Problem 5. We want to construct a cylindrical can whose volume is 105 mm. The material for the wall of the can costa $3/mm", the material for the bottom of the can costs $7/mm' and the material for the top of the can costs $2/mmº. Determine the radius and the height of the can that will minimize the cost of the materials nooded to construct the can. Problom 6. We have a piece of cardboard that is 50 cm ly 20 cm and we are going to ent a square out of the corners and fold up the sides to form a base. Determine the height of the box that will give a maximum volume.
Answers
Let's start by setting up the equation for the volume of a cylinder:
V = πr²h
We are given that V = 105 mm³, so we can substitute that in:
105 = πr²h
To minimize the cost of materials, we need to consider the cost of the wall, bottom, and top separately.
The cost of the wall is given as $3/mm, so the cost of the wall for the entire can would be:
Cost of wall = 2πrh * 3
The cost of the bottom is given as $7/mm², so the cost of the bottom of the can would be:
Cost of bottom = πr² * 7
The cost of the top is given as $2/mm², so the cost of the top of the can would be:
Cost of top = πr² * 2
The total cost of materials would be the sum of these three costs:
Total cost = 2πrh * 3 + πr² * 7 + πr² * 2
Simplifying this equation, we get:
Total cost = 2πr(3h + 2r) + 7πr²
Now we need to find the values of r and h that will minimize the total cost. We can use the equation for the volume of the can to solve for h in terms of r:
h = 105 / (πr²)
Substituting this into the equation for total cost, we get:
Total cost = 2πr(3(105/πr²) + 2r) + 7πr²
Simplifying this equation, we get:
Total cost = 630/r + 6πr² + 7πr²
To minimize this equation, we need to take the derivative with respect to r and set it equal to zero:
d(Total cost)/dr = -630/r² + 13πr = 0
Solving for r, we get:
r = √(630/13π)
r ≈ 5.18 mm
Now we can use the equation for h to find the height of the can:
h = 105 / (π(5.18)²)
h ≈ 3.27 mm
Therefore, to minimize the cost of materials, we should construct a cylindrical can with a radius of approximately 5.18 mm and a height of approximately 3.27 mm.
Problem 6:
We want to find the maximum volume of a box that can be made from a piece of cardboard that is 50 cm by 20 cm.
Let's start by drawing a diagram:
```
+--------+ h
| |
| |
| |
| |
| |
| |
+--------+
b
```
We are going to cut out equal squares from each corner of the cardboard, so the length of each side of the square will be x.
After cutting out the squares and folding up the sides, we will have a box with height h and base dimensions (50-2x) by (20-2x).
The volume of the box is given by the equation:
V = (50-2x)(20-2x)h
We want to maximize this equation, so we need to take the derivative with respect to x and set it equal to zero:
dV/dx = -4xh + 40h + 100h/x - 4hx = 0
Simplifying this equation, we get:
x³ - 30x² + 500x - 2000 = 0
We can use a graphing calculator or a numerical method to find the value of x that maximizes this equation. Using a graphing calculator, we get:
x ≈ 7.5 cm
Substituting this value of x into the equation for the volume of the box, we get:
V ≈ 3750 cm³
Therefore, the height of the box that will give a maximum volume is h = 7.5 cm, and the maximum volume is approximately 3750 cm³.
Problem 5: To minimize the cost of constructing the cylindrical can, we need to find the optimal radius (r) and height (h). The volume of the cylinder is given by V = πr^2h = 105 mm³. The cost of the materials is given by the sum of the cost of the wall, bottom, and top: C = 3(2πrh) + 7(πr^2) + 2(πr^2).
To minimize the cost, we'll use calculus. First, we can express h in terms of r using the volume equation: h = 105/(πr^2). Then, substitute this into the cost equation and differentiate with respect to r. Finally, set the derivative equal to zero and solve for r. After obtaining the optimal r, find the corresponding h using the volume equation. These values will minimize the cost of constructing the can.
Problem 6: For this problem, let x be the length of the square cut out from each corner. The dimensions of the folded box will be (50-2x) cm x (20-2x) cm x x cm. The volume of the box is V = x(50-2x)(20-2x). To maximize the volume, differentiate V with respect to x, set the derivative equal to zero, and solve for x. This will give you the height (x) that results in the maximum volume for the box.
