Step-by-step explanation:
we use a function as a kind of recipe or template for an actual calculation (or actual "ingredients").
as long as we don't handle the actual food items or actual numbers, it all stays theoretical. variables have no actual value and represent every possible case with every possible value.
but as soon as we get an actual unit value (like 8 in our case), we can put it in place of the variables (that are really nothing else but placeholders for actual values) and simply caucuses the result.
so,
when the question asks what is f(8), it really means what is the result when x = 8.
therefore,
f(8) = 8 - 7 = 1
that's it. that is the whole thing. no mystery, magic or genius strikes necessary.
Answer:
[tex] \sf \: f(8) = 1[/tex]
Step-by-step explanation:
Given function,
→ f(x) = x - 7
Now we have to,
→ Find the required value of f(8).
We have to use,
→ x = 8
Then the value of f(8) will be,
→ f(x) = x - 7
→ f(8) = 8 - 7
→ [ f(8) = 1 ]
Hence, the value of f(8) is 1.
Which point on the number line has the least absolute value?
The point with the least absolute value on the number line is always the point zero.
The absolute value of a number is the distance that number is from zero on the number line. Therefore, the point on the number line with the least absolute value is the point closest to zero. This point is located at zero itself, as it is the point on the number line that is equidistant from both the positive and negative numbers.
To further explain, consider the following examples:
- The point 3 is 3 units away from zero, but the point -3 is also 3 units away from zero.
- The point 5 is 5 units away from zero, but the point -5 is also 5 units away from zero.
- The point 0 is 0 units away from zero, making it the point with the least absolute value on the number line.
In conclusion, the point with the least absolute value on the number line is always the point zero.
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Which statement is true about all perfect cubes?
A
A perfect cube represents 3 times the area of a face of the cube.
B
A perfect cube represents the sum of 9 edge lengths of the cube.
A perfect cube represents the volume of a cube with equal integer side lengths.
D
A perfect cube represents the surface area of a cube with equal integer side lengths.
C
Ello beautiful people how are you doing today and wyd this weekend
Hello beautiful people! I hope you're all doing well today. As for me, I'm feeling great and ready for the weekend! I plan on spending some quality time with my loved ones, exploring new places, and trying out new activities.
I believe that weekends are meant for rest, relaxation, and rejuvenation, so I'm looking forward to taking a break from my busy work schedule.
One of the things I love about weekends is the opportunity to disconnect from the stresses of everyday life and focus on things that bring me joy. Whether it's going for a hike, trying out a new recipe, or catching up with friends over a cup of coffee, there's always something to look forward to.
this weekend I plan on making the most of my free time by doing things that make me happy and help me recharge. I hope you all have a great weekend as well, and that you find time to do the things you love with the people you care about. Remember, life is short, so let's make the most of every moment we have!
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2 A model of (CH₂O)4 was created using colored beads. Carbon atoms were
represented by black beads, hydrogen atoms by red beads, and oxygen atoms
by blue beads. Which of the following combinations of beads shows an accurate
model of (CH₂O)4?
A 4 black, 8 red, and 4 blue
B 1 black, 2 red, and 1 blue
C 4 black, 6 red, and 4 blue
D
1 black, 8 red, and blue
The correct number of beads is; 4 black, 8 red, and 4 blue. Option A
What is a molecular model?A molecular model is a depiction of molecules or chemical compounds made physically, visually, or mathematically in order to comprehend their behavior and characteristics. These models can range from real models made of plastic or metal to computer-generated graphics or mathematical formulae, and they can be straightforward or sophisticated.
There are four carbon atoms, eight hydrogen atoms and four oxygen atoms.
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3) Find the maximum and minimum values of f(x,y) = xyon the region inside the triangle whose vertices are (6,2), (0,3), and (6.0).
Therefore, the maximum value of f(x,y) inside the triangle is 80/9, which occurs along the line y = (-1/2)x + 4 at the point (8/3, 10/3), and the minimum value is -32, which occurs at the critical point (-8,4).
