Answer:
always true
Step-by-step explanation:
Adding 4-b to both sides of the equation gives ...
ax = 4
Then dividing by "a" gives ...
x = 4/a
This is always true when a≠0.
The paraboloid z = 6 − x − x2 − 5y2 intersects the plane x = 2 in a parabola. Find parametric equations in terms of t for the tangent line to this parabola at the point (2, 2, −20).
Answer:
x = 2
y = 2 + t
z = -20 -20t
Step-by-step explanation:
First, we are going to find the equation for this parabola. We replace x = 2 in the equation of the paraboloid, thus:
[tex]z = 6-x-x^{2} -5y^{2}[/tex]
if x = 2, then
[tex]z = 6-(2)-2^{2}-5y^{2}[/tex]
[tex]z = -5y^{2}[/tex]
Now, we calculate the tangent line to this parabola at the point (2,2,-20)
The parametrization of the parabola is:
x = 2
y = t
[tex]z = -5t^{2}[/tex] since [tex]z = -5y^{2}[/tex]
We calculate the derivative
[tex]\frac{dx}{dt}= 0[/tex]
[tex]\frac{dy}{dt}= 1[/tex]
[tex]\frac{dz}{dt}= -10t[/tex]
we evaluate the derivative in t=2, since at the point (2,2,-20) y = 2 and y = t
Thus:
[tex]\frac{dx}{dt}= 0[/tex]
[tex]\frac{dy}{dt}= 1[/tex]
[tex]\frac{dz}{dt}= -10(2)= -20[/tex]
Then, the director vector for the tangent line is (0,1,-20)
and the parametric equation for this line is:
x = 2
y = 2 + t
z = -20 -20t
The parametric equation of the tangent line is [tex]L(t)=(2,2+t,-20-20t)[/tex]
Parabola :The equation of Paraboloid is,
[tex]z =6-x-x^{2} -5y^{2}[/tex]
Equation of parabola when [tex]x = 2[/tex] is,
[tex]z=6-2-2^{2} -5y^{2} \\\\z=-5y^{2}[/tex]
The parametric equation of parabola will be,
[tex]r(t)=(2,t,-5t^{2} )[/tex]
Now, we have to find Tangent vector to this parabola is,
[tex]T(t)=\frac{dr(t)}{dt}=(0,1,-10t)[/tex]
We get, the point [tex](2, 2, -20)[/tex] when [tex]t=2[/tex]
The tangent vector will be,
[tex]T(2)=(0,1,-20)[/tex]
The tangent line to this parabola at the point (2, 2, −20) will be,
[tex]L(t)=(2,2,-20)+t(0,1,-20)\\\\L(t)=(2,2+t,-20-20t)[/tex]
Learn more about the Parametric equation here:
https://brainly.com/question/21845570
A scoop of ice cream has a 3 inch radius. How tall should the ice cream cone of the same radius be in order to contain all of the ice cream inside the cone?
Answer:
12cm
Step-by-step explanation:
The scoop of Ice Cream is in the shape of a circular solid which is a Sphere.
For the ice cream to fit into the cone, the volume of the cone must be equal to that of the sphere.
Radius of the Sphere=3cm
Volume of a Sphere = [tex]\frac{4}{3}\pi r^3[/tex]
Volume of a Cone=[tex]\frac{1}{3}\pi r^2h[/tex]
[tex]\frac{1}{3}\pi X 3^2h=\frac{4}{3}\pi X 3^3\\\frac{1}{3}h=\frac{4}{3} X 3\\\frac{1}{3}h=4\\h=4 X 3=12cm[/tex]
The Cone of same radius must be 12cm tall.
The caldwells are moving across the country. Mr Caldwell leaves 3 hours before Mrs Caldwell. If he averages 45 mph and she averages 65 mph, how many hours will it take Mrs Caldwell to catch mr. Caldwell
Mrs. Caldwell will travel 135 miles at an additional 20 mph to catch up to Mr. Caldwell. Therefore, it will take Mrs. Caldwell 6.75 hours to catch up to Mr. Caldwell.
