Two cars leave the same parking lot, one heading north and the other east. After several minutes, the eastbound car traveled 5 kilometers. If the two cars are now a straight-line distance of 13 kilometers apart, how far has the northbound car traveled?
Answers
Based on the information given, we can use the Pythagorean theorem to determine the distance traveled by the northbound car.
Let's denote the distance traveled by the northbound car as 'x' kilometers.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse (the straight-line distance between the two cars) is equal to the sum of the squares of the other two sides (the distances traveled by each car).
In this case, the northbound car's distance is 'x' kilometers and the eastbound car's distance is 5 kilometers.
So we have the equation:
x^2 + 5^2 = 13^2
Simplifying, we get:
x^2 + 25 = 169
Subtracting 25 from both sides, we get:
x^2 = 144
Taking the square root of both sides, we get:
x = 12
So the northbound car has traveled 12 kilometers.
Answer: 12 km
Step-by-step explanation:
As seen in the figure the distance that the northbound car traveled equals to the distance from point N to the parking lot.
pythagorean: [tex]\sqrt{13^{2}-5^{2} }=12km[/tex]
Related Questions
Choose scales for the coordinate plane shown so that you can graph the points J (5, 50), K(3, 50), L (4, -40), M (-2, 40), and N (-5, -10). on the x-axis use a scale of ____ units for each grid square. on the y-axis use a scale of ____ units of each grid square. complete the explanation for using these scales for each axis. the x-coordinates range from ____ to ____, and the y-coordinates range from ____ to ____.
Answers
On the x-axis use a scale of 2 units for each grid square. on the y-axis use a scale of 10 units of each grid square. The x-coordinates range from -5 to 5. y-coordinates range from -40 to 50.
What is coordinate plane?In mathematics, points and functions are represented and graphed on the coordinate plane, commonly known as the Cartesian plane. The x-axis and y-axis are two perpendicular number lines that meet at the origin to form the plane (0,0).
On the coordinate plane, points are denoted by ordered pairs (x, y), where x denotes the point's separation from the y-axis and y denotes its separation from the x-axis.
For the given coordinates the range of x and y coordinates are:
x-coordinates range from -5 to 5
y-coordinates range from -40 to 50.
Thus, we use a scale of 2 on the x-axis and 10 units on the y axis.
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An economy is operating with output $400 billion above its natural level, and fiscal policymakers want to close this expansionary gap. The central bank agrees to adjust the money supply to hold the interest rate constant, so there is no crowding out. The marginal propensity to consume is 3/4, and the price level is completely fixed in the short run.
to close the expansionary gap, the government would need to increase or decrease spending by $ billion.
Answers
Hence, to close the expansionary gap, the government would need to decrease spending by $100 billion.
To close the expansionary gap, the government would need to decrease spending by $100 billion.
The formula to calculate the change in equilibrium output due to a change in government spending is:
∆Y = (∆G / (1 - MPC))
Where:
∆Y = change in equilibrium output
∆G = change in government spending
MPC = marginal propensity to consume
Here, the output is $400 billion above its natural level, and the central bank agrees to adjust the money supply to hold the interest rate constant, so there is no crowding out. Therefore, we can assume that the change in government spending (∆G) would have a one-to-one effect on the equilibrium output (∆Y).
Given that MPC = 3/4, we can plug in the values into the formula:
400 = (∆G / (1 - 3/4))
∆G = (1 - 3/4) * 400
∆G = (1/4) * 400
∆G = 100
Hence, to close the expansionary gap, the government would need to decrease spending by $100 billion.
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solve y''+y=t using laplace inverse with y(0)=1 and y'(0)=-2
Answers
The solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.
What is equation?Equation is a mathematical statement that expresses the equality of two expressions. It shows the relationship between two or more variables and can be written using symbols, numbers, and operations. Equations are used to describe physical laws, to make calculations, and to solve problems. Examples of equations include the Pythagorean theorem, Newton's laws of motion, and linear equations.