To find the maximum and minimum values of f(x,y) = xy on the region inside the triangle whose vertices are (6,2), (0,3), and (6,0), we use the method of Lagrange multipliers.
First, we need to find the critical points of f(x,y) subject to the constraint that (x,y) lies inside the triangle. We can express this constraint using the equations of the lines that form the sides of the triangle:
y = (-1/2)x + 4
y = (3/2)x
y = 0
Next, we set up the Lagrange multiplier equation:
∇f = λ∇g
where g(x,y) is the equation of the constraint, i.e., the triangle.
We have:
f(x,y) = xy
∇f = <y, x>
g(x,y) = y - (-1/2)x - 4 = 0
∇g = <-1/2, 1>
Setting ∇f = λ∇g, we get:
y = (-1/2)λ
x = λ
Substituting these into the constraint equation, we get:
(-1/2)λ - 4 = 0
Solving for λ, we get:
λ = -8
Substituting this into y = (-1/2)λ and x = λ, we get:
x = -8 and y = 4
Therefore, the only critical point of f(x,y) inside the triangle is (-8,4).
Next, we need to check the values of f(x,y) at the vertices and along the sides of the triangle.
At the vertices:
f(6,2) = 12
f(0,3) = 0
f(6,0) = 0
Along the line y = (3/2)x:
f(x, (3/2)x) = (3/2)x^2
Using the vertex (6,2) and the x-intercept (4/3, 2), we can see that the maximum value of (3/2)x^2 on this line occurs at x = 4. Therefore, the maximum value of f(x,y) along this line is:
f(4,6) = 24
Along the line y = (-1/2)x + 4:
f(x, (-1/2)x + 4) = (-1/2)x^2 + 4x
Using the vertex (6,2) and the x-intercept (8,0), we can see that the maximum value of (-1/2)x^2 + 4x on this line occurs at x = 8/3. Therefore, the maximum value of f(x,y) along this line is:
f(8/3,10/3) = 80/9
Finally, we need to check the values of f(x,y) at the critical point (-8,4). We have:
f(-8,4) = -32
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Integrate fover the given curve. f(x,y) = x+ y, C: x^2 + y^2 = 4 in the first quadrant from
(2,0) to (0,2)
The integral of f(x, y) = x + y over the given curve is 8.
To integrate the function f(x, y) = x + y over the curve C: x² + y² = 4 in the first quadrant from (2, 0) to (0, 2), we will use the line integral. Since the curve is a circle, we can parameterize it using polar coordinates as follows:
x = 2cos(θ)
y = 2sin(θ)
Now, let's find the derivatives:
dx/dθ = -2sin(θ)
dy/dθ = 2cos(θ)
Next, we substitute x and y in f(x, y):
f(x, y) = 2cos(θ) + 2sin(θ)
Now, we can set up the line integral:
∫[f(x, y) * ||dr/dθ||]dθ
Since ||dr/dθ|| = sqrt((-2sin(θ))^2 + (2cos(θ))^2) = 2, the line integral becomes:
∫[2cos(θ) + 2sin(θ)] * 2 dθ
To find the limits of integration, we can use the points (2, 0) and (0, 2). In polar coordinates, these points correspond to θ = 0 and θ = π/2.
So, the line integral becomes:
∫[4cos(θ) + 4sin(θ)]dθ from 0 to π/2
Now, we can integrate and evaluate:
[4sin(θ) - 4cos(θ)] from 0 to π/2 = [4(1) - 4(0)] - [4(0) - 4(1)] = 8
Thus, the integral of f(x, y) = x + y over the given curve is 8.
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the top of the farm silo is a hemisphere with a radius of 9ft. the bottom of the silo is a cylinder with a height of 35ft. how many cubic feet of grain can the solo hold? use 3.14 for pi and round your answer to the nearest cubic foot.
To find the total volume of the silo, we need to add the volume of the hemisphere on top to the volume of the cylinder at the bottom.