Explanation:This is a rate time distance problem in mathematics, typically learned in middle school. To calculate how long it will take Mrs. Caldwell to catch up with Mr. Caldwell, we need to compare the distance traveled by each person in the same time. Because rate equals distance over time (r=d/t), we know that the distance each person traveled is rate x time.
Mr. Caldwell left 3 hours before Mrs. Caldwell, so he traveled at 45 mph for 3 hours, or 135 miles. Once Mrs. Caldwell leaves, she needs to cover these 135 miles at a faster speed to catch up. Her speed is 20 mph greater than Mr. Caldwell’s. We divide the distance that Mr. Caldwell has covered (135 miles) by the difference in their speeds (20 mph) to find it will take Mrs. Caldwell 6.75 hours to catch up to him.
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A pure acid measuring x liters is added to 300 liters of a 20% acidic solution. The concentration of acid, f(x), in the new substance is equal to the liters of pure acid divided by the liters of the new substance, or . Which statement describes the meaning of the horizontal asymptote? The greater the amount of acid added to the new substance, the more rapid the increase in acid concentration. The greater the amount of acid added to the new substance, the closer the acid concentration is to one-fifth. As more pure acid is added, the concentration of acid approaches 0. As more pure acid is added, the concentration of acid approaches 1.
Answer:
the answer is d
Step-by-step explanation:
Why is the law of cosines a stronger statement than the pythagorean theorem?
Answer:
Answer in explanation
Step-by-step explanation:
The two laws are mathematical laws which are used in navigating problems which involves triangles. While the Pythagorean theorem is used primarily and exclusively for right angled triangle, the cosine rule is used for any type of triangle.
So, why is the cosine rule a stronger statement? The reason is not far fetched. As said earlier, the cosine rule can be used to resolve any triangle type while the Pythagorean theorem only works for right angled triangle. In fact, we can say the Pythagorean theorem is a special case of cosine rule. The reason why the expression is different is that, for the expression, cos 90 is zero, which thus makes our expression bend towards the Pythagorean expression view.
The explanation regarding the law of cosines is the stronger statement if compared with the Pythagorean theorem is explained below.
Difference between the law of cosines be the stronger statement if compared with the Pythagorean theorem:The Pythagorean theorem is used when there is the right-angled triangle, while on the other hand, the cosine rule is used for any type of triangle. Here the Pythagorean theorem should be considered for the special case of cosine rule. Due to this the cosine law should be stronger if we compared it with the Pythagorean theorem.
Learn more about cosine here;https://brainly.com/question/16299322
PLZ HELPPPPPPPPPPPPPPPP
At 9:00, Paula has x cups of food in a container. Pamula pours 2 1/2 cups of food into the container. Then she removes 3/4 cups of food to feed her dog. Now there are 5 1/4 of food in the container. What is the equation?
Answer:
The equation representing the scenario is [tex]x+\frac{5}{4}-\frac{3}{4} = \frac{21}{4}[/tex].
Step-by-step explanation:
Given:
Amount of food in the container = [tex]x \ cups[/tex]
Amount of food added in the container = [tex]2\frac12\ cups[/tex]
We will now convert the mixed fraction into Improper fraction by Multiplying the whole number part by the fraction's denominator and then add that to the numerator and then write the result on top of the denominator.
[tex]2\frac12\ cups[/tex] can be rewritten as [tex]\frac{5}{4}\ cups[/tex]
Amount of food added in the container = [tex]\frac{5}{4}\ cups[/tex]
Amount of food removed from container to feed dog = [tex]\frac34 \ cups[/tex]
Amount of food remaining = [tex]5\frac{1}{4} \ cups[/tex]
[tex]5\frac{1}{4} \ cups[/tex] can be Rewritten as = [tex]\frac{21}{4}\ cups[/tex]
Amount of food remaining = [tex]\frac{21}{4}\ cups[/tex]
We need to write the equation for above scenario.
Solution:
Now we can say that;
Amount of food remaining is equal to Amount of food in the container plus Amount of food added in the container minus Amount of food removed from container to feed dog.
framing in equation form we get;
[tex]x+\frac{5}{4}-\frac{3}{4} = \frac{21}{4}[/tex]
Hence the equation representing the scenario is [tex]x+\frac{5}{4}-\frac{3}{4} = \frac{21}{4}[/tex].
