Two boats on opposite banks of a river start moving towards each other. They first pass each other 1400 meters from one bank. They each continue to the opposite bank, immediately turn around and start back to the other bank. When they pass each other a second time, they are 600 meters from the other bank. We assume that each boat travels at a constant speed all along the journey. Find the width of the river?
S1 = speed of boat 1
t1 = time to do 1400 meters (boat 1)
S1*t1 = 1400
S2 = speed of boat 2
1400 + S2*t1 = X
t2 = time to do X + 600 (boat 2)
S1*t2 = X + 600
S2*t2 = 2X - 600
S1 = 1400/t1
S2 = (X-1400)/t1
T = t2/t1
1400*T = X + 600
X*T - 1400*T = 2X - 600
X = 3600 meters
River is 3600 meters wide
Andrew has 48 country CDs and 72 rock CDs. He wants to arrange them on shelves so that the two types of CDs are separate, but he wants the largest number of CDs on a shelf as possible. How many CDs should he put on each shelf?
1. What is the remainder when x^2+4 is divided by x-2?
2. Evaluate f(-1) using substitution: f(x)=2x^3-3x^2-18x-8
3. The point (1,0) lies on the graph of p(x)=x^4-2x^3-x+2
True or false
4. If x-1 is a factor of p(x)=x^3-7x^2+15x-9, which of the following represents the complete factorization for p(x)?
A. (x-3)(x+4)(x+1)
B. (x-3)(x-3)(x-1)
C. (x-3)(x+3)(x-1)
D. (x-3)(x+3)(x+1)
Answer:
1) 8
2) 5
3) False
4) Option B
Step-by-step explanation:
1) We have to find the remainder when we divide (x² + 4) by (x - 2)
To get the remainder we will put x = 2 in (x² + 4)
= (2)² + 4
= 8
2). We have to evaluate f(-1) using substitution in f(x) = 2x³ - 3x² - 18x - 8
f(-1) = 2(-1)³ - 3(-1)²- 18(-1) - 8
= 2(-1) - 3 + 18 -8
= -2 - 3 + 18 - 8
= 5
3) The point (1, 0) lies on the graph of p(x) = [tex]x^{4}-2x^{3}-x+2[/tex]
If this point lies on the graph then p(1) should be equal to zero.
p(1) = 1³ - 7(1)² + 7(1) - 9
= 1 - 7 + 7 - 9
= -8 ≠ 0
Therefore, It's false.
4). (x - 1) is a factor of p(x) = x³ - 7x² + 15x - 9
Now we will factorize it further when (x - 1) is a zero factor.
By Synthetic division
1 | 1 - 7 15 -9
1 -6 9
-----------------------------
1 -6 9 0
Now we have got the expression as (x - 1)(x²- 6x + 9)
Or (x -1)(x² - 6x + 9) = (x - 1)(x - 3)(x - 3)
Therefore, Option B. is the answer.
Find the number of degrees in an angle which is 42 degrees less than its complement
The required angle can be found by solving the equation as 24°.
How to solve a linear equation?A linear equation can be solved by equating the LHS and RHS of the equation following some basic rules such as by adding or subtracting the same numbers on both sides and similarly, doing division and multiplication with the same numbers.
Suppose the measure of required angle be x.
Then, its complement can be written as 90 - x.
As per the question, the following equation can be formed as,
x + 42 = 90 - x
⇒ 2x = 48
⇒ x = 24
Hence, the measure of the required angle is 24°.
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Use the chain rule to find dz/dt calculator
The chain rule allows you to find the derivative of a function that is a composition of other functions. With an example, we calculated dz/dt with z = y^3 and y = 2t+1. While calculators can help, understanding the process is key.
Explanation:The chain rule in calculus is a theorem that allows you to find the derivative of a composite function. If a variable z depends on the variable y, which itself depends on another variable t, then z, is a function of t. The chain rule could be mathematically expressed as dz/dt = (dz/dy) × (dy/dt).
For instance, if z = y^3 and y = 2t+1, we can calculate dz/dt by first finding the derivatives dz/dy and dy/dt, which are 3y² and 2, respectively. We then substitute y = 2t+1 into dz/dy to get dz/dy = 3(2t+1)². Therefore, using the chain rule, dz/dt = 3(2t+1)² × 2.
While your request mentions a calculator, knowing these steps should enable you to perform the operation using most scientific calculators. However, understanding the process is very important before relying on technology.
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The final expression for [tex]\frac{dz}{dt}[/tex] incorporates the partial derivatives and the derivatives of x and y with respect to t.
Given the function z = xy⁷ - x²y, where x = t² + 1 and y = t² - 1, we are to find [tex]\frac{dz}{dt}[/tex].