The volume of a hemisphere is given by:
V_hemi = (2/3)πr^3
where r is the radius of the hemisphere.
Substituting r = 9ft, we get:
V_hemi = (2/3)π(9ft)^3
= 1521π ft^3
The volume of a cylinder is given by:
V_cyl = πr^2h
where r is the radius of the cylinder and h is its height.
Substituting r = 9ft and h = 35ft, we get:
V_cyl = π(9ft)^2(35ft)
= 2673π ft^3
Therefore, the total volume of the silo is:
V_silo = V_hemi + V_cyl
= 1521π + 2673π
= 4194π ft^3
≈ 13160 ft^3
Rounding to the nearest cubic foot, the silo can hold approximately 13160 cubic feet of grain.
If the coordinates of two points are P (-2, 3) and Q (-3, 5), then find (abscissa of P) – (abscissa of Q)
The difference between the abscissa of P and Q is 1.
The abscissa of a point is its x-coordinate, or horizontal distance from the origin (usually measured along the x-axis).
In the given problem, the abscissa of point P is -2, which means it is located 2 units to the left of the origin on the x-axis. The abscissa of point Q is -3, which means it is located 3 units to the left of the origin on the x-axis.
To find the difference between the abscissas of P and Q, we simply subtract the abscissa of Q from the abscissa of P:
(abscissa of P) - (abscissa of Q) = (-2) - (-3) = -2 + 3 = 1
Therefore, the difference between the abscissas of P and Q is 1 unit.
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The table shows the amount of pet food in cups remaining in an automatic feeder as a function of the number of meals the feeder has dispensed.
number of meals dispensed. n. 1. 3. 6. 7. amount of pet food remaining . f of n. cups. 21. 15. 6. 3.
based on the table, which function models this situation?
The function that models this situation is f(n) = -3n + 24.
To find the function, we need to analyze the relationship between the number of meals dispensed (n) and the amount of pet food remaining (f(n)).
1. Observe the change in f(n) when n increases by 1 meal. From n=1 to n=3, f(n) decreases from 21 to 15, a change of -6. From n=6 to n=7, f(n) decreases from 6 to 3, a change of -3.
2. The decrease in f(n) is not constant, so the function is not linear. However, the decrease becomes smaller as n increases.
3. Consider the average rate of change in f(n) per meal: (-6/2) = -3, (-3/1) = -3.
4. Since the average rate of change is constant, the function is linear.
5. The function has the form f(n) = -3n + b. To find b, plug in the value of n and f(n) from the table: 21 = -3(1) + b, which gives b = 24.
6. Therefore, the function that models this situation is f(n) = -3n + 24.
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i need help fast!!!!
Answer:
1st choice: 1/4(y - 10) = 2/3
Step-by-step explanation:
the "variable" is y
"is" means "=" (equals sign)
one fourth = 1/4
"difference of" means subtract
Answer: 1/4(y - 10) = 2/3
240:360=?:120 (Please quickly)
Answer:
? equals 80
Step-by-step explanation:
In the relations v=u+at,findv,when u=6 a=10 t=2
The value of v in the equation is 26
How to calculate the value of v in the equation?The equation is given as
v= u + at
The parameters given are
u= 6
a= 10
t= 2
v= 6 + 10(2)
v= 6 + 20
v= 26
Hence the value of v in the equation is 26
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A) Construct an appropriate tabular representation/summary of the random variable Number of years in operation and provide an interpretation.
b) Construct a cross-tabulation of the data on Daily Income and Type of service and provide an interpretation. Hint: Use a class width of N$ 500 for Daily Income.
c) Calculate and interpret relative measures of variability for the Daily Income for each of the three categories of Type of service
Un terreno de forma rectangular tiene un perímetro de 105 metros. Si el ancho es la mitad del largo, ¿Cuáles son las medidas del terreno? *
Sea "l" la medida del largo del terreno y "a" la medida del ancho del terreno.