HELP HOW DO I FIND THE B VALUE OF THIS
Answer:
b = [tex]\frac{8}{3}[/tex]
Step-by-step explanation:
period = [tex]\frac{2\pi }{b}[/tex], that is
b = [tex]\frac{2\pi }{period}[/tex] = [tex]\frac{2\pi }{\frac{3\pi }{4} }[/tex] = 2π × [tex]\frac{4}{3\pi }[/tex] = [tex]\frac{8}{3}[/tex]
Answer:
f(x) = 4cos(8/3)x - 3.
The missing space is 8/3.
Step-by-step explanation:
The general form is f(x) = Acosfx + B where A = the amplitude, f = frequency and B is the vertical shift..
Here A is given as 4, B is - 3 and the frequency f = 2 π / period =
2π / (3π/4)
= 8/3.
So the answer is f(x) = 4cos(8/3)x - 3.
(1 point) A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at a rate of 4 feet per second, how fast is the circumference changing when the radius is 18 feet?
Answer:
8pi feet per second
Or, 25.1 feet per second (3 sf)
Step-by-step explanation:
C = 2pi×r
dC/dr = 2pi
dC/dt = dC/dr × dr/dt
= 2pi × 4 = 8pi feet per second
dC/dt = 25.1327412287
sin= 5/13, and cos b= 3/5, evaluate cos(a-b).
cos (a - b) is 56/65
Step-by-step explanation:
Step 1:Given sin a = 5/13, find cos a.
sin a = opposite side/hypotenuse = 5/13
The adjacent side can be found using Pythagoras Theorem.
Hypotenuse² = Opposite Side² + Adjacent Side²
⇒ Adjacent side² = Hypotenuse² - Opposite Side²
= 13² - 5² = 169 - 25 = 144
∴ Adjacent Side = 12
⇒ cos a = adjacent side/hypotenuse = 12/13
Step 2:Given cos b = 3/5, find sin b.
cos b = adjacent side/hypotenuse = 3/5
The opposite side can be found using Pythagoras Theorem.
Hypotenuse² = Opposite Side² + Adjacent Side²
⇒ Opposite side² = Hypotenuse² - Adjacent Side²
= 5² - 3² = 25 - 9 = 16
∴ Opposite Side = 4
⇒ sin b = opposite side/hypotenuse = 4/5
Step 3:Find cos(a - b).
cos(a - b) = cos a cos b + sin a sin b
= 12/13 × 3/5 + 5/13 × 4/5
= 36/65 + 20/65 = 56/65
Look at the proof. Name the postulate you would use to prove the two triangles are congruent.
A. AAA Postulate
B. SSS Postulate SAS
C. SAS Postulate
Answer:
Option C, SAS Postulate
Step-by-step explanation:
I think that it is option C because it does not give you 3 angles or 3 sides, it gives you 2 angles and 1 side.
Answer: Option C, SAS Postulate
PLZ HELP WILL MARK BRAINLIEST
Using the distance formula, d = √(x2 - x1)2 + (y2 - y1)2, what is the distance between point (3, 2) and point (5, 4) rounded to the nearest tenth?
5.3 units
1 unit
10 units
2.8 units
Answer:
2.8 units
You should get 2√(2) or √(8) which is less than 3 and greater than 2.
Answer: 2.8 units
Step-by-step explanation:
The formula for determining the distance between two points on a straight line is expressed as
Distance = √(x2 - x1)² + (y2 - y1)²
Where
x2 represents final value of x on the horizontal axis
x1 represents initial value of x on the horizontal axis.
y2 represents final value of y on the vertical axis.
y1 represents initial value of y on the vertical axis.
From the given points
x2 = 5
x1 = 3
y2 = 4
y1 = 2
Therefore,
Distance = √(5 - 3)² + (4 - 2)²
Distance = √2² + 2² = √4 + 4 = √8
Distance = 2.8 units
Find a degree 3 polynomial with real coefficients having zeros 3 and 3−3i and a lead coefficient of 1. Write P in expanded form.
