A police car traveling south toward Sioux Falls, Iowa, at 160km/h pursues a truck east away from Sioux Falls at 140 km/h. At time t=0, the police car is 60km north and the truck is 50km east of Sioux Falls. Calculate the rate at which the distance between the vehicles is changing at t=10 minutes, (Use decimal notation. Give your answer to three decimal places)
Answers
The rate of change of the distance between the police car and the truck after 10 minutes is approximately 193.66 m/s
What is a rate of change of a function?The rate of change of a function is the rate at which the output of the function is changing with regards to the input.
The velocity of the police car = 160 km/h south
The velocity of the truck = 140 km/h east
Distance of the police car from Sioux Falls = 60 km north
Distance of the truck from Sioux Falls = 50 km east
Required;
The rate at which the distance between the vehicles is changing at t = 10 minutes
Solution;
Let d represent the distance between the vehicles, we have;
d² = x² + y²
Where;
x = The distance of the truck from Sioux falls
y = The distance of the police car from Sioux Falls
Which gives;
[tex] \displaystyle{ \frac{d}{dt} d^2= \frac{d}{dt}(x^{2}) +\frac{d}{dt} (y^{2}) }[/tex]
Which gives;
[tex] \displaystyle{ 2 \cdot d \cdot \frac{d}{dt} d=2 \cdot x \cdot \frac{dx}{dt} + 2 \cdot y \cdot\frac{dy}{dt} }[/tex]
After 10 minutes, we have;
y = 60 - (10/60)×160 = 100/3
x = 50 + (10/60)×140 = 220/3
d = √((100/3)² + (220/3)²) = 20•(√146)/3
[tex] \displaystyle{ \frac{d}{dt} d=\frac{2 \cdot x \cdot \frac{dx}{dt} + 2 \cdot y \cdot\frac{dy}{dt} }{2 \cdot d }}[/tex]
Which gives;
[tex] \displaystyle{ \frac{d}{dt} d= \frac{2 \times \frac{220}{3} \times 140 + 2 \times \frac{100}{3} \times 160}{2 \times \frac{20 \times \sqrt{146}}{3}}}[/tex]
[tex] \displaystyle{ \frac{2 \times \frac{220}{3} \times 140 + 2 \times \frac{100}{3} \times 160}{2 \times \frac{20 \times \sqrt{146}}{3}}\approx 193.66}[/tex]
Therefore;
[tex] \displaystyle{ \frac{d}{dt} d \approx 193.66}[/tex]
The rate of change of the distance between the vehicles with time, [tex] \displaystyle{ \frac{d}{dt} d}[/tex] after 10 minutes is approximately 193.660 m/sLearn more about rate of change of a function here:
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Related Questions
Which equation has the same value as X 2/3(6x+12)=-24
Answers
The equations that has the same value as x in 2/3(6x+12)=-24 are 4x+8 = -24 and 4x = -32 , option(a) and (e) are correct .
In the question ,
the equation 2/3(6x+12)=-24 is given
on solving for x ,we get
4x+8 = -24
4x = -24-8
4x = -32
x = -8 .
Solving option(a)
4x+8 = -24
4x = -24-8
4x = -32
x = -8
Solving for option(b)
9x+18 = -24
9x = -24-18
9x = -42
x = -42/9
solving for option(c)
4x = -16
x = -4
solving for option(d)
(18x+36)/2 = -24
18x + 36 = -48
18x = -48-36
18x = -84
x = -84/18
solving for option(e)
4x = -32
x = -32/4
x = -8
we can see that only option (a) and option(e) , given the value of x as -8 .
Therefore , the equations that has the same value as x in 2/3(6x+12)=-24 are 4x+8 = -24 and 4x = -32 , option(a) and (e) are correct .
The given question is incomplete , the complete question is
Which equation has the same value as X 2/3(6x+12)=-24 ?
Select two options
(a) 4x+8 = -24
(b) 9x+18 = -24
(c) 4x = -16
(d) (18x+36)/2 = -24
(e) 4x = -32
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8.12 Midpoint formula: find the endpoint EUW
Learn with an example
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The midpoint of PQ is M(4, 0). One endpoint is P(6, 0). Find the coordinates of the other
endpoint Q.
Write the coordinates as decimals or integers.
‹=OD
Work
Answers
The coordinates of the other endpoint Q.=(2,0)
How to calculate the coordinates of the other endpoint Q ?