We solve this differential equation using Laplace inverse, with the initial conditions y(0)=1 and y'(0)=-2. First, we take the Laplace transform of the equation:
L[y''+y]=L[t]
Using the properties of Laplace transform, we can write this as:
s2Y(s)-sy(0)-y'(0)+Y(s)= (1/s)
Substituting the initial conditions and rearranging terms, we have:
Y(s)= (1/s) + (2/s2) + (1/s2)
We can then invert the Laplace transform to get the solution of the original equation:
y(t)= 1-2t+te-t
Therefore, the solution of the differential equation y'' + y = t with the initial conditions y(0)=1 and y'(0)=-2 is y(t)= 1-2t+te-t.
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find the area and perimeter of each figure below.
Answers
Answer:
finding the perimeter, you sumthe distance all round that is 7+7.5+17.8+6=38.3
38.3 is the perimeter
Elliot read a report from a previous year saying that 6 % 6%6, percent of adults in his city biked to work. He wanted to test whether this had changed, so he took a random sample of 240 240240 adults in his city to test H 0 : p = 0.06 H 0 :p=0.06H, start subscript, 0, end subscript, colon, p, equals, 0, point, 06 versus H a : p ≠ 0.06 H a :p =0.06H, start subscript, start text, a, end text, end subscript, colon, p, does not equal, 0, point, 06, where p pp is the proportion of adults in Elliot's city that bike to work. The sample results showed 21 2121 adults who biked to work, and the corresponding test statistic was z ≈ 1.79 z≈1.79z, approximately equals, 1, point, 79. Assuming that the necessary conditions are met, what is the approximate P-value for Elliot's significance test?
Answers
Answer: To find the P-value, we first need to determine the direction of the alternative hypothesis. Since the alternative hypothesis is two-tailed, we need to divide the significance level (α) by 2 before finding the critical values and the rejection region. Therefore, we have:
H0: p = 0.06
Ha: p ≠ 0.06
α = 0.05/2 = 0.025
The test statistic is z = 1.79, which represents the number of standard errors that the sample proportion is from the hypothesized population proportion under the null hypothesis. We can find the P-value by looking up the area in the standard normal distribution table beyond the test statistic in both tails:
P-value = P(Z ≤ -1.79 or Z ≥ 1.79)
= P(Z ≤ -1.79) + P(Z ≥ 1.79)
= 0.0364 + 0.0364
= 0.0728
Therefore, the approximate P-value for Elliot's significance test is 0.0728. Since the P-value is greater than the significance level of 0.05, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the proportion of adults who bike to work in Elliot's city has changed significantly from the previous year.
Step-by-step explanation:
I need help
A population of bacteria is growing according to the equation p(t)=800e^0.14t Estimate when the population will exceed 1151.
t= ---------
Answers
Answer: t=2.59
Step-by-step explanation:
This is a matter of clearing out the equation
set 1151=800e^0.14t
1151/800=e^0.14t
ln(1151/800)/0.14=t
t=2.59
1. Dorie Sparrow, assistant manager of The Clothes Horse, Inc., must mark all clearance rack dresses back
to their regular selling price. She had marked all of them down 70%. What regular selling price does Dorie
need to sell a dress for that had been marked down to $104.98?
Answers
Therefore, Dorie needs to sell the dress for $349.93 in order to mark it back to its regular selling price.
What is selling price?
Selling price refers to the price at which a product or service is sold to customers. It is the amount of money that a buyer pays to the seller in exchange for the product or service. The selling price is usually higher than the cost price, which is the amount that the seller paid to acquire or produce the product or service. The difference between the selling price and the cost price is called the profit margin, and it is the profit that the seller makes on the sale of the product or service.
If a dress had been marked down 70%, this means that it is being sold for only 30% of its original selling price.
Let P be the original selling price of the dress. Then:
0.3P = $104.98
Solving for P, we get:
P = $104.98 / 0.3
P = $349.93
Therefore, Dorie needs to sell the dress for $349.93 in order to mark it back to its regular selling price.
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I need help solving this thank you
Answers
The negation is the fourth option.