De acuerdo con el problema, el ancho es la mitad del largo, es decir, a = l/2.
El perímetro de un rectángulo se calcula sumando las longitudes de sus cuatro lados, por lo que en este caso:
Perímetro = 2l + 2a = 2l + 2(l/2) = 3l
Sabemos que el perímetro es de 105 metros, entonces:
3l = 105
l = 105/3 = 35
Por lo tanto, el largo del terreno es 35 metros. Y, como el ancho es la mitad del largo, entonces:
a = l/2 = 35/2 = 17.5
Por lo tanto, el ancho del terreno es de 17.5 metros.
En resumen, las medidas del terreno son 35 metros de largo y 17.5 metros de ancho.
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Determine the unique solution, y(x), to the differential equation that satisfies the given initial condition. dy/dx = 8x⁷/y⁴, y(0) = 4
y(x) = ...
The unique solution, y(x), to the given differential equation with initial condition y(0) = 4 is
:
y(x) = [-(6x⁸ - 64)]¹/³
Determine the unique solution?
To determine the unique solution, y(x), to the given differential equation with initial condition y(0) = 4, we first need to separate the variables and integrate both sides with respect to x and y, respectively.
dy/y⁴ = 8x⁷ dx
Integrating both sides, we get:
-1/3y³ = 2x⁸ + C
where C is the constant of integration.
Now we can use the initial condition y(0) = 4 to solve for C:
-1/3(4)³ = 2(0)⁸ + C
C = -64/3
Substituting C back into the previous equation, we get:
-1/3y³ = 2x⁸ - 64/3
Multiplying both sides by -3 and taking the cube root, we get:
y(x) = [-(6x⁸ - 64)]¹/³
Therefore, the unique solution, y(x), to the given differential equation with initial condition y(0) = 4 is:
y(x) = [-(6x⁸ - 64)]¹/³
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Find the derivative of the vector function r(t) = ln(7-t^2)i + sqrt(13+tj – 4e^{9t} r’(t) =
The derivative of the vector function is: r'(t) = (-2t/(7-t^2)) i + (1/(2sqrt(13+t))) j - 36e^(9t) k
We are given a vector function r(t) = ln(7-t^2)i + sqrt(13+t)j – 4e^(9t)k, and we need to find its derivative r'(t).
The derivative of a vector function is obtained by differentiating each component of the vector function separately.
So, let's differentiate each component:
r(t) = ln(7-t^2)i + sqrt(13+t)j – 4e^(9t)k
r'(t) = (d/dt) ln(7-t^2) i + (d/dt) sqrt(13+t) j - (d/dt) 4e^(9t) k
Using the chain rule of differentiation, we have:
r'(t) = -2t/(7-t^2) i + 1/(2sqrt(13+t)) j - 36e^(9t) k
Therefore, the derivative of the vector function is:
r'(t) = (-2t/(7-t^2)) i + (1/(2sqrt(13+t))) j - 36e^(9t) k
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A function is a rule that assingns each value of independent variable to exactly value of the dependent variable
A function is a rule that assingns each value of independent variable to exactly one value of the dependent variable.
A function is a mathematical concept that relates two sets of values, known as the domain and the range. The domain is the set of independent variables, while the range is the set of dependent variables. A function is a rule that assigns to each value in the domain exactly one value in the range.
For example, if we have a function f(x) = 2x + 3, the domain would be any possible value of x, and the range would be any possible value of 2x + 3. So if we put x = 2, then f(x) = 2(2) + 3 = 7. Therefore, the function assigns the value of 7 to the value of 2 in the domain.
Functions are used in various branches of mathematics, science, and engineering to model and analyze relationships between two or more variables. They are an important concept in calculus, where they are used to study rates of change and optimization problems.
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Now, take the square root of both sides of the equation (c^2 = n^2) and write the resulting equation. Is there any way for this equation to be true? How?