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Related Questions
An jar contains 5 green marbles and 9 purple marbles. A marble is drawn and dropped back into the jar. Both marbles are green. If another marble is drawn, what is the probability that it will be green?
Answers
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the formula for Probability
\(Probability=\frac{number\text{ of required events}}{number\text{ of total possible outcomes}}\)STEP 2: Write the different outcomes
\(\begin{gathered} n(green)=5 \\ n(purple)=9 \\ n(Total)=5+9=14 \end{gathered}\)STEP 3: Find the probability that the third marble drawn is green
Since it can be seen from the question that the selection was done with replacement, this means that the sample space of 5 green marbles and 9 purple marbles are not affected. The probability of this outcome using the formula in step 1 will be given by:
\(\begin{gathered} P(green)=\frac{n(green)}{n(Total)} \\ \\ P(green)=\frac{5}{14} \end{gathered}\)The probability that it will be green is 5/14
Brandon take a rectangular piece of fabric and make a diagonal cut from one corner to the oppoite corner. The cut he make i 13 inche long and the width of the fabric i 5 inche. What i the fabric' length?
Answers
The length of the fabric which Brandon formed a rectangle, is 11 inches.
To find the length of the fabric, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the length of the fabric is one of the other two sides, and the diagonal cut is the hypotenuse. So, we can write the equation:
\(L^2 + 5^2 = 13^2\)
where L is the length of the fabric.
Solving for L, we get:
\(L^2 = 144 - 25 = 119, and L =\sqrt{119} = 11.\)
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the set of all positive integers that are divisible by both 15 and 35 is infinite. what is the least positive integer in this set?550105210525
Answers
The least positive integer that will be divisible by both 15 and 35 will be 105.
The given integers are 15 and 35.
As we know that the least positive integer will be divisible by both 15 and 35 will be LCM ( least common factor) of 15 and 35.
The least common multiple of LCM of two integers is the common multiple of two numbers such that it is the least among all common multiples.
For example, LCM of 3 and 5 will be 15 because among all common multiples of 3 and 5, 15 will be least common multiple.
For LCM of 15 and 35, let's write multiples of both the numbers.
Multiples of 15 = 15,30,45,60,75,90,105,120,135,150,165,180,195,210,....
Multiples of 35= 35,70,105,140,175,210,...
We can see that 105 and 210 are two common multiples out of which 105 is the least multiple.
Therefore, the least positive integer that will be divisible by both 15 and 35 will be 105.
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which table represents viable solutions for y=5x, where x is the number of tickets sold for the school play a y is the amount of money collected for the tickets?
Answers
Given:
The given equation is:
\(y=5x\)
Where x is the number of tickets sold for the school play a y is the amount of money collected for the tickets.
To find:
The correct table of values from the given options.
Solution:
We know that the number of tickets and amount of money collected for the tickets cannot be negative. So, options A and B are incorrect.
We have,
\(y=5x\)
For \(x=0\),
\(y=5(0)\)
\(y=0\)
For \(x=10\),
\(y=5(10)\)
\(y=50\)
For \(x=51\),
\(y=5(51)\)
\(y=255\)
For \(x=400\),
\(y=5(400)\)
\(y=2000\)
In option C, all the ordered pairs (0,0), (10,50), (51,255), (400,2000) are in the table. So, option C is correct.
For \(x=65\)
\(y=5(65)\)
\(y=325\)
Since the ordered pair (65,350) does not satisfy the given equation, therefore the option D is incorrect.
Hence, the correct option is C.
help me pls with this one its my last question
Answers
Answer:
The answer is 3.