Given:
Midpoint = PQ = M(4,0)
One endpoint = P = [tex](x_1,y_1)[/tex] = (6,0)
Other endpoint Q= [tex](x_2,y_2)[/tex]
We know that,
[tex]\text{Midpoint PQ}=(\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2} )\\\\=(\frac{6+x_2}{2} ,\frac{0+y_2}{2} )=(4,0)\\\\= > \frac{6+x_2}{2} =4\\\\= > 6+x_2= 8\\\\= > x_2=2[/tex]
[tex]\text{Similarly},\\\\\frac{0+y_2}{2} =0\\\\= > y_2=0[/tex]
So, the Other endpoint Q=(2,0)
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pnereThe radius, R, of a sphere is 5.8 m. Calculate the sphere's volume, V.Use the value 3.14 for it, and round your answer to the nearest tenth. (Do not round any intermediate computations.)
Answers
The volume of a sphere is given by the following formula:
[tex]V=\frac{4}{3}\pi r^3[/tex]Where r is the radius of the sphere r=5.8m.
Replace this value and solve for V:
[tex]\begin{gathered} V=\frac{4}{3}\cdot3.14\cdot(5.8m)^3 \\ V=\frac{4}{3}\cdot3.14\cdot195.112m^3 \\ V=\frac{2450.61m^3}{3} \\ V=816.9m^3 \end{gathered}[/tex]The volume of the sphere is 816.9 cubic meters.
Help me pls it would be nice thank youuuuu
Answers
X: -2, -1, 0, 1, 2
Y: -2, 1, 2, 1, -2
Hopefully this is helpful
can someone help me? thank you so much
Answers
image has the answers
(3z + 2) (6-z) = 0 is there more then one solution to this if so can you tell me and use commas?
Answers
Answer:
Yes, [tex]z=-\frac{2}{3}, 6[/tex]
Step-by-step explanation:
Using the zero-product property,
[tex]3z+2=0, 6-z=0 \\ \\ \implies z=-\frac{2}{3}, 6[/tex]
g (x)=5x-5,find g (-6)
Answers
Answer: -35
Step-by-step explanation:
set the every x in the equation to (-6)
g(-6) = 5(-6)-5
g(-6) = -30-5
g(-6) = -35
Please answer this quickly, I don’t need no explanation or work, just the letter, thank you!
Answers
Solution:
Given the graph;
If a vertical line intersects a curve on an xy-plane more than once then for one value of x the curve has more than one value of y, and so, the curve does not represent a function.
ANSWER: B
Find the length of side QS. Round to the nearest tenth.Q39Р16°R28°S
Answers
QS = 20.7
Explanation:To get length of side QS, we will consider triangle SQR (right angle triangle)
opposite (the side opposite the angle 28°) = QS
adjacent = base = 39
We apply tangent ratio:
tan28 = opposite/adjacent
tan28 = QS/39
QS = 39 × tan28
QS = 39 × 0.5317
QS = 20.7363
To the nearest tenth, QS = 20.7
Jacob bought a book that cost $11.95 and magazines that cost $4.95 each. Isabella bought a book that cost $15.79 and magazines that cost $3.99 each. Jacob spent the same amount or less money than Isabella. Write and solve an inequality to find m, the number of magazines for which Jacob spent less than or the same amount as Isabella.
Answers
Answer:
16.90≤19.78
Step-by-step explanation:
Given any triangle ABC labeled as shown, the law of sines states: B B AN b sino sin 4 sin B А. sin 4 sin B В. sino sin 4 sin B C. sino a sin A sino sin B D.
Answers
Answer: According to the law of sines, we have the following:
Therefore, according to this, the answer is Option (B).
36. Hiking You hiked 8.3 miles in Denali
National Park which is 3 miles farther than
you hiked yesterday. How far did you hike
yesterday?
Answers
Answer:
You hiked 5.3 miles yesterday
An investment offers a total return of 17 percent over the coming year. Powell Arms thinks the total real return on this investment will be only 12 percent. What does Powell believe the inflation rate will be over the next year?
Answers
Powell believe that the inflation rate will be 4.46% over the next year .
In the question ,
it is given that
total return offered by the investment((nominal rate) = 17% = 0.17
return according to Powell Arms(real rate) = 12% = 0.12
let the inflation rate be x.
The inflation rate over the next year can be calculated using the formula .
