of the bench?
The cost of the bench is $ 375
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
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 data ,
Let the cost of the bench be = B
Now , the equation will be
Let the cost of the table be = T
Now , the total cost of the table and bench = $ 799
Substituting the values in the equation , we get
B + T = 799 be equation (1)
And , garden table costs $49 more than the bench
So , T = 49 + B be equation (2)
Substituting the value of T from equation (2) in equation (1) , we get
B + 49 + B = 799
2B + 49 = 799
Subtracting 49 on both sides of the equation , we get
2B = 750
Divide by 2 on both sides of the equation , we get
B = $ 375
Therefore , the value of B is $ 375
Hence , The cost of the bench is $ 375
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Solve the equation
5x - 5 = 3x - 9
A. 1
B. -2
C. -3
D. -1
A basket contains 13 pieces of fruit: 5 apples, 5 oranges, and 3 bananas. Jonas takes a piece of fruit at random from the basket, and then Beth takes a piece at random. What is the probability that Jonas will get an orange and Beth will get an apple?
An article with a net weight of 10 lb (pounds) is packaged in a box that weighs ½ lb. If 20 of these boxed articles are put into one freight container that weighs 15 lb, what is the gross weight?
Solve for the variable "a" in this equation. Show work:
c=8πab
Six times a number decreased by ten is eight. Find the number.
A) 2
B) 3
C) 4
D) 5
What type of transformation takes the graph of
f(x)=|x|
to the graph of
g(x)=â3+|x|
?
Â
vertical translation of 3 units up
horizontal translation of 3 units right
vertical translation of 3 units down
horizontal translation of 3 units left
The sum of 4 times mike's age and 3 times susan's age is 79. mike is 1 year more than 3 times as old as susan. find each of their ages.
Smart enough to answer this question? Gracie, Mary, and Nancy each have a small collection of seashells. Gracie has 5 more than 1 1/4 times the number of shells Mary has. Nancy has 1 more than 1 1/2 times the number of shells Mary has. Gracie and Nancy have the same number of shells. If x is the number of shells Mary has, identify the equation that represents this situation and identify its solution. (isnt fraction is one and one fourths, second fraction is 1 and one halfs)
For the points in the table, does pressure vary directly with volume? If so, identify k and write an equation. No, this is not a direct variation. Yes, this is a direct variation with k = . Yes, this is a direct variation with k = 1. Yes, this is a direct variation with k = 3.
Answer:
it is A "No, this is not a direct variation."
Step-by-step explanation:
i jus took the test
Pressure and volume have an inverse relationship, known as Boyle's Law, where if pressure increases, volume decreases and vice-versa. This is represented by the mathematical equation P.V = k, where k is a constant. This is graphically represented as a hyperbola.
Explanation:From the details available, it can be concluded that the relationship between pressure (P) and volume (V) is not a direct variation, but an inverse one. This relationship is also known as Boyle's Law, which states that the product of the pressure and volume for a given amount of gas at a constant temperature remains constant. Mathematically this can be written as P = k·1/V or P.V = k, where k is a constant.
This means that when the pressure increases, volume decreases and vice-versa. If you were to graphically represent this, you would get a hyperbola, not a straight line, which distinguishes it as an inverse relationship rather than a direct one.
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Which of the following is a solution to the system shown below? y = 3x 2 - 8x - 2 and 2x - y = 5
Answer:
(1/3, -13/3)
Step-by-step explanation:
I know this is correct because I got this question right on my quiz. Exact question, word for word. Hope this helps!
find the digit that makes 2,72_ divisible by 6
To find the missing digit to make 2,72_ divisible by 6, add the digit 4.
To find the digit that makes 2,72_ divisible by 6, we need to focus on the divisibility rules for 6. In this case, a number is divisible by 6 if it is divisible by both 2 and 3. For 2, the last digit must be even, and for 3, the sum of the digits must be a multiple of 3.
Given the number 2,72_, the blank digit needs to be 4 to make the number divisible by 6. This is because when the digit 4 is added, the number becomes 2,724, which is divisible by both 2 and 3.
To determine the number of trout in a lake, a conservationist catches 180 trout, tags them and throws them back into the lake. later, 48 trout are caught; 12 of them are tagged. how many trout would the conservationist expect to be in the lake?
What is -2,3 rotated 90 degrees counterclockwise around the origin?
Answer(-3,2)
Step-by-step explanation:
The accepted value of a length measurement is 200 cm and the experimental value is 198 cm what is the percent error
The percent error of the given problem is 1% inaccuracy.
How to calculate the percent error?Suppose the actual value and the estimated values after the measurement are obtained. Then we have:
Error = Actual value - Estimated value
We have been given that accepted value of a length measurement is 200 cm and the experimental value is 198 cm.
Actual value = 200
Estimated value = 198
The percent accuracy,
198/200 x 100%
= 99%
Therefore,
Error = Actual value - Estimated value
Percent Inaccuracy: 100 - 99%
= 1% inaccuracy
Hence, the percent error of the given problem is 1% inaccuracy.