Step-by-step explanation:
6x - 48 = -15y2x + 5y = 16
Answers
Let's begin by listing out the information given to us:
\(\begin{gathered} 6x-48=-15y-----1 \\ 2x+5y=16-------2 \end{gathered}\)We will solve these equation simultaneously following these steps:
I. We will pick a number to multiply the equation such that the values of corresponding variables are equal
\(\begin{gathered} 6x-48=-15y-----1 \\ 2x+5y=16-------2 \\ \text{Multiply equation 2 by 3, so we can eliminate the x variable} \\ 6x+15y=48------3 \\ \text{Combine equations 1 \& 3, we have:} \\ 6x-48=-15y-----1 \\ 6x+15y=48------3 \end{gathered}\)II. Subtract equation 3 from 1, we have:
\(\begin{gathered} 6x-48=-15y-----1 \\ \operatorname{Re}arranging\text{ equation 1, we have:} \\ 6x+15y=48------1 \\ 6x+15y=48------3 \\ \text{Equation 1 - 3 is:} \\ 6x-6x+15y-15y=48-48 \\ 0+0=0\Rightarrow0 \end{gathered}\)The solution for these equations is non-existent
A rectangle has sides measuring (3x + 5) units and (6x + 11) units. Part a: what is the expression that represents the area of the rectangle? show your work to receive full credit. Part b: what are the degree and classification of the expression obtained in part a? part c: how does part a demonstrate the closure property for polynomials?.
Answers
Part a: The expression that represents the area of the rectangle is A = (3x + 5)(6x + 11). To calculate the area, we must multiply the two sides of the rectangle together. The length of the rectangle is 3x + 5 units and the width of the rectangle is 6x + 11 units. Therefore, the area is (3x + 5)(6x + 11).
Part b: The degree of the expression obtained in part a is 2 and the classification is a binomial.
Part c: The closure property for polynomials states that the result of combining two polynomials is also a polynomial. In part a, we combined two polynomials, 3x + 5 and 6x + 11, to form the expression (3x + 5)(6x + 11). This expression is also a polynomial, demonstrating the closure property for polynomials. The closure property for polynomials allows us to combine two polynomials and still have a polynomial as the result. This is important in mathematics because it allows us to simplify equations and solve problems more quickly and efficiently.
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An angle measures . What is the measure of its complement?
Answers
Step-by-step explanation:
What does the angle measure so I can answer your question
Solve for x
-5(-2-5)-x-1=-12
Answers
Answer:
46
Step-by-step explanation:
-5(-2-5)-x-1=-12
-5(-7)-x-1=- 12
35 - x - 1 = -12
34 - x = -12
-x = -12 - (-34)
-x = - 46
x = 46
Answer:
\(x=46\)
Step-by-step explanation:
\(-5\left(-2-5\right)-x-1=-12\)
⟹ Subtract -2 and 5 = -7
\(5\left(-2-5\right)\)\(5\left(-7\right)\)⟹ Remover parentheses:
\(5\times7\)\(-35\)\(-\left(-35\right)-x-1=-12\)
⟹ Apply rule: \(-\left(-a\right)=a\)
\(35-x-1=-12\)
⟹ Combine like terms:
\(-x+35-1=-12\)\(\:35-1=34\)\(-x+34=-12\)⟹ Subtract 34 from both sides:
\(-x+34-34=-12-34\)\(-x=-46\)⟹ Divide both sides by -1:
\(\frac{-x}{-1}=\frac{-46}{-1}\)\(x=46\)___________________
OAmalOHopeO
___________________
-3|x|+2x-1 if x=-5
i need help with this or else i will not get my computer
Answers
Answer:
-26
Step-by-step explanation:
Hello There!
For this problem all we have to do is plug in -5 into x and calculate the value
so..
-3|-5|+2(-5)-1
so first things first | | means absolute value
The absolute values of -5 is 5 so the equation would be
-3(5) + 2(-5) - 1
-3*5=-15
2*-5=-10
-15-10=-25
-25-1=-26
so the answer would be -26
problem e.6 find the general solution to the following system of linear differential equations. the independent variable is ????. one eigenvalue and the corresponding eigenvector are provided. x1 ′
Answers
The general solution to the given system of linear differential equations is:
X(t) = c₁\(e^{-t}\)[a₁, a₂, a₃]ᵀ + c₂\(e^t\)[1, -1, 0]ᵀ + c₃\(e^{2t}\)[2, 1, -1]ᵀ. Here, c₁, c₂, and c₃ are constants determined by the initial conditions, and [a₁, a₂, a₃]ᵀ is the given eigenvector.