6 + 3 ≠ 9 or 6 - 3 ≠ 9
How to write the negation?The negation of an equation is an inequality such that we just change the equal sign, by the "≠" sign.
Here we start with the two equations.
6 + 3 = 9 or 6 - 3 = 9
Just change the equal signs for different signs:
6 + 3 ≠ 9 or 6 - 3 ≠ 9
That is the negation, fourth option.
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Find the area of a rectangle with length 4.3 cm and breadth 3.9 cm
Answers
Answer:
16.77 cm²
Step-by-step explanation:
Area of rectangle = L x W
L = 4.3 cm
Width = 3.9 cm
Let' solve
4.3 x 3.9 = 16.77 cm²
So, the area of the rectangle is 16.77 cm²
What is the range of the relation whose graph is shown?
Answers
The range of the relation whose graph is shown include the following: C. -1 ≤ y ≤ 1.
What is a domain?In Mathematics and Geometry, a domain is the set of all real numbers for which a particular function is defined.
Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = {-4, 4} or -4 ≤ x ≤ 4.
Range = {-1, 1} or -1 ≤ y ≤ 1.
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(a) What is the value of x? Show your work.
(b) What is the measure of angle C? Show your work.
Answers
In triangle ABC
a) The value of x = 29⁰
b) The angle c equal to 93⁰
What is a triangle?A triangle is a closed plane figure that is formed by connecting three line segments, also known as sides, at their endpoints. The three endpoints, or vertices, where the sides of the triangle meet are not collinear. Triangles are important in mathematics and geometry because they are the simplest polygon that can exist in two-dimensional space.
According to the given informationIn a triangle, the sum of all interior angles is always 180 degrees. Therefore, we can use this fact to find the value of x and angle c.
We know that:
angle a = 35⁰
angle b = 52⁰
angle c = 3(x+2)⁰
Using the fact that the sum of all interior angles in a triangle is 180 degrees, we can write:
angle a + angle b + angle c = 180
Substituting the values we know, we get:
35 + 52 + 3(x+2) = 180
Simplifying the equation, we get:
87 + 3x + 6 = 180
3x + 93 = 180
3x = 87
x = 29
Therefore, x = 29⁰
To find angle c, we can substitute the value of x into the equation we were given for angle c:
angle c = 3(x+2)
angle c = 3(29+2)
angle c = 3(31)
angle c = 93
Therefore, angle c is equal to 93⁰.
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A waiter made $288 in tips after waiting on 12 tables. What was the waiter’s average tip per table?
A$24
B$13
C$15
D$16
Answers
Answer:
24
Step-by-step explanation:
Answer:
A. $24
Step-by-step explanation:
Given:
total of $288number of values is 12 tablesSolve for average tip:
Average is the same as the meanTo solve you have add up the total and divide by the total number of values288 / 12 = 24Answer:
Thus, the waiter's average tip per table is $24.
The answer is A.
In art class students are mixing blue and red paint to make purple paint. Deondra
mixes 6 cups of blue paint and 7 cups of red paint. Arun mixes 2 cups of blue paint
and 3 cups of red paint. Use Deondra and Arun's percent of red paint to determine
whose purple paint will be redder.
Deondra percent of red paint (to nearest whole number) =
Arun percent of red paint (to nearest whole number) =
O Deondra's purple paint will be redder.
O Arun's purple paint will be redder.
o The two purple paints will be equally red.
Submit Answer
%
%
attempt 1 out of 2
Answers
Arun's purple paint will be redder.
Define percentagePercentage is a way of expressing a proportion or a fraction as a number out of 100. It is represented by the symbol "%". For example, if you say that 20% of students in a class scored an A grade in a test, it means that 20 out of every 100 students received an A grade.
Deondra mixed 6 cups of blue paint and 7 cups of red paint, so the percent of red paint in her mixture is:
7 / (6 + 7) × 100% = 53.8%, which rounds to 54%.
Arun mixed 2 cups of blue paint and 3 cups of red paint, so the percent of red paint in his mixture is:
3 / (2 + 3) × 100% = 60%.