Answer:
P = x³ − 9x² + 36x − 54
Step-by-step explanation:
Complex roots come in conjugate pairs. So if 3−3i is a zero, then 3+3i is also a zero.
P = (x − 3) (x − (3−3i)) (x − (3+3i))
P = (x − 3) (x − 3 + 3i) (x − 3 − 3i)
P = (x − 3) ((x − 3)² − (3i)²)
P = (x − 3) ((x − 3)² + 9)
P = (x − 3)³ + 9 (x − 3)
P = x³ − 9x² + 27x − 27 + 9x − 27
P = x³ − 9x² + 36x − 54
My Notes Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution. (Enter your answer using interval notation.)t(t−4)y"+3ty'+4y=2,y(3)=0,y'(3)=−1
Answer:
The answer to the question is
The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is (-∞, 4)
Step-by-step explanation:
To apply look for the interval, we divide the ordinary differential equation by (t-4) to
y'' + [tex]\frac{3t}{t-4}[/tex] y' + [tex]\frac{4}{t-4}[/tex]y = [tex]\frac{2}{t-4}[/tex]
Using theorem 3.2.1 we have p(t) = [tex]\frac{3t}{t-4}[/tex], q(t) = [tex]\frac{4}{t-4}[/tex], g(t) = [tex]\frac{2}{t-4}[/tex]
Which are undefined at 4. Therefore the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution, that is where p, q and g are continuous and defined is (-∞, 4) whereby theorem 3.2.1 guarantees unique solution satisfying the initial value problem in this interval.
The existence and uniqueness theorems for ODEs determine that the longest interval where the initial value problem has a unique and twice-differentiable solution is (0, 4), avoiding discontinuities at t=0 and t=4.
Explanation:The initial value problem provided is a second-order linear ordinary differential equation (ODE) of the form:
t(t-4)y"+3ty'+4y=2, with initial conditions y(3)=0 and y'(3)=-1.
To determine the longest interval in which the solution is guaranteed to be unique and twice-differentiable, we need to consider the existence and uniqueness theorems for ODE's, which are predicated on the functions of the equation being continuous over the interval considered. Here, the coefficients of y" and y' are t(t-4) and 3t respectively. The problematic points occur where the coefficient of y" is zero because it will make the equation not well-defined, which occurs at t=0 and t=4. Therefore, the longest interval around the initial condition t=3 that avoids these points is (0, 4). Within this interval, the coefficients are continuous, and hence, the conditions for the existence and uniqueness of the solution are satisfied.
y = x + 2y = -2x + 2y = -3x + 2y = -5x + 2y = -
3
2
x + 2y = -
5
2
x + 2y = -x + 2y = 2x + 2y = 5x + 2y =
5
2
x + 2
0
Answer:
i need more context
Step-by-step explanation:
When I count as a principal of $1000 and earns 4% simple interest per year and other account as a principal $1000 and earns 4% interest compounded annually which account has the greater balance at the end of four years
Answer: the account that earned compound interest has the greater balance at the end of four years.
Step-by-step explanation:
The formula for determining simple interest is expressed as
I = PRT/100
Where
I represents interest paid on the amount invested.
P represents the principal or amount invested.
R represents interest rate
T represents the duration of the investment in years.
From the information given,
P = 1000
R = 4%
T = 4 years
I = (1000 × 4 × 4)/100 = 160
Total amount earned is
1000 + 160 = $1160
The formula for determining compound interest is expressed as
A = P(1+r/n)^nt
Where
A = total amount in the account at the end of t years
r represents the interest rate.
n represents the periodic interval at which it was compounded.
P represents the principal or initial amount deposited
From the information given,
P = 1000
r = 4% = 4/100 = 0.04
n = 1 because it was compounded once in a year.
t = 4 years
Therefore,.
A = 1000(1+0.04/1)^1 × 4
A = 1000(1.04)^4
A = $1170
Which inequality can Josh use to determine x, the minimum number of visits he needs to earn his first free movie ticket?