To do this, we apply the chain rule for derivatives:[tex]\frac{dz}{dt}[/tex] = ([tex]\frac{dz}{dx}[/tex]) × ([tex]\frac{dx}{dt}[/tex]) + ([tex]\frac{dz}{dy}[/tex]) × ([tex]\frac{dy}{dt}[/tex]).
First, we calculate [tex]\frac{dx}{dt}[/tex] and [tex]\frac{dy}{dt}[/tex].Then, we find the partial derivatives [tex]\frac{dz}{dx}[/tex] and [tex]\frac{dz}{dy}[/tex].Finally, we combine these results to find dz/dt using the chain rule.Let's find [tex]\frac{dx}{dt}[/tex] and [tex]\frac{dy}{dt}[/tex]:
⇒ [tex]\frac{dx}{dt}[/tex] = d(t² + 1) ÷ dt = 2t
⇒ [tex]\frac{dy}{dt}[/tex] = d(t² - 1) ÷ dt = 2t
Next, we compute the partial derivatives of z with respect to x and y:⇒ [tex]\frac{dz}{dx}[/tex] = y⁷ - 2xy
⇒ [tex]\frac{dz}{dy}[/tex] = 7xy⁶ - x²
Now, we substitute x and y into these partial derivatives:x = t² + 1, y = t² - 1
⇒ [tex]\frac{dz}{dx}[/tex] = (t² - 1)⁷ - 2(t² + 1)(t² - 1)
⇒ [tex]\frac{dz}{dy}[/tex] = 7(t² + 1)(t² - 1)⁶ - (t² + 1)²
Finally, we combine these to find [tex]\frac{dz}{dt}[/tex]:
⇒ [tex]\frac{dz}{dt}[/tex] = ([tex]\frac{dz}{dx}[/tex]) × ([tex]\frac{dx}{dt}[/tex]) + ([tex]\frac{dz}{dy}[/tex]) × ([tex]\frac{dy}{dt}[/tex]).
Substituting everything into this formula:
⇒ [tex]\frac{dz}{dt}[/tex] = [(t² - 1)⁷ - 2(t² + 1)(t² - 1)] × 2t + [7(t² + 1)(t² - 1)⁶ - (t² + 1)²] × 2t
Complete question:
Use the Chain Rule to find [tex]\frac{dz}{dt}[/tex].
z = xy⁷ - x²y, x= t² + 1, y = t² - 1
.....Simplify (8^4)^–16.
Malcom uses a marker to draw a straight line on a piece of paper. The line is 2/3 ft long. He wants to divide the line into sections that are each 1/9 ft long. How many sections will the line be divided into? please help ill mark brainliest
Answer:
6 sections.
Step-by-step explanation:
Malcom uses a marker to draw a straight line on a piece of paper.
The line is [tex]\frac{2}{3}[/tex] feet long.
He wants to divide the line into sections of [tex]\frac{1}{9}[/tex] feet long.
The number of sections would be = [tex]\frac{\frac{2}{3}}{\frac{1}{9} }[/tex]
= [tex]\frac{2}{3}[/tex] × [tex]\frac{9}{1}[/tex]
= [tex]\frac{18}{3}[/tex]
= 6 sections
There would be 6 sections.
The frequency distribution shows the lengths, in inches, of available baseball bat at a batting cage. what is the mean of the frequency distribution? Baseball bat length| 27 28 31 33 34 F| 2 5 3 1 4
if r is five less than three times L, then r=
Answer: r=L-3
Step-by-step explanation:
How do you write 11/ 4 as a percentage?
Suppose △ABC≅△EFG. Which congruency statement is true? AB¯¯¯¯¯≅FG¯¯¯¯¯ BC¯¯¯¯¯≅FG¯¯¯¯¯ AC¯¯¯¯¯≅EF¯¯¯¯¯ AB¯¯¯¯¯≅EG¯¯¯¯¯
A = E
B = F
C = G
Look for the equation which has the same placement. Your answer should be B) BC=FG
Answer: Second option [tex]BC\cong FG[/tex] is the correct option.
Explanation:
If two triangles are congruent then their corresponding sides are congruent.
In [tex]\triangle ABC[/tex] and [tex]\triangle EFG[/tex], AB, BC, and CA are corresponding to EF, FG and GE respectively.
Therefore, if [tex]\triangle ABC\cong \triangle EFG[/tex] then,
[tex]AB\cong EF[/tex],
[tex]BC\cong FG[/tex],
And, [tex]CA\cong GE[/tex]
Therefore, only second option is correct.
Find the percent of decrease. 140 to 84.
Evaluate the function for x = –4c if c = –2.
f(x) = -3^2 - 4x
A.
–160
B.
–256
C.
–224
D.
160
A video store charges $5 per movie, and the fifth movie is free. How much do you actually pay per movie?
The amount paid per movie is $4.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
Charge per movie = $5
The fifth movie charge = $0
Now,
The charge for 5 movies.