(1+Real rate ) = (1+nominal rate )/ (1+ inflation rate )
Substituting the values , we get
(1+0.12) = (1+0.17)/(1+x)
1.12 = 1.17/(1+x)
1+x = 1.17/1.12
1 + x = 1.0446
x = 1.0466-1
x = 0.0446
x = 4.46%
Therefore , Powell believe that the inflation rate will be 4.46% over the next year .
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Given the right triangle ABC with altitude BD drawn to the hypotenuse AC. If AC=6 and DC=4, what is the length of BC in simplest radical form ?
Answers
This problem is an application of the Geometric mean theorem. It says that
[tex]\frac{6}{x}=\frac{x}{4}[/tex]Comment: In other words, it says that the length of BC (x) is the geometric mean between the lengths of AC and DC.
Then,
[tex]x^2=6\cdot4=24[/tex][tex]x=\sqrt[]{24}=2\cdot\sqrt[]{6}[/tex]................................................................................................................................................................
Let's talk a little about the simplest radical form of a square root
[tex]\sqrt[]{a}[/tex]The first step to finding it is to write the number within the root as a product of prime powers, such product is called its integer factorization. Let's do that for 24:
Then, the integer factorization of 24 is
[tex]24=2^3\cdot3[/tex]Thus,
[tex]\sqrt[]{24}=\sqrt[]{2^3\cdot3}[/tex]The idea now is to take out of the root all we can. The rule is that we can only take out powers of 2 (for our root is a square root). In the expression
[tex]2^3\cdot3[/tex]There is only one power of 2, within 2^3. We can write it as
[tex]2^2\cdot2\cdot3[/tex]How are we going to take out it? We are going to take out the base of the power, which is 2 in this case. Then,
[tex]\sqrt[]{24}=2\cdot\sqrt[]{2\cdot3}=2\cdot\sqrt[]{6}[/tex]In simple terms, the simplest radical form of a root is what results after taking out the root all that can be taken out.
The answer has to be a geometric proof. Thank you!
Answers
Given data:
The given triangle in which AD is on perpendicular bisector on BC.
In triangle ABD and ACD.
[tex]\begin{gathered} \angle ADB=\angle\text{ADC}=90^{\circ} \\ BD=CD(\text{given)} \\ AD=AD\text{ (common)} \\ \Delta ABD\cong\Delta ACD(\text{SAS)} \end{gathered}[/tex]Simmilary triangle BED and triangle CED.
[tex]\begin{gathered} \angle BDE=\angle CDE \\ BD=CD \\ ED=ED \\ \Delta BED\cong\Delta CED(SAS) \end{gathered}[/tex]The fisr expression can be written as,
[tex]\begin{gathered} \Delta ABD\cong\Delta ACD \\ \Delta\text{ABE}+\Delta BED\cong\Delta ACE+\Delta\text{CED} \end{gathered}[/tex]Substitute CED in place of BED.
[tex]\begin{gathered} \Delta ABE+\Delta CED\cong\Delta ACE+\Delta CED \\ \Delta ABE\cong\Delta ACE \end{gathered}[/tex]Thus, the triangle ABE is congruent to trriangle ACE.
For each scenario below, choose the graph that gives the best representation.
(a). The daily amount of calories that a rat eats increases steadily from birth to about age 3 months. Then, the amount levels off until about age 6 months, when it increases again.
(b). Jane leaves her house on her bike. She rides at a constant speed until she reaches a lemonade stand, where she parks her bike and takes a rest. Then she turns around and bikes home as fast as she can.
Answers
A) The second graph gives the best representation of the daily amount of calories consumed by the rat.
B)The fourth graph gives the best representation of Jane's journey.
The first slope of the graph represents the steady increase in the consumption of calories by the rat until the age of 3. The horizontal line represents the level off until the age of 6. The slopes after the horizontal line represent the increase in consumption of calories after age 6.
The first slope of the graph represents the steady increase in the distance from her home as she is traveling away from her home. The horizontal line that comes after the slope represents the time she rested at the lemonade stand. And the negative slope that comes after the horizontal line represents the decreasing distance to her home as she travels back to her home.
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The graph shows the mass of the bucket containing liquid depends on the volume of liquid in the bucket. Use the graph to find the range of the function.
Answers
From the graph, the range of the function in the graph is; 0 ≤ M ≤ 6.5
What is the range of the graph Function?
The range of a function is the set of all possible output values for which the function still exists.