Answer:
3.5x + 15 ≥ 55
Step-by-step explanation:
I think the question below contains the missing information.
Josh has a rewards card for a movie theater. - He receives 15 points for becoming a rewards card holder. - He earns 3.5 points for each visit to the movie theatre. - He needs at least 55 points to earn a free movie ticket. Which inequality can Josh use to determine x, the minimum number of visits he needs to earn his firs free movie ticket?
My answer:
Becoming a member = 15 pointsVisiting the moving theater = 3.5 pointsTotal points needed for a free movie ticket = 55Let x is the number of times he visits = 3.5x
Total points = Points received on becoming a member + Points received on x visits
So,
Total Points = 15 + 3.5x
We know the total points must be at least 55 for a free movie ticket. This can be expressed as:
3.5x + 15 ≥ 55
A right cylindrical solid is cut in half to form the figure shown. If the length is 20 cm and the diameter is 8 cm, what is the surface area?
(80π + 160) cm2
(96π + 160) cm2
320π cm2
(320π + 160) cm2
Answer:
(96π + 160) cm2
Step-by-step explanation:
What are the solutions to the system of equations?
{y=2x2−8x+5
{y=x−2
Final answer:
To find the solutions to the system of equations, use the substitution method. The solutions are (1/2, -3/2) and (7, 5).
Explanation:
To find the solutions to the system of equations, we can use the substitution method. First, solve one of the equations for y in terms of x. Let's solve the second equation for y:
y = x - 2
Now substitute this expression for y into the first equation:
x - 2 = 2x^2 - 8x + 5
Now we have a quadratic equation. Rearrange it into standard form:
2x^2 - 9x + 7 = 0
Next, factor the quadratic equation:
(2x - 1)(x - 7) = 0
Set each factor equal to zero and solve for x:
2x - 1 = 0, x - 7 = 0
x = 1/2, x = 7
Now substitute these values of x back into either of the original equations to find the corresponding values of y:
For x = 1/2: y = 1/2 - 2 = -3/2
For x = 7: y = 7 - 2 = 5
So the solutions to the system of equations are (1/2, -3/2) and (7, 5).
Jerry manages a local car dealership. At the beginning of the month, his lot had m vehicles. During the month his salesman sold n vehicles, and he purchased p vehicles more. How many vehicles did the dealership have at the end of the month?
Answer:
m+p-n
Step-by-step explanation:
given that Jerry manages a local car dealership. At the beginning of the month, his lot had m vehicles. During the month his salesman sold n vehicles, and he purchased p vehicles more
We are to find the number of vehicles did the dealership have at the end of the month
At the end of the month the dealer would have
no of vehicles at the start of the month- sales of the vehicle in that month+Purchase of vechicles during that month
No of vehicles at the start of the month = m
Purchase during month =p
Total vehicles including purchase = m+p
LESS: Vehicles sold in the month = n
No of vehicles at the end = m+p-n
Trevor Once to buy a car that cost 23600 he has 5000 for down payment how much more will Trevor O the car right solve and create an equation for his situation define the variable
Answer:
5000 + x = 23600
Step-by-step explanation:
a car that cost = 23600
down payment = 5000
So he needs to pay: 23600 - 5000 = 18600 more to get the car
Let x represent the amount he needs to pay more, an equation for his situation:
5000 + x = 23600
What do you know about the solution(s) to the system of equations?
A. There is no solution.
B. The solution is (2,0).
C. The solution is (0,−1).
D. There are infinitely many solutions.
Answer:
A because the linesnever cross.
Step-by-step explanation:
Answer:
There is no solution
Step-by-step explanation:
A scientist measured the exact distance between two points on a map and came up with the following number: 0.04000 km.
Which digits are the significant figures in this measurement?
Explain your answer.
Answer:
The first zero after decimal point and 4 only
Step-by-step explanation:
Despite having 5 decimal points, the rules of significant figures dictate that unless there is a digit other than zero after, the only significant numbers are those that come before zero. For this case, the significant digits are only 0.04 but if it was 0.0400005 then all the other zeros would have also be considered significant.