To find the general solution to the given system of linear differential equations, let's start by rewriting the system in matrix form:
X' = AX + B
where X = [x₁, x₂, x₃]ᵀ, X' is the derivative of X with respect to t, and A and B are matrices defined as follows:
A = [[-1, 2, 4],
[-1, 0, 1],
[2, 1, 1]]
B = [2, 0, 0]ᵀ
To find the general solution, we need to compute the eigenvalues and eigenvectors of matrix A. You mentioned that one eigenvector is provided, so let's denote it as v₁.
Eigenvalues (λ₁, λ₂, λ₃) and eigenvectors (v₁, v₂, v₃) of matrix A are:
λ₁ = -1 (provided)
v₁ = [a₁, a₂, a₃]ᵀ
To find the remaining eigenvalues and eigenvectors, we can solve the characteristic equation:
|A - λI| = 0
where I is the identity matrix.
Let's solve for the remaining eigenvalues (λ₂, λ₃) and eigenvectors (v₂, v₃).
Using the eigenvalue λ = -1 and eigenvector v₁, we have:
(A + I)v₁ = 0
Substituting the values of A and λ, we get:
[0, 2, 4]ᵀa = 0
Solving this system of equations, we find that the eigenvector v₁ is:
v₁ = [1, -2, 1]ᵀ
Now, let's find the remaining eigenvalues and eigenvectors by solving the characteristic equation:
|A - λI| = 0
Substituting the values of A and λ, we get:
|[-1, 2, 4],
[-1, 1, 1],
[2, 1, 2] - λ[I]| = 0
Expanding the determinant and solving, we find the remaining eigenvalues:
λ₂ = 1
λ₃ = 2
To find the corresponding eigenvectors, we solve the equations:
(A - λ₂I)v₂ = 0
(A - λ₃I)v₃ = 0
Solving these systems of equations, we find the eigenvectors:
v₂ = [1, -1, 0]ᵀ
v₃ = [2, 1, -1]ᵀ
Now that we have the eigenvalues and eigenvectors, we can write the general solution:
X(t) = c₁\(e^{\lambda_1t}\)v₁ + c₂\(e^{\lambda_2t}\)v₂ + c₃\(e^{\lambda_3t}\)v₃
where c₁, c₂, and c₃ are constants determined by the initial conditions.
By substituting the values of the eigenvalues and eigenvectors, we obtain the final general solution for X(t):
X(t) = c₁\(e^{-t}\)[a₁, a₂, a₃]ᵀ + c₂\(e^t\)[1, -1, 0]ᵀ + c₃\(e^{2t}\)[2, 1, -1]ᵀ
The complete question is:
Find the general solution to the system of linear differential equations. The independent variable is t. All of the eigenvalues and one of the eigenvectors are provided for you.
2x₁ + 2x₂ + 4x₃ = 2-x₁'
- x₁ + x₂' = 2x₂ + x₃
x₃' = 2x₁ + x₂ + x₃
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Express the number 3,440,000 in Scientific Notation.
Answers
Answer:
3.44 * 10^6
Step-by-step explanation:
3,440,000
3.44 * 10^6
Answer: It would be 3.44 × 10^6.
What is 2x + 2x + 2 = 4x + 2
Answers
Answer:
I think the answer is 33 I'm not sure but that is what I got
If the speed of an object in motion is doubled, its kinetic energy becomes how many times the original kinetic energy
Answers
Answer: Becomes four times
Step-by-step explanation:
Given
Speed is doubled for a moving object
Suppose initial speed is u
Increased speed is 2u
Kinetic Energy is given by
\(\Rightarrow K=0.5mu^2\)
When speed is doubled
\(\Rightarrow K'=0.5m(2u)^2\\\Rightarrow K'=(0.5mu^2)\times 4\\\Rightarrow K'=4K\)
Kinetic energy becomes four times
Ken thought that the distance between his new home and work was 32 miles. Upon verifying it, he found the distance was 41.5 miles. Find Ken’s percent error.