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The table represents the height of a ball thrown up from the roof of a building, h(t), in meters, t seconds after it is thrown upward.
Answer:
A. The ball is at the same height as the building between 8 and 10 seconds after it is thrown.
C. The ball reaches its maximum height about 4 seconds after it is thrown
Step-by-step explanation:
edg.
What is the solution of the linear quadratic system or equations?
Y=x^2+5x-3
Y-x=2
The solution to the system of equations are (1, 3) and (-5, -3)
Given the following functions
Y=x^2+5x-3
Y-x=2; y = x + 2
Equating both outputs will give
x^2+5x-3 = x + 2
x^2+5x-3 - x - 2 = 0
x^2 + 4x - 5 = 0
Factorize the equation
x^2 + 5x - x - 5=0
x(x+5) - 1(x+5) = 0
(x-1)(x+5) = 0
x = 1 and -5
Recall that y = x + 2
y = 3 and -3
Hence the solution to the system of equations are (1, 3) and (-5, -3)
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The solution to a linear quadratic system of equations can be found by substitifying one equation into the other and solving. In this case, the solutions are (1,3) and (-5,-3).
Explanation:To find the solution of a linear quadratic system of equations, you need to substitute one equation into the other. In this case, for the equations [tex]Y = x^2 + 5x - 3[/tex] and [tex]Y - x = 2[/tex], you would first isolate Y in the linear equation, resulting in [tex]Y = x+2.[/tex]
Then, you substitute (x + 2) for Y in the quadratic equation, leading to [tex]x + 2 = x^2 + 5x - 3.[/tex]
Next, rearrange the equation to get it in the standard quadratic form: 0 = [tex]x^2 + 5x - x - 3 - 2.[/tex] Simplifying this results in the equation [tex]0 = x^2 + 4x - 5.[/tex]
Finally, solve this equation by factoring or using quadratic formula, which gives the solutions as x = 1, -5. Substituting both 1 and -5 into the equation Y = x + 2 gives the corresponding y-values, Y = 3 and -3. Therefore, the solutions to the system of equations are (1,3) and (-5,-3).
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What is the length of each side of a square room if the area is 400 ft²?
Suppose that the length of a certain rectangle is 5 meters less than four times its width. the perimeter of the rectangle is 40 meters. find the length and width of the rectangle.
Richard borrowed $950 from his parents to make some repairs on his car. He promised to repay the loan by giving his parents at least $85 from his paycheck each week. Assuming richard sticks to his promise, which of the following is not a possible scenario for the amount of debt Richard has remaining?
I think it's B but I'm not 100% sure
Answer:
B
Step-by-step explanation:
To solve this, you need to know first how many weeks will take Richard replacing the money.
We will call X the number of weeks so we say
$85 * X =$950
X= 11.17
This means it will take at least ≈12 weeks to pay his debt.
To know which scenario it's not possible, use the equation to choose the right option.
A: $85*5= $425
After 5 weeks, he paid $425, this mean he's $525 in debt.
B: $85*6= $510
After 6 weeks, he paid $510, this means he's $440 in debt.
C. $85*7= $595
After 7 week,s he paid $595, this means he's $355 in debt.
D. $85*8= $680
After 7 weeks, he paid $680, this means he's $270 in debt.
The not possible scenario is B because he will be paying at least $85 or more so he will debt $440 or less al week 6.
What are the steps for using a compass and straightedge to construct a square? Drag and drop the steps in order from start to finish. Use a straightedge to draw line a and label a point on the line as point P. Without changing the compass width, place the compass point on point S and draw an arc in the interior of ∠SPT . Use the straightedge to draw QS¯¯¯¯¯ and QT¯¯¯¯¯. Construct a line perpendicular to line a through point P. Label a point on this line as point R. With the compass open to the desired side length of the square, place the compass point on point P and draw an arc on line a and an arc on PR←→ . Label the points of intersection as points S and T. Keeping the same compass width, place the compass on point T and draw an arc in the interior of ∠SPT to intersect the previously drawn arc. Label the point of intersection as point Q.
Answer: The steps for using a compass and straightedge to construct a square are as follows :-
1. Use a straightedge to draw line a and label a point on the line as point P.
2. Construct a line perpendicular to line a through point P.
3. Label a point on this line as point R.
4. With the compass open to the desired side length of the square, place the compass point on point P and draw an arc on line a and an arc on PR←→.
5. Label the points of intersection as points S and T.
6. Keeping the same compass width, place the compass on point T and draw an arc in the interior of∠SPT to intersect the previously drawn arc.
7. Label the point of intersection as point Q.
8. Without changing the compass width, place the compass point on point S and draw an arc in the interior of∠SPT .
9. Use the straightedge to draw QS ,QT,SR and RT.
Here we have finished with the square RTQS.