100 pyramid shaped chocolate candies with a square base of 12 mm size and height of 15 mm are melted in a cylinder coil pot if the part has a radius of 75 mm what is the height of the melted candies in the pot.
Answer: the height of the melted candies in the pot is 12.2 mm
Step-by-step explanation:
The formula for determining the volume of a square base pyramid is expressed as
Volume = area of base × height
Area of the square base = 12² = 144 mm²
Volume of each pyramid = 15 × 144 = 2160 mm³
The volume of 100 pyramid shaped chocolate candies is
2160 × 100 = 216000 mm³
The formula for determining the volume of a cylinder is expressed as
Volume = πr²h
Since the pyramids was melted in the cylindrical pot whose radius is 75 mm, it means that
216000 = 3.14 × 75² × h
17662.5h = 216000
h = 216000/17662.5
h = 12.2 mm
Answer:
The height of the melted candies in the pot is 4.07mm
Step-by-step explanation:
H= 100*1/3(12)^2(15)/π(75)^2=64/5π=4.07
If Naomi were to paint her living room alone, it would take 5 hours. Her sister Jackie could do the job in 8 hours. How many hours would it take them working together? Express your answer as a fraction reduced to lowest terms, if needed.
Answer:
40/13
The decimal form is going to be 3.076
Select the correct answer. Solve -9 2/7 -(-10 3/7) . A. -1 1/7 B. 1 1/7 C. 19 1/7 D. 19 5/7
Answer:
B. 1 1/7
Step-by-step explanation:
-9 2/7-(-10 3/7)
=-9 2/7+10 3/7
=1 1/7
Therefore, B. 1 1/7
Answer:
The answer is B
Step-by-step explanation:
B. 1 1/7
Tara bought Three boxes of dog treats with 40 truth in each box two boxes of cat treats with 20 trees in each box simplify the expression below to find the total number of trees are bought
Answer:
Tara bought a total of 160 treats.
Step-by-step explanation:
We are given the following in the question:
Number of boxes of dog treats = 3
Number of treats in each dog box = 40
Total number of treats in dog box =
[tex]40 \times 3 = 120[/tex]
Number of boxes of cat treats = 2
Number of treats in each cat box = 20
Total number of treats in cat box =
[tex]20\times 2 = 40[/tex]
Total number of treats Tara brought =
Total number of treats in dog box + Total number of treats in cat box
[tex](40\times 3)+(20\times 2)\\= 120 + 40\\=160[/tex]
Thus, Tara bought a total of 160 treats.
In order to develop a more appealing cheeseburger, a franchise uses taste tests with 15 different buns, 8 different cheeses, 3 types of lettuce, and 4 types of tomatoes. If the taste tests were done at one restaurant by one tester who takes 10 minutes to eat each cheeseburger, approximately how long would it take the tester to eat all possible cheeseburgers?
Answer:
There would be
13 x 8 x 4 x 3 = 3,276 different cheeseburger combinations
If the taste tester takes 10 minutes to eat a cheeseburger, then it would take him
3276 x 10 = 32760 minutes
Eating round the clock, it would take him
32760 / 60 = 546 hours
546 / 24 = 22 days 18 hours
Now these are the numbers I'm seeing
1313 buns
88 cheese
44 lettuces
33 tomatoes
There would be
1313 x 88 x 44 x 33 = 167,769,888 different cheeseburger combinations
If the taste tester takes 10 minutes to eat a cheeseburger, then it would take him
167769888 x 10 = 1,677,698,880 minutes
1677698880 / 60 = 27,961,647 hours
27961647 / 24 = 1,165,068.62 days
1165068.62 / 365 = 3,191.97 years
Step-by-step explanation:
A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec2 is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10.
What is the total distance the car travels in this 30 second interval? Your must show your work but you may use your calculator to evaluate. Give 3 decimal places in your answer and include units.
Im not really sure how to go about this? Would I use the trapezoidal rule i dont know please help.