Answers
Show that ∑
i=1
6
(dx
i
+e)=d(∑
i=1
6
x
i
)+6e 2. Show the equation below in a Sigma operator notation: (5x
3
+4)+(5x
3
+5)+(5x
3
+6)+(5x
3
+7)+(5x
3
+8)+(5x
3
+9)
Answers
Since both sides are equal, we have shown that ∑(i=1 to 6) (dx_i + e) = d(∑\((i=1 to 6) x_i\)) + 6e. This represents the summation of the terms \((5x_3 + i)\) for i = 1 to 6.
To show that ∑(i=1 to 6) \((dx_i + e)\)= d(∑(i=1 \(x_i\)) + 6e, we can expand both sides and compare.
Left-hand side:
∑\((i=1 to 6) (dx_i + e) = (dx_1 + e) + (dx_2 + e) + (dx_3 + e) + (dx_4 + e) +\)(dx_5 + \(e) + (dx_6 + e)\)
= \(dx_1 + dx_2 + dx_3 + dx_4 + dx_5 + dx_6 + e + e + e + e\)+ e + e
= \((dx_1 + dx_2 + dx_3 + dx_4 + dx_5 + dx_6) + 6e\)
Right-hand side:
d(∑\((i=1 to 6) x_i)\)+ 6e = d\((x_1 + x_2 + x_3 + x_4 + x_5 + x_6)\)+ 6e
Now, let's compare the two sides:
Left-hand side: \((dx_1 + dx_2 + dx_3 + dx_4 + dx_5 + dx_6)\) + 6e
Right-hand side: d\((x_1 + x_2 + x_3 + x_4 + x_5 + x_6)\) + 6e
Since both sides are equal, we have shown that ∑\((i=1 to 6) (dx_i + e)\) = d(∑(i=1 to 6)\(x_i)\) + 6e.
To represent the equation\((5x_3 + 4) + (5x_3 + 5) + (5x_3 + 6) + (5x_3 +\)7) + \((5x_3 + 8) + (5x_3 + 9)\) using a Sigma operator notation, we can write it as:
∑\((i=1 to 6) (5x_3 + i)\)
This represents the summation of the terms \((5x_3 + i)\)for i = 1 to 6.
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please answer
12a=6a+24
Answers
Answer:
a=4
Step-by-step explanation:
Hope this helps! Plz give brainliest!
Answer:
Subtract 6a6a from both sides.
12a-6a=24
12a−6a=24
2 Simplify 12a-6a12a−6a to 6a6a.
6a=24
6a=24
3 Divide both sides by 66.
a=\frac{24}{6}
a=
6
24
4 Simplify \frac{24}{6}
6
24
to 44.
a=4
a=4
Step-by-step explanation:
Julio left the science museum and traveled toward his friend's house at an average speed of 60 km/h. Kristin left sometime later traveling in the opposite direction with an average speed of 60 km/h. After Julio had traveled for three hours they were 300 km apart. Find the number of hours Kristin traveled.
Answers
Answer: 150
Step-by-step explanation: Half of 300 is 150.
the answer is :150
Step-by-step explanation: Half of 300 is 150. Half of 300 is 150.
*Best answer will get brainliest*
Find the product.
-8x5y2 · 6x2y
A. -8x10y2
B. -8x7y3
C. -48x7y3
D. -48x10y2
Answers
Explanation:
The coefficients are -8 and 6, which are to the left of the variable expressions. The coefficients multiply to -48. Based on this alone, the answer is between choice C or choice D.
For the variables, we add the exponents. The x terms multiply to x^5*x^2 = x^(5+2) = x^7
The y terms multiply to y^2*y = y^2*y^1 = y^(2+1) = y^3
Putting everything together, we end up with -48x^7y^3
\( \huge \boxed{\mathfrak{Question} \downarrow}\)
Find the product of -8x⁵y² · 6x²y.\( \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}\)
\( \sf \: - 8 x ^ { 5 } y ^ { 2 } \cdot 6 x ^ { 2 } y\)
Use the rules of exponents to simplify the expression.
\( \sf\left(-8\right)^{1}x^{5}y^{2}\times 6^{1}x^{2}y^{1} \)
Use the Commutative Property of Multiplication.
\( \sf\left(-8\right)^{1}\times 6^{1}x^{5}x^{2}y^{2}y^{1} \)
To multiply powers of the same base, add their exponents.