Now, from the graph, we can see that it is a linear graph that starts on the vertical axis with a coordinate of approximately (0, 1) which denotes 1 kg when the volume is 0 liters.
Now, we see that the line of the graph stops at the coordinate (7.5, 6.5) which denotes 6.5 kg when the volume is 7.5 liters.
Therefore the maximum mass is 6.5 kg while the minimum is 0 kg. Thus,;
Range; (0 ≤ M ≤ 6.5)
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Find the constant of proportionality from a graph
Answers
Answer: 2
Step-by-step explanation:
The constant of proportionality is the same as the slope of the line. Using the slope formula with the points (0, 0) and (2, 4), [tex]\frac{4-0}{2-0}=2[/tex].
(03.06) 5.)Choose the point-slope form of the equation below that represents the line that passes through the point (6, -3) and has a slope of 1/2OPTIONS: A.) y-6 = 1/2(x+3)B.) y = 1/2x - 6C.) y+3 = 1/2(x-6)D.) y-2y = 12I NEED THIS DONE ASAP!
Answers
Answer the point-slope form of the equation below that represents the line that passes through the point (6, -3) and has a slope of 1/2
A linear equation can have the form
y = mx + b
The slope is m
The point-slope form is
(y- y1) = m(x -x1) (1)
the point (x1, y1) = (6, -3)
______________
Replacing in (1) the slope and the point (x1, y1)
(y- y1) = m(x -x1)
(y- (-3)) = 1/2(x -6)
y+3 = 1/2(x-6)
__________________
So, checking the options the correct answer is C
A.) y-6 = 1/2(x+3)
B.) y = 1/2x - 6
C.) y+3 = 1/2(x-6)
D.) y-2y = 12
A) graph the function: f(x) = -2^xB) domain of the function?C) range of the function?D) Equation of the asymptote?E) y-intercept of the graph?
Answers
We are given the following function:
[tex]y=-2^x[/tex]Part A. We are asked to draw the graph of the function. This is an exponential function with a negative sign, this means that the graph is reflected across the x-axis. Therefore, the graph is:
Part B. The domain of a function is the values that the fuction can take as an input. Since the function is an exponential function, it can take any value of "x" therefore, the domain is all the real numbers, we write this as follows:
[tex]D=(-\infty,\infty)[/tex]Part B. The range of a function is the values that the function outputs, The range of an exponential function are the values that are greater than zero, but since the given function is reflected across the x-axis, this means that the rage is the negative real numbers, therefore, the range is:
[tex]R=(-\infty,0)[/tex]Part D. For an exponential function of the form:
[tex]y=a(b^x)[/tex]The asymptote is x-axis, since zero is never an output of the function. Therefore the equation of the asymptote is:
[tex]y=0[/tex]Part E. The y-intercept is the value of the function when "x = 0", therefore, substituting in the function we get:
[tex]f(0)=-2^0[/tex]Solving the operations:
[tex]f(0)=-1[/tex]Therefore, the y-intercept is -1
What is the average rate of change of f(x), represented by the graph, over the interval [-1, 2]?
Answers
The estimate (to one decimal place) of the average rate of change f is 2.3
How to estimate the average rate of change f?The interval is given as
[-1, 2]
This can be rewritten as
x = -1 to x = 2
This can also be represented as
(a, b) = (-1, 2)
From the attached graph, we have
f(-1) = -5
f(2) = 2
The estimate (to one decimal place) of the average rate of change f is
Rate = [f(b) - f(a)]/[b - a]
This gives
Rate = [f(2) - f(-1)]/[2 + 1]
So, we have
Rate = [2 + 5]/[2 + 1]
Evaluate
Rate = 2.3
Hence, the average rate of change f is 2.3
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the product of 5 and y
Answers
To get the product we multiply.
[tex] = 5 \times y \\ = 5y[/tex]
Solve the system by substitution. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, enter the general solution in terms of x, where x is any real number.)3x − 2y = 545x + 10y = −30(x, y) =
Answers
Solution:
Given:
[tex]\begin{gathered} 3x-2y=54 \\ 5x+10y=-30 \end{gathered}[/tex]Using the graphical method,
The solution is the point of intersection of the two lines.
Therefore, the solution is;
[tex](x,y)=(12,-9)[/tex]Please help will give Brainliest!!
Kevin took $45 with him to spend on snack for himself and his friends at the movie theater. The price for each bucket of popcorn was $4. The price of each drink was half the price of a bucket of popcorn.