Since Arun's mixture has a higher percentage of red paint, his purple paint will be redder than Deondra's.
Therefore, the answer is Arun's purple paint will be redder.
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Alberto gets a fish tank
Answers
Jackson is on point. Of all fish, catfish make up one-third of the total.
How much gravel does Alberto add to the tank in pounds?The United States of America, Great Britain, and other English-speaking countries predominately utilise the pound as a unit of weight.
Any quantity associated with something is assumed to contain one pound if it does. The pound is the most widely used weight unit in the English-speaking world, including Britain, America, and other nations.
Zebra danios make up half of all the fish.
More fancy guppies than catfish, by a factor of two.
x = guppies
2x = catfish
x + 2x = 1/2 of the fish population
3x = 1/2 of the fish population
3x * 2 = 6x
x = 1/6 number of fancy guppies
catfish:
2x = 2(1/6) = 2/6 = 1/3 catfish.
Jackson is on point. Of all fish, catfish make up one-third of the total.
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The complete question is,
Alberto puts 3 kinds of fish in his fish tank: zebra danios, fancy guppies, and cat fish. Half of Alberto's fish are zebra danios. He has 2 times as many catfish as he has fancy guppies. Alberto's friend Jackson says 1/3 of Alberto's fish are catfish. Is Jackson correct?
7. Can the following be the lengths of the sides of a triangle?
a. 20 cm, 40 cm, 50 cm
b. 20 cm, 40 cm, 60 cm
c. 41 cm, 250 mm, 12 cm
Answers
Answer:
B
Step-by-step explanation:
THE TWO SMALLER SIDES = THE LARGER SIDE HENCE ITS A TRIANGLE
Which of the numbers 0, 1, 2, 3 or 4 make the equation 8/y2 + 2 true?
Answers
None of the given numbers make the equation 8/y² + 2 true.
What is algebra?
Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.
To solve this problem, we can substitute each of the given numbers (0, 1, 2, 3, 4) for y in the equation 8/y² + 2 and see if the equation is true.
Substituting y=0 would make the denominator of the fraction zero, which is undefined, so y=0 is not a valid choice.
Substituting y=1 would give us:
8/1² + 2 = 8 + 2 = 10
So, 1 is not the answer.
Substituting y=2 would give us:
8/2² + 2 = 8/4 + 2 = 2 + 2 = 4
So, 2 is not the answer.
Substituting y=3 would give us:
8/3² + 2 = 8/9 + 2 = 0.888 + 2 = 2.888
So, 3 is not the answer.
Substituting y=4 would give us:
8/4² + 2 = 8/16 + 2 = 0.5 + 2 = 2.5
So, 4 is not the answer.
Therefore, none of the given numbers make the equation 8/y² + 2 true.
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A drawer contains 10 blue pens, 12 black pens, and 3 red pens. Without looking, Mr. Lopez is going to take one pen from the drawer, use it, and then put it back into the drawer. Then he is going to take another pen from the drawer to use. What is the probability of Mr. Lopez taking a red pen first and then taking a blue pen?
Answers
Answer: 4.8%
Step-by-step explanation: the total amount of pens in the drawer is (10+12+3) = 25
the amount of red pens in the drawer is 3
the probability of picking out a red pen from the drawer = 3/25
the amount of blue pens in the drawer is 10
the probability of picking out a red pen from the drawer = 10/25
the probability of picking out a red pen then a blue pen afterwards = (10/25 x 3/25) = 4.8%
help please! state the key features for the graph
Answers
Answer:
Axis of symmetry =1
vertex =(1,2)
y intercept =0
min/max= -6,2
domain= 0,1,2
range =y≥1,2
Will mark brainliest if answer is correct
Answers
Answer:
[tex]3( {2}^{2} ) - {2}^{2} + 4 = 12[/tex]
[tex] {2}^{3} + b( {2}^{2} ) + 43(2) - 126 = 4b - 204[/tex]
[tex]4b - 32 = 12[/tex]
[tex]4b = 44[/tex]
[tex]b = 11[/tex]
For this value of b, these graphs will intersect at (2, 12). Please use your graphing calculator to confirm that this is the only point of intersection.