\( \sf\left(-8\right)^{1}\times 6^{1}x^{5+2}y^{2+1} \)
Add the exponents 5 and 2.
\( \sf\left(-8\right)^{1}\times 6^{1}x^{7}y^{2+1} \)
Add the exponents 2 and 1.
\( \sf\left(-8\right)^{1}\times 6^{1}x^{7}y^{3} \)
Multiply -8 times 6 to get -48.
\( \boxed{ \boxed{\bf \: C) \: -48x^{7}y^{3} }}\)
would 3x+2>-4 be a
open circle —->
open circle <——
closed circle—->
closed circle<—-
Answers
.Martin is an after-school math tutor. He noticed that 6 people still needed help and only Two-fifths of the tutoring session time was left. Since each person is to be given an equal amount of time, Martin wrote the expression below to find the fraction of the tutoring session each person who still needed help would be allotted.
Two-fifths divided by 6
Which expression is equivalent to Martin’s expression?
StartFraction 6 Over 15 EndFraction divided by 6
StartFraction 6 Over 5 EndFraction divided by 2
Five-halves times 6
Two-fifths times 6
Answers
Answer:
6/15 divided by 6
Step-by-step explanation:
\(\frac{2}{5} / 6\\= \frac{6}{15} / 6\\\)
find the volume 9m by 5m by 9m
Answers
Answer:
405m³
Step-by-step explanation:
I will assume these are the dimensions of a square or rectangle. To find the volume, we can use the formula [ v = b * h * l ].
v = 9 * 5 * 9
v = 45 * 9
v = 405m³
Best of Luck!
Step-by-step explanation:
volume=length×breadth×height
=9×5×9
405
can anyone help me!!
Answers
6 : AB = tan30
0.58AB = 6
AB ≈ 10.3
Answer:
Step-by-step explanation:
by writing the formula of tan(30°), you'll get the answer.
Let's write it :
tan(x)=\(\frac{opposite}{adjacent\\}\)
we have a 30° angle so x=30° :
tan(30°)=\(\frac{AB}{AC} = \frac{AB}{6} =\frac{1}{\sqrt{3} }\)
⇒ √3 * AB = 6 ⇒ AB= \(\frac{6}{\sqrt{3} }\)≅ 3.46
using dijkstra’s algorithm, find the sink tree rooted at vertex 7.
Answers
Dijkstra's algorithm calculates the shortest path from vertex 7 to all other vertices in the graph, forming a tree structure where vertex 7 is the root.
Dijkstra's algorithm is a graph traversal algorithm used to find the shortest path between two vertices in a weighted graph. To find the sink tree rooted at vertex 7, we can apply Dijkstra's algorithm starting from vertex 7. The algorithm proceeds by iteratively selecting the vertex with the smallest distance from the current set of vertices and updating the distances to its adjacent vertices.
Starting from vertex 7, we initialize the distance of vertex 7 as 0 and the distances of all other vertices as infinity. Then, we explore the adjacent vertices of vertex 7 and update their distances accordingly. We repeat this process, selecting the vertex with the smallest distance each time, until we have visited all vertices in the graph.
The result of applying Dijkstra's algorithm to find the sink tree rooted at vertex 7 is a tree structure that represents the shortest paths from vertex 7 to all other vertices in the graph. Each vertex in the tree is connected to its parent vertex, forming a directed acyclic graph. This sink tree provides a clear visualization of the shortest paths and their corresponding distances from vertex 7 to each vertex in the graph.
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Evaluate x4 + 4x3 - 2x2 + 11x - 6 for x = 3
Answers
Step-by-step explanation:
\(f(x) = {x}^{4} + 4 {x}^{3} - 2 {x}^{2} + 11x - 6\)
\(f(3) = {3}^{4} + 4. {3}^{3} - 2. {3}^{2} + 11.3 - 6\)
\(f(3) = 81 + 4.27 - 2.9 + 33 - 6\)
\(f(3) = 81 + 108 - 18 + 33 - 6\)
\(f(3) = 198\)
Answer:
198Step-by-step explanation:
x4 + 4x3 - 2x2 + 11x - 6 for x = 3
3^4 + 4 * 3^3 - 2 * 3^2 + 11 * 3 - 6 =
81 + 4 * 27 - 2 * 9 + 11 * 3 - 6 =
81 + 108 - 18 + 33 - 6 =
198
two ladders, one that is 6 6 feet long and one that is 9 9 feet long, are leaning up against a building. both ladders are leaning so that the angle they make with the ground is the same. the shorter ladder touches the wall at a point that is 5 5 feet 9 9 inches above the ground. how much higher above the ground does the second ladder touch the wall above the shorter ladder?