Sketch the graph that represents the situation and label the intercepts. Use one axis to represent the number of bucketw of popcorn and the other axis to represent the number of drinks.
Explain your graph.
Answer:
(Score for Question 2: ___ of 5 points)
Graph the absolute value function.
f(x)=|1/3 x+2|-4
Answer:
(Score for Question 3: ___ of 4 points)
(a) Graph a linear function of your choice. On the same graph, graph a linear function transformed 2 units up and 3 units down.
(b) What was the equation of your linear function in slope-intercept form?
(c) What was the equation of the transformed function in slope-intercept form?
Answers
The required answer would be the inequality 4x + 2y ≥ 45, the graph has been attached which represents the given situation.
What is inequality?Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
We have been given that the price for each bucket of popcorn was $4. The price of each drink was half the price of a bucket of popcorn.
The total amount of spend is $45
As per the given condition, the required inequality would be as
4x + 2y ≥ 45
The situation's representation is in the attached graph with labels for the intercepts.
Here y-axis represents the number of drinks, and the x-axis represents the number of buckets of popcorn.
Thus, the required answer would be the inequality 4x + 2y ≥ 45, the graph has been attached which represents the given situation.
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What is a number that when you divide it by 2 and subtract 3.8 from the quotient, you get 7?
Answers
Let the number x.
Then the number divided by 2 gives
[tex]\frac{x}{2}[/tex]Subtract 3.8 from the quotient gives
[tex]\frac{x}{2}-3.8[/tex]Hence,
[tex]\frac{x}{2}-3.8=7[/tex][tex]\begin{gathered} \frac{x}{2}-3.8=7 \\ \frac{x}{2}=7+3.8=10.8 \\ \Rightarrow x=2\times10.8=21.6 \end{gathered}[/tex]x = 21.6
3 sisters age combined are 57. Jenny is 6 years
older then Lynn. Kim is 5 less then twice the age of
Lynn age. What are each sisters ages
Answers
Answer: Jenny = 6 | Kim = 7 | (Unknown Sister because no name) = 44
Step-by-step explanation:
Jenny = 6 years (that we already know)
6 x 2 = 12 - 5 = 7 years for Kim
7 + 6 = 13
To find the unknown age of the last sister, we will subtract 57 - 13 = 44
So the last sister is 44.
(I am very sorry if this is wrong, I am only in Middle School).
Which of the following numbers is a square number? Check all that apply.A. 62B. 64C. 100D. 116
Answers
the answer
a perfect square is a number that when two exact same number multiply each other, it produce
example 2 * 2 = 4
4 is a square number and the square root is 2
[tex]\begin{gathered} \text{square root of } \\ a.62=\sqrt{62}=7.87 \\ b.64=\sqrt{64}=8 \\ c.100=\sqrt{100}=10 \\ d.116=\sqrt{116}=10.77 \end{gathered}[/tex]the answer to your question is 64 and 100 which is option B and C
Four hundred yards of fence is to be used to endose a rectangular area next to a straight river. The river bank acts as one side of the rectangle, and the fence is used to make the other three sides ofthe rectangle. Suppose the width w in yards of the rectangle is along the river bank.(a) Express the height of the rectangle in terms of w.b) Express the area of the rectangle in terms of w.
Answers
400 yards of fence = total fencing
400 = 2 widths + 2 heights
Since one side (width) is along the river, and the river bank acts as one side of the rectangle, the fence will be used in 3 sides:
400 = w+2h
Solving for h:
400-w=2h
(400-w)/2 = h
h= (400-w)/2 (a)
h= 200-1/2w (simplified)
To express the area:
Area of a rectangle : height x width
Since h = 200-1/2w
A = (200-1/2w) w
A = 200w-1/2w^2 (b)
Determining SimilarityAre the following triangles similar?YesNoExplain which similarity condition you used andjustify completely
Answers
Given data:
The given figure of the triangles.
In triangle STH and triangle GFH.
[tex]\begin{gathered} \angle T=\angle G \\ \angle THS=\angle GHF(vertically\text{ opposite)} \\ \end{gathered}[/tex]Thus, yes the triangles are similar by angle-angle (AA) property.
Solve the following equation 12+5x=2x+3
Answers
Answer:
-3
Step-by-step explanation:
im assuming u want to find x
1. move all the numbers with x to the left side to make life easier
2. 5x-2x= 3x
3. 3-12= -9
4. you want the x to be solo
5. -9/3 = -3