Only answer if you can properly answer
Yes, there is a way for this equation to be true in now, take the square root of both sides of the equation (c^2 = n^2) and write the resulting equation
To take the square root of both sides of the equation (c^2 = n^2), you would perform the following steps:
1. Take the square root of both sides:
√(c^2) = √(n^2)
2. Simplify the square roots:
c = n
The resulting equation is c = n.
This equation can be true if both c and n have the same value. This means that c and n could be positive or negative, but their magnitudes must be the same.
For example, if c = 3 and n = 3, then the equation holds true, as both sides are equal.
Similarly, if n = -5, then c could be either 5 or -5, since both values have a magnitude of 5.
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To conserve water, many communities have developed water restrictions. The water utility charges a fee of $34, plus an additional $1.36 per hundred cubic feet (HCF) of water. The recommended monthly bill for a household is between $60 and $85 dollars per month. If x represents the water usage in HCF in a household, write a compound inequality to represent the scenario and then determine the recommended range of water consumption. (Round your answer to one decimal place.
60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 19.1 and 37.5 HCF.
Hown to write the inequalityThe correct compound inequality to represent the scenario is:
60 ≤ 1.36x + 34 ≤ 85
To solve for x, we need to isolate it in the middle of the inequality:
60 - 34 ≤ 1.36x ≤ 85 - 34
26 ≤ 1.36x ≤ 51
Finally, we divide by 1.36 to isolate x:
19.12 ≤ x ≤ 37.5
Therefore, the recommended range of water consumption is between 19.1 and 37.5 HCF. The answer is (D) 60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 19.1 and 37.5 HCF.
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complete question
To conserve water, many communities have developed water restrictions. The water utility charges a fee of $34, plus an additional $1.36 per hundred cubic feet (HCF) of water. The recommended monthly bill for a household is between $60 and $85 dollars per month. If x represents the water usage in HCF in a household, write a compound inequality to represent the scenario and then determine the recommended range of water consumption. (Round your answer to one decimal place.)
60 ≤ 1.36x − 34 ≤ 85; To stay within the range, the usage should be between 69.1 and 87.5 HCF.
60 ≤ 1.36x − 34 ≤ 85; To stay within the range, the usage should be between 44.1 and 87.5 HCF.
60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 37.5 and 44.1 HCF.
60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 19.1 and 37.5 HCF.
There are 30 skittles in a box, for every 5 green there are 7 yellow, how many yellows are there in the box
There are 42 yellow skittles in the box.
Based on the given information, we know that the ratio of green skittles to yellow skittles is 5:7. This means that for every 5 green skittles, there are 7 yellow skittles.
To find out how many yellow skittles are in the box, we need to know how many sets of 5 green skittles there are. We can do this by dividing the total number of skittles in the box (30) by 5 (since there are 5 green skittles for every set).
30 ÷ 5 = 6
This means there are 6 sets of 5 green skittles in the box.
Now we can use the ratio of 5:7 to find out how many yellow skittles there are in each set:
5 green skittles : 7 yellow skittles
Since there are 7 yellow skittles in each set, we can find the total number of yellow skittles by multiplying 7 by the number of sets (6):
7 x 6 = 42
There are 42 yellow skittles in the box.
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) Nadia buys 4 1/5 pounds of plums. Nadia used a 55 cent coupon off her entire purchase. Her total after the coupon was $3. 23. If c represents the cost per pound for the plums, create and solve an equation to determine the cost per pound for the plums
If c represents the cost per pound for the plums, the cost per pound for the plums is $0.90.
First, we need to determine the total cost of the plums before the coupon was applied.
4 1/5 pounds can be written as a mixed number:
4 1/5 = 21/5
So, the total cost of the plums without the coupon can be found by multiplying the cost per pound (c) by 21/5:
Total cost = c * 21/5
Now we can create an equation to represent the total cost after the coupon was applied:
Total cost - coupon = $3.23
Substituting the expression for total cost:
c * 21/5 - 0.55 = 3.23
To solve for c, we can start by adding 0.55 to both sides:
c * 21/5 = 3.78
Then, we can isolate c by multiplying both sides by the reciprocal of 21/5:
c = 3.78 / (21/5)
c = 0.90
Therefore, the cost per pound for the plums is $0.90.