Answer:
666.667 feet
Step-by-step explanation:
Slope = -1
Intercept = 10
y = -t + 10
y is the acceleration
Integrate y fornv
v = -t²/2 + 10t + c
At t=0, v=0 so c = 0
v = -t²/2 + 10t
Turns when v = 0,
-t²/2 + 10t = 0
t = 0, 20
Integrate v for s
s = -t³/6 + 5t² + c
At t = 0, s = 10
10 = c
s = -t³/6 + 5t² + 10
s at t=30,
-(30³)/6 + 5(30)² + 10
= 10m
(Back to starting point)
At t = 20,
Displacement in
-(20³)/6 + 5(20)² + 10
= 343.333
Total distance = 2(343.333-10)
= 666.6667
The total distance the car travels in this 30 second interval is 1333.34 units
From the graph, we have the following points
(0, 10) and (10, 0).
Start by calculating the slope (m) of the graph
[tex]m = \frac{y_2 - y_1}{x_2 -x_1}[/tex]
So, we have:
[tex]m = \frac{0 - 10}{10-0}[/tex]
[tex]m =- \frac{10}{10}[/tex]
[tex]m =- 1[/tex]
The equation is then calculated as:
[tex]y = m(x -x_1) + y_1[/tex]
This gives
[tex]y = -1(x -0) + 10[/tex]
[tex]y = -1x+ 10[/tex]
[tex]y = -x+ 10[/tex]
The above equation represents the acceleration (y) as a function of time (x).
Integrate to get the velocity (v)
[tex]v = -\frac{x\²}{2} + 10x + c[/tex]
From the question, we have:
The velocity (v) of the car is 0, when the time (x) is 0.
So, we have:
[tex]0 = -\frac{0\²}{2} + 10(0) + c[/tex]
This gives
[tex]c = 0[/tex]
So, the equation becomes
[tex]v = -\frac{x\²}{2} + 10x + 0[/tex]
[tex]v = -\frac{x\²}{2} + 10x[/tex]
Set v = 0.
So, we have:
[tex]-\frac{x\²}{2} + 10x = 0[/tex]
Multiply through by -2
[tex]x^2 -20x = 0[/tex]
Factorize
[tex]x(x -20) = 0[/tex]
Split
[tex]x = 0\ or\ x -20 = 0[/tex]
Solve for x
[tex]x = 0[/tex] or [tex]x = 20[/tex]
Integrate velocity (v) to get the displacement (d)
[tex]v = -\frac{x\²}{2} + 10x[/tex]
[tex]d = -\frac{t\³}{6} + 5t\² + c[/tex]
From the question, we have:
The position (d) of the car is 10, when the time (x) is 0.
So, we have:
[tex]-\frac{(0)\³}{6} + 5(0)\² + c = 10[/tex]
[tex]c = 10[/tex]
So, the equation becomes
[tex]d = -\frac{t\³}{6} + 5t\² + 10[/tex]
The position at 30 seconds is:
[tex]d = -\frac{(30)\³}{6} + 5(30)\² + 10[/tex]
[tex]d = 10[/tex]
The position at 20 seconds is:
[tex]d = -\frac{(20)\³}{6} + 5(20)\² + 10[/tex]
[tex]d = 676.667[/tex]
The total distance is then calculated as:
[tex]Total = 2 \times (d_2 -d_1)[/tex]
This gives
[tex]Total = 2 \times (676.667 -10)[/tex]
[tex]Total = 2 \times 666.667[/tex]
[tex]Total = 1333.34[/tex]
Hence, the total distance is 1333.34 units
Read more about distance at:
https://brainly.com/question/2239252
One number is 20 times another number. The product of the two numbers is 180. Write an equation and use it to find all pairs of numbers that satisfy the
Answer: the Smaller number is 3
The larger number is 60
Step-by-step explanation:
Let x represent the smaller of the numbers.
Let y represent the larger number.
One number is 20 times another number. This means that
y = 20x
The product of the two numbers is 180. This means that
xy = 180- - - - - - - - - -1
Substituting y = 2x into equation 1, it becomes.
x × 20x = 180
20x² = 180
x² = 180/20 = 9
x = √9
x = 3
y = 20x = 20 × 3
y = 60