Which ordered pairs lie on the graph of the exponential function f(x)=−32x+5
?
Answers
The ordered pair (0, 2) lies on the graph.
The ordered pair (-1, 0) lies on the graph.
The ordered pair (-1, 0) lies on the graph.
What is function?
In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.
The exponential function f(x) = -32x+5 can be written in the form of
f(x) = [tex]a(b^{x})+c[/tex], where a, b, and c are constants. Comparing with the given function, we have a = -3, b = 2, and c = 5.
To find the ordered pairs that lie on the graph of the exponential function, we can plug in different values of x and calculate the corresponding values of y using the function. Here are some examples:
When x = 0, we have f(0) = [tex]-3(2^{0})[/tex] + 5 = 2. Therefore, the ordered pair (0, 2) lies on the graph.
When x = 1, we have f(1) = [tex]-3(2^{1})[/tex] + 5 = -1. Therefore, the ordered pair (1, -1) lies on the graph.
When x = -1, we have f(-1) = [tex]-3(2^{-1})[/tex] + 5 = 6/2 - 3 = 0. Therefore, the ordered pair (-1, 0) lies on the graph.
We can continue this process to find more ordered pairs that lie on the graph of the exponential function.
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a right triangle has a hypotenuse of length 7 inches. If one angle is 38 degrees, find the length of each leg.
Answers
Check the picture below.
[tex]\sin(38^o )=\cfrac{\stackrel{opposite}{x}}{\underset{hypotenuse}{7}}\implies 7\sin(38^o)=x\implies 4.31\approx x \\\\[-0.35em] ~\dotfill\\\\ \cos(38^o )=\cfrac{\stackrel{adjacent}{y}}{\underset{hypotenuse}{7}}\implies 7\cos(38^o)=y\implies 5.52\approx y[/tex]
Make sure your calculator is in Degree mode.
Answer:
4.31 in5.52 inStep-by-step explanation:
You want the leg measures in a right triangle with a hypotenuse of 7 inches and an angle of 38°.
Trig functionsThe mnemonic SOH CAH TOA reminds you of the relations between side lengths and trig functions:
Sin = Opposite/Hypotenuse ⇒ Opposite = Hypotenuse×Sin
Cos = Adjacent/Hypotenuse ⇒ Adjacent = Hypotenuse×Cos
ApplicationThe given triangle will have opposite and adjacent sides of ...
opposite = (7 in)sin(38°) ≈ 4.31 in
adjacent = (7 in)cos(38°) = 5.52 in
The leg lengths of the triangle are 4.31 inches and 5.52 inches.
Use the slip and y-intercept to identify the equation of this line
Answers
Answer:
y=2x
Step-by-step explanation:
equation formula is y=mx+b
m is slope
b is y-intercept
you know that slope is 2
y intercept is 0
so plug it into the equation
y=2x+0 or just y=2x
if you dilate triangle ABC by a scale factor of 3 and (0,0) is the center, what will be the length of AB?
Answers
the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center would be 3 times the original length of AB.
How to solve the question?
To find the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center, we can use the following formula:
AB' = AB x 3
where AB' is the length of AB after the dilation, and AB is the original length of AB.
However, we need to first determine the length of AB in the original right triangle ABC. To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's assume that AB is the hypotenuse of the right triangle ABC, and that AC and BC are the other two sides. Then we have:
AB²= AC²+ BC²
Without more information about the lengths of AC and BC, we cannot determine the value of AB. However, once we have determined the length of AB, we can use the formula above to find the new length of AB after dilation.
Assuming we know the length of AB in the original right triangle ABC, we can now use the formula for dilation to find the new length of AB:
AB' = AB x 3
For example, if AB is 5 units long in the original triangle, then after dilation, AB' would be:
AB' = 5 x 3 = 15 units
Therefore, the new length of AB after a dilation by a scale factor of 3 with (0,0) as the center would be 3 times the original length of AB.