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A designer is planning a trai
l mix box that is
shaped l
ike a rectangular prism. The front of
the box must have the width and height shown.
The volume of the box must be 162 cubic
inches. What must be the depth, d, of the box?
HELP
The depth of trail mix box that is shaped like a rectangular prism must be 2.4 inches.
How can we estimate the depth of the box?We are going to use the formula for determining volume to work out the depth of the rectangular prism.
The volume of a rectangular prism is given by the formula:
V = l × w × h
where:
V = the volume
l = the length
w = the width
h = the height.
Given:
w = 7.5 inches
h = 9 inches
V = 162 cubic inches
Now, we are to find the value of d, the depth of the box, which corresponds to the length of the rectangular prism.
Substituting the values into the formula for volume:
162 = d × 7.5 × 9
Simplifying the right-hand side:
162 = 67.5d
Dividing both sides by 67.5, we get:
d = 162 ÷ 67.5
d = 2.4 inches
Therefore, the depth of the box shaped as rectangular prism = 2.4 inches.
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Determine whether the two figures are similar. If so, give the similarity ratio of the smaller figure to the larger figure. The figures are not drawn to scale.
*
Captionless Image
Yes; 3:5
Yes; 2:3
Yes; 2:5
No they are not similar
Determine whether the two figures are similar: D. No, the two figures are not similar.
What are the properties of quadrilaterals?In Geometry, two (2) quadrilaterals are similar when the ratio of their corresponding sides are equal in magnitude and their corresponding angles are congruent.
Additionally, two (2) geometric figures such as quadrilaterals are considered to be congruent only when their corresponding side lengths are congruent (proportional) and the magnitude of their angles are congruent;
Ratio = 12/8 = 10/6 = 10/6
Ratio = 3/2 ≠ 5/3 = 5/3
In conclusion, the two figures are not similar because the ratio of their corresponding sides is not proportional.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Omar Cuts A Piece of wrapping paper with the shape and dimensions as shown.Find The Area Of The Wrapping Paper.Round Your Answer To The Nearest Tenth Of Needed
Answer:
72.5 square inches
Step-by-step explanation:
See attachment.
The areas of the 2 shapes are in blue, but when added together:
60+12.5=72.5
Hope this helps!
Find m∠A. PLEASEEEEEEEEEEE HELP ASAP WILLING TO DO ANYTHING PLEASEEEEEE
I have gotten 113
Step-by-step explanation:
The right angle triangle has three angles and the value of its two angles are 90°and30°.We need to find the third angle which is the sum of 90°and 30°subtract from 180°=60°.60° is vertically opposite to angle C. We'll be having a quadrilateral whose angles add up to 360° .Subtract the sum of the three angles from 360° and you'll get 113°
A corporation earned a profit of $ 2.5 × 1 0 4 $2.5×10 4 for 200 days in a row. What was the corporation’s total profit during this time period? Express your answer in scientific notation.
Answer: hopefully the image helps
Step-by-step explanation:
The Anderson family went on a trip to see the Paul Bunyan and Blue Ox statue near Lake Bemidji. It took the family 6 hours to travel 330 miles to the statue. What was the Anderson family's average miles per hour (mph)?
btw I don't know how to mark people brainiest so if you tell me how I will to if you help me.
a police car is parked 40 feet due north of a stop sign on straight road. a red car is travelling towards the stop sign from a point 160 feet due east on the road. the police radar reads that the distance between the police car and the red car is decreasing at a rate of 100 feet per second. how fast is the red car actually traveling along the road?
The red car is actually traveling along the road at a speed of approximately 26.67 feet per second.