Answers
The second ladder touches the wall approximately 11 feet higher than the shorter ladder, or equivalently, around 8 feet 8 inches higher.
Let's denote the height at which the second ladder touches the wall as h. We can set up a proportion based on the similar right triangles formed by the ladders and the building:
(6 6 feet) / (h) = (9 9 feet) / (5 5 feet 9 9 inches + h)
To solve for h, we can cross-multiply and solve the resulting equation:
(6 6 feet) * (5 5 feet 9 9 inches + h) = (9 9 feet) * (h)
Converting the measurements to inches:
(66 inches) * (66 inches + h) = (99 inches) * (h)
Expanding and rearranging the equation:
4356 + 66h = 99h
33h = 4356
Solving for h:
h = 4356 / 33 = 132 inches
Converting back to feet and inches:
h ≈ 11 feet
Therefore, the second ladder touches the wall approximately 11 feet higher than the shorter ladder, or equivalently, around 8 feet 8 inches higher.
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Which division problem does the number line below best illustrate?
0 1 2 3 4 5 6 7 8 9 10 11 12 13
O 12-3-4
O 9-3-3
O 12-2-6
o 16-4-4
Answers
Answer:
12/3=4 ..............
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how can you test to see if a given point is in the solution set of a system of inequalities?
Answers
Answer:
you can plug the point into the inequality and see if it works
Step-by-step explanation:
If I have xy>4 and I want to test the point 4,6 I would take xy which would be 4 and 6 because those are the numbers from the coordinates. when writing coordinates x is the first number and y is the second. 4 times 6 because x times y which is xy is equal to 24 now we have 24>4 and this is true
Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A
Answers
An invertible matrix P and a diagonal matrix D such that A = PDP⁻¹ is shown below.
What is a matrix?A matrix is a rectangular array or table of numbers, symbols, or expressions that are arranged in rows and columns to represent a mathematical object or a property of such an object in mathematics. For instance, consider a matrix with two rows and three columns.To find an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹:
Let \(\lambda\) be an eigenvalue of A. Then:
\(\begin{aligned}0 &=|A-\lambda I| \\&=\left|\begin{array}{ccc}-11-\lambda & 3 & -9 \\0 & -5-\lambda & 0 \\6 & -3 & 4-\lambda\end{array}\right| \\&=-(0)\left|\begin{array}{cc}3 & -9 \\-3 & 4-\lambda\end{array}\right|+(-5-\lambda)\left|\begin{array}{cc}-11-\lambda & -9 \\6 & 4-\lambda\end{array}\right|-(0)\left|\begin{array}{cc}-11-\lambda & 3 \\6 & -3\end{array}\right| \\&=(-5-\lambda)[(-11-\lambda)(4-\lambda)+54] \\&=-(5+\lambda)^{2}(2+\lambda)\end{aligned}\)
Therefore, \(\lambda = -2\) or \(\lambda = -5\).