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Determine the circumference and approximate area of the
given circle, using 3.14 for pie.
Answers
The circumference and approximate area of the given circle is 69.08 inches & 380.14 square inches.
What is circumference?
Circumference is the distance around the edge of a circular object or a round shape. It is the length of the boundary or perimeter of the circle. The formula is given by C = 2πr, where C is the circumference, r is the radius of the circle, and π is a mathematical constant approximately equal to 3.14.
The circumference of a circle is given by the formula:
C = 2πr
where r is the radius of the circle and π (pi) is a mathematical constant approximately equal to 3.14.
Using this formula and plugging in the given value of radius:
C = 2 x 3.14 x 11
C = 69.08 inches (rounded to two decimal places)
So the circumference of the circle with 11 inches radius is approximately 69.08 inches.
The area of a circle is given by the formula:
A = πr²
Again, using the given value of radius and approximating π to 3.14:
A = 3.14 x 11²
A = 3.14 x 121
A = 380.14 square inches (rounded to two decimal places)
So the approximate area of the circle with 11 inches radius is approximately 380.14 square inches.
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I BEG U FOR HELP WILL GIVE BRAINLIEST PLLSSSS
Answers
Answer:
3,4,5 is the answer
Step-by-step explanation:
for the explanation using pythagoras theoem
[tex] {3}^{2} + {4}^{2} = {5 \\ }^{2} \\ 3 \times 3 + 4 \times 4 = 5 \times 5 \\ 9 + 16 = 25 \\ 25 = 25[/tex]
may you give me branliest as you promised
A machine in a factory makes chairs at rate of two chairs every 10 minutes how much time does the machine take to make five chairs how many minutes would it take for the factory to fulfill an order for 32 chairs show your work or explain how you determine your answers
Answers
To make 5 chairs, the machine will need 5 times as much time, which is 5 x 5 = 25 minutes.
To fulfill an order for 32 chairs, the machine will need to make 32/2 = 16 sets of 2 chairs.
This means the machine will take 16 x 10 = 160 minutes to make 32 chairs.
I obtained these answers by using basic math. First, I found out how many chairs the machine makes in one minute by dividing the rate of production (2 chairs every 10 minutes) by the number of minutes (10). That gave me the production rate of 1 chair every 5 minutes.
Using that production rate, I could then multiply it by the number of chairs I needed to find out how many minutes it would take for the machine to produce them.
For the order of 32 chairs, I used the production rate to figure out how many sets of two chairs the machine would need to make to reach the total amount. And then I multiplied that number by the time it takes to make two chairs (10 minutes) to get the total time it would take to produce 32 chairs.
Help Please...
You have 67 coins consisting of half-dollars and quarters. The number of quarters is 7 more than three times the number of half-dollars.
How many quarters do you have?
How many half -dollars do you have?
Answers
There are 52 quarters and 15 half-dollars
To solve this problem
Let's represent the number of half-dollars as "x" and the number of quarters as "y".
From the problem statement, we know that:
x + y = 67 (because there are a total of 67 coins)
y = 3x + 7 (because the number of quarters is 7 more than three times the number of half-dollars)
We can use substitution to solve for x:
x + (3x + 7) = 67
4x + 7 = 67
4x = 60
x = 15
So there are 15 half-dollars. We can use this to find the number of quarters:
y = 3x + 7
y = 3(15) + 7
y = 52
So there are 52 quarters.
Therefore, there are 52 quarters and 15 half-dollars.
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Suppose that the functions fand g are defined as follows.
f(x)=2x-1
g(x)=√3x-5
Answers
The composite functions (f/g)(x) and (f-g)(x) are (2x-1)/√(3x-5) and (2x-1) -√(3x-5)
Calculating the composite functions (f/g)(x) and (f-g)(x)To calculate (f/g)(x), we need to divide f(x) by g(x):
(f/g)(x) = f(x)/g(x) = (2x-1)/√(3x-5)
The domain of (f/g)(x) is the set of all x-values for which the denominator √(3x-5) is not equal to zero and non-negative
3x-5 ≥ 0, or x ≥ 5/3
Therefore, the domain of (f/g)(x) is x ≥ 5/3.
To calculate (f-g)(x), we need to subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x) = (2x-1) - √(3x-5)
The domain of (f-g)(x) is the set of all x-values for which the expression inside the square root is non-negative:
3x-5 ≥ 0, or x ≥ 5/3
Therefore, the domain of (f-g)(x) is x ≥ 5/3.
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Find an equation of the osculating plane and an equation of the normal
plane of the curve x = sin 2t, y = t, z = cos 2t at the point (0, π, 1).
Answers
The equation of the normal plane is 4y = 4π, or equivalently, y = π.
What is osculating plane?The word osculate comes from the Latin osculatus, which is a past participle of the verb osculari, which means "to kiss." Thus, an osculating plane is one that "kisses" a submanifold.
To find the osculating plane and normal plane of the curve x = sin 2t, y = t, z = cos 2t at the point (0, π, 1), we need to follow these steps:
Find the first and second derivatives of the curve with respect to t.Evaluate the derivatives at t = π to get the velocity, acceleration, and curvature vectors at the point (0, π, 1).Use the velocity and acceleration vectors to find the normal vector of the osculating plane.Use the normal vector and the point (0, π, 1) to find the equation of the osculating plane.Use the curvature vector to find the normal vector of the normal plane.Use the normal vector and the point (0, π, 1) to find the equation of the normal plane.Step 1: Find the first and second derivatives of the curve with respect to t.
x' = 2cos2t
y' = 1
z' = -2sin2t
x'' = -4sin2t
y'' = 0
z'' = -4cos2t
Step 2: Evaluate the derivatives at t = π.
x'(π) = 2cos2π = 2
y'(π) = 1
z'(π) = -2sin2π = 0
x''(π) = -4sin2π = 0
y''(π) = 0
z''(π) = -4cos2π = -4
So the velocity vector at the point (0, π, 1) is v = ⟨2, 1, 0⟩, the acceleration vector is a = ⟨0, 0, -4⟩, and the curvature vector is κv = ⟨0, 4, 0⟩.
Step 3: Use the velocity and acceleration vectors to find the normal vector of the osculating plane.
The normal vector of the osculating plane is given by the cross product of the velocity and acceleration vectors:
n = v × a = ⟨2, 1, 0⟩ × ⟨0, 0, -4⟩ = ⟨4, 0, 0⟩
Step 4: Use the normal vector and the point (0, π, 1) to find the equation of the osculating plane.
The equation of the osculating plane is given by:
4(x - 0) + 0(y - π) + 0(z - 1) = 0
Simplifying, we get:
4x - 4 = 0
So the equation of the osculating plane is 4x = 4, or equivalently, x = 1.
Step 5: Use the curvature vector to find the normal vector of the normal plane.
The normal vector of the normal plane is given by the curvature vector:
n' = κv = ⟨0, 4, 0⟩
Step 6: Use the normal vector and the point (0, π, 1) to find the equation of the normal plane.
The equation of the normal plane is given by:
0(x - 0) + 4(y - π) + 0(z - 1) = 0
Simplifying, we get:
4y - 4π = 0
So, the equation of the normal plane is 4y = 4π, or equivalently, y = π.
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Suppose that the function h is defined as follows.
if
-2
-1
h(x)= 0
1
2
Graph the function h.
-3.5
if-2.5
if-1.5
if -0.5
if 0.5 ≤x≤1.5
Answers
Suppose that the function h is defined as follows. -2 -1 h(x)= if – 2, the graph is given below: (see image)
What is a Graph?A graph is a mathematical structure used to represent relationships between objects or entities. It consists of a set of vertices (also known as nodes) and a set of edges that connect pairs of vertices.
In a graph, the vertices represent the objects or entities being studied, while the edges represent the connections or relationships between them.
For example, in a social network graph, the vertices might represent individual users, and the edges might represent their connections (e.g. friendships) with other users.
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