We can start by drawing a diagram of the situation:
P (police car)
|
|
|
40 | S (stop sign)
-------|--------------------
| 160
| R (red car)
Let's use the Pythagorean theorem to find the distance between the police car and the red car at any time t:
d(t)² = 40² + (160 - v*t)²
Where v is the speed of the red car in feet per second, and d(t) is the distance between the police car and the red car at time t.
We want to find how fast the red car is actually traveling along the road, so we need to find v when the distance between the police car and the red car is decreasing at a rate of 100 feet per second:
d'(t) = -100
We can take the derivative of the equation for d(t) with respect to time:
2d(t)d'(t) = 0 + 2(160 - v*t)(-v)
Simplifying and plugging in d'(t) = -100, we get:
-4000 + 2v²t = -100(160 - vt)
Solving for v, we get:
v = 80/3 ≈ 26.67 feet per second
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Francium is a radioactive element discovered by Marguerite Perey in 1939 and named after her country. Francium has a half-life of 22 minutes.
a) Write an exponential function that models the mass how many grams remain from a 480-gram sample after t minutes.
b) How many grams remain after 2 hours?
After 2 hours, approximately 4.38 grams of Francium remain from the 480-gram sample.
What is Algebraic expression ?
An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. It may contain one or more terms, with each term separated by a plus or minus sign. Algebraic expressions are used in algebra to represent mathematical relationships and formulas.
a) To write an exponential function that models the mass of Francium remaining after t minutes, we can use the formula:
N = N0 * [tex](1/2)^{(t / t1/2)}[/tex]
where N is the amount remaining after time t, N0 is the initial amount, t1/2 is the half-life, and (t/t1/2) means raised to the power of t/t1/2.
In this case, the initial amount is 480 grams, the half-life is 22 minutes, and we want to find the amount remaining after t minutes. Therefore, the exponential function that models the mass of Francium remaining after t minutes is:
N = 480 * [tex](1/2)^{t/22}[/tex]
b) 2 hours is equal to 120 minutes. To find how many grams of Francium remain after 2 hours, we can substitute t = 120 into the exponential function we found in part a):
N = 480 *[tex](1/2)^{ (120 / 22) }[/tex] ≈ 4.38 grams
Therefore, after 2 hours, approximately 4.38 grams of Francium remain from the 480-gram sample.
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Use Euler's method with step size 0.5 to compute the approximate y-values yi, y(0.1), y(0,2), of the solution of the initial-value problem y' = 1 – 2x – 2y, y(0) = – 3. y1 = y2 = y3 = y4 =
The approximate values of y at x = 0.1, 0.2, 0.3, and 0.4 are all equal to y1 = y2 = y3 = y4 = 0.5, as we only used the first step of Euler's method.
We can use Euler's method with a step size of 0.5 to approximate the solution of the given initial-value problem as follows:
First, we need to find the slope at the initial point (0, -3):
y' = 1 - 2x - 2y
y'(0, -3) = 1 - 2(0) - 2(-3) = 7
Using Euler's method, we can approximate the solution at x = 0.5:
y(0.5) ≈ y(0) + hy'(0, -3) = -3 + 0.57 = 0.5
Next, we can use the approximate value y(0.5) to approximate the solution at x = 1:
y(1) ≈ y(0.5) + hy'(0.5, 0.5) = 0.5 + 0.5(1 - 2(0.5) - 2(0.5)) = -0.5
Similarly, we can use the approximate value y(1) to approximate the solution at x = 1.5:
y(1.5) ≈ y(1) + hy'(1, -0.5) = -0.5 + 0.5(1 - 2(1) - 2(-0.5)) = -1.25
Finally, we can use the approximate value y(1.5) to approximate the solution at x = 2:
y(2) ≈ y(1.5) + hy'(1.5, -1.25) = -1.25 + 0.5(1 - 2(1.5) - 2(-1.25)) = -2.4375
Therefore, the approximate values of y at x = 0.1, 0.2, 0.3, and 0.4 are all equal to y1 = y2 = y3 = y4 = 0.5, as we only used the first step of Euler's method.
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