= 5 x 4 + 0
= 20
This means,
5 movies cost $20
The cost of one movie.
= 20/5
= $4
Thus,
The cost per movie is $4.
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Solve for x. 45(15x+20)−7x=56(12x−24)+6
The value of x is 559.5.
An algebraic equation can be defined as mathematical statement in which two expressions are set equal to each other.
To solve the equation [tex]\(45(15x+20)7x=56(12x*24)+6\)[/tex], follow these steps:
First, distribute the multiplication on both sides of the equation:
[tex]\(45 \cdot 15x + 45 \cdot 20 - 7x = 56 \cdot 12x - 56 \cdot 24 + 6\)[/tex]
[tex]\(675x + 900 - 7x = 672x - 1344 + 6\)[/tex]
Next, combine like terms on both sides:
[tex]\(668x + 900 = 672x - 1338\)[/tex]
Now, move all terms involving [tex]\(x\)[/tex] to one side and constants to the other side:
[tex]\(668x - 672x = -1338 - 900\)[/tex]
[tex]\(-4x = -2238\)[/tex]
To find [tex]\(x\)[/tex], divide both sides by [tex]\(-4\)[/tex]:
[tex]\(x = \frac{-2238}{-4}\)[/tex]
[tex]\(x = 559.5\)[/tex]
So, the value of x is 559.5.
Sam is a painter. He has 2 1/4 gallons of paint, and it takes 3/4 of a gallon to paint a room. How many rooms can he paint?
A.1
B.2
C.3
D.4
Of a group of 500 people, 60 are fit enough to climb Mount Shasta in California. The group has 300 women and 200 men. 12% of the men are fit to climb the mountain. What is the probability that a person picked at random from the group is a male or is fit to climb Mount Shasta?
The probability that a person picked at random from the group is a male or is fit to climb Mount Shasta is 47.2%.
What is probability?Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.
Probability = Number of favourable outcomes / Number of sample
Take 12 of 200 to determine the number of fit men,
Fit men = 12% × 200
Fit men = 0.12 × 200
Fit men = 24
Since they are asking males and the number of people fits enough, you must subtract the fit men from the fit people so you are not doubling the men
60 - 24 = 36
Add the fit people to the male group,
36+200 = 236
The probability will be calculated as,
P = 236/500
P = 118/250
P = 59/125
P = 47.2%
The probability is 47.5%.
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Use the table and diagram to answer this question.
(I attached the picture)
A right triangle has a hypotenuse of 50 feet and a leg of 14 feet. Which is the length of the other leg? 24 feet 30 feet 42 feet 48 feet
Answer:
The length of the other leg is 48 feet
Step-by-step explanation:
To find the length of the other leg, we will simply follow the steps below;
We will use the Pythagoras theorem formula to solve;
Write down the Pythagoras theorem formula
opposite² + adjacent² = hypotenuse²
Let the value of the other leg be x
From the question given;
hypotenuse = 50, a leg, say the opposite = 14
Lets substitute into our formula;
opposite² + adjacent² = hypotenuse²
14² + x² = 50²
196 + x² = 2500
subtract 196 from both-side of the equation
196 - 196 + x² = 2500 - 196
x² = 2304
Take the square root of both-side of the equation
√x² = √2304
x = 48 feet
The length of the other leg is 48 feet
How do i solve the square root of 54x^9y^6?
Solve for x.
2/5≤x−4
Find the constants m and b in the linear function f(x) = mx + b so that f(4) = 1 and the straight line represented by f has slope -14
What is the density of a 7000 gram brick that is 4 inches x 5 inches x 3 inches?
How can I combine like terms ?
Step-by-step explanation: To understand how to combine like terms, let's look at an example.
9c + 2c
The two parts of this expression, +9c and +2c, are called terms. The numbers in front of the variables, +9 and +2, are called coefficients.
Since the variables, c and c, are identical, the two terms in this problem are called like terms.
Like terms can be added together by simply adding their coefficients.
So in this problem, since 9 + 2 = 11, 9c + 2c is simply 11c.
Use your calculator to find an interval of length 0.01 that contains a root. (enter your answer using interval notation.)
in the problem 4 times 12 equals 48 which numbers are the factors
My sister is 77 years older than i am. the sum of our ages is 3535. find our ages.
what is the x an what is the slope of angle lmn
since line MO bisects, that means both angles are the same
6x-22 = 2x +34
6x = 2x +56
4x=56
x = 56/4 = 14
x=14
6(14)-22 = 62
2(14)+34 = 62
62+62 = 124
angle LMN = 124 degrees
Find the measure of each complementary angles with measure 4x and 5x - 18 degrees
Find the value of p that makes the linear graph y=p-3x pass through the point where the lines 4x-y=6 and 2x-5y=12