Let \(\mathbf{v}=\left[\begin{array}{l}a \\b \\c\end{array}\right]\) be an eigenvector associated with the eigenvalue \(\lambda\).\(\lambda = -2\): From \((A+2 I) \mathbf{v}=\mathbf{0}\) we obtain:
\(\left[\begin{array}{ccc}-9 & 3 & -9 \\0 & -3 & 0 \\6 & -3 & 6\end{array}\right]\left[\begin{array}{l}a \\b \\c\end{array}\right]=\left[\begin{array}{l}0 \\0 \\0\end{array}\right]\)
Applying elementary row operations to find the reduced echelon form of the coefficient matrix, we obtain:
\(\left[\begin{array}{ccc}-9 & 3 & -9 \\0 & -3 & 0 \\6 & -3 & 6\end{array}\right] \stackrel{\left(-\frac{1}{9}\right) R_{1} \rightarrow R_{1}}{\longrightarrow}\left[\begin{array}{rrr}1 & -\frac{1}{3} & 1 \\0 & -3 & 0 \\6 & -3 & 6\end{array}\right] \stackrel{(-6) R_{1}+R_{3} \rightarrow R_{3}}{\longrightarrow}\left[\begin{array}{rrr}1 & -\frac{1}{3} & 1 \\0 & -2 & 0 \\0 & -1 & 0\end{array}\right] \stackrel{\left(-\frac{1}{2}\right) R_{2} \rightarrow R_{2}}{\longrightarrow}\)
\(\left[\begin{array}{ccc}1 & -\frac{1}{3} & 1 \\0 & 1 & 0 \\0 & -1 & 0\end{array}\right] \stackrel{\left(\frac{1}{3}\right) R_{2}+R_{1} \rightarrow R_{1}}{\stackrel{R_{2}+R_{3} \rightarrow R_{3}}{\longrightarrow}}\left[\begin{array}{ccc}1 & 0 & 1 \\0 & 1 & 0 \\0 & -1 & 0\end{array}\right] \stackrel{R_{2}+R_{3} \rightarrow R_{3}}{\longrightarrow}\left[\begin{array}{ccc}1 & 0 & 1 \\0 & 1 & 0 \\0 & 0 & 0\end{array}\right]\)
Hence, we have a + c = 0 and b = 0. Let c = -1 and a = 1.So, \(\left[\begin{array}{c}1 \\0 \\-1\end{array}\right]\) is an eigenvector associated with \(\lambda = -2\).
\(\lambda = -5\): From \((A+5 I) \mathbf{v}=\mathbf{0}\) we obtain:
\(\left[\begin{array}{ccc}-6 & 3 & -9 \\0 & 0 & 0 \\6 & -3 & 9\end{array}\right]\left[\begin{array}{l}a \\b \\c\end{array}\right]=\left[\begin{array}{l}0 \\0 \\0\end{array}\right]\)
Applying elementary row operations to find the reduced echelon form of the coefficient matrix, we obtain:
\(\left[\begin{array}{ccc}-6 & 3 & -9 \\0 & 0 & 0 \\6 & -3 & 9\end{array}\right] \stackrel{\left(-\frac{1}{6}\right) R_{1} \rightarrow R_{1}}{\longrightarrow}\left[\begin{array}{ccc}1 & -\frac{1}{2} & \frac{3}{2} \\0 & 0 & 0 \\6 & -3 & 9\end{array}\right] \stackrel{(-6) R_{1}+R_{3} \rightarrow R_{3}}{\longrightarrow}\left[\begin{array}{ccc}1 & -\frac{1}{2} & \frac{3}{2} \\0 & 0 & 0 \\0 & 0 & 0\end{array}\right]\)
Hence, we have a - 1/2b + 3/2c = 0. Let b = 2 and c = 0.Then a = 1. Let b = 0 and c = -2. Then a = 3. Therefore,
\(\left[\begin{array}{l}1 \\2 \\0\end{array}\right],\left[\begin{array}{c}3 \\0 \\-2\end{array}\right]\)
are two linearly independent vectors associated with \(\lambda = -5\)
Matrices P and D: Let
\(P=\left[\begin{array}{ccc}1 & 1 & 3 \\0 & 2 & 0 \\-1 & 0 & -2\end{array}\right]\)
be the matrix whose columns are the eigenvectors obtained in the previous step. Set
\(Q=\left[\begin{array}{ccc}-2 & 0 & 0 \\0 & -5 & 0 \\0 & 0 & -5\end{array}\right]\)
to be the diagonal matrix whose diagonal entries are the eigenvalues.
Thus we have, A = PDP⁻¹.
Therefore an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹ is shown.
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The correct question is given below:
Diagonalize matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹.
\(A=\left[\begin{array}{ccc}-11 & 3 & -9 \\0 & -5 & 0 \\6 & -3 & 4\end{array}\right]\)
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Answers
Answer:
(-4, 3) 2nd quadrant.
Step-by-step explanation: