Finding the rate of change in something with respect to something is known as differentiation. For Example, Change in y with respect to x is known as the derivative of y with respect to x.
How to solve it?(a) [tex]9{x}^2-{y}^2 = 1[/tex]
Differentiating equation with respect to x :
[tex]9*2x - 2y\frac{dy}{dx} = 0\\ 18x = 2y\frac{dy}{dx}\\9\frac{x}{y} =\frac{dy}{dx}[/tex]
(b)
[tex]9{x}^2-{y}^2 = 1\\9{x}^2-1 = {y}^2 \\\sqrt[2]{9{x}^2-1} = y\\[/tex]
differentiating with respect to x gives:
[tex]\frac{1}{2\sqrt{9{x}^2-1} } *(18x) = \frac{dy}{dx}\\\frac{1}{\sqrt{9{x}^2-1} } *(9x) = \frac{dy}{dx}[/tex]
(c) [tex]9\frac{x}{y} =\frac{dy}{dx}[/tex]
substituting value of y
[tex]\sqrt[2]{9{x}^2-1} = y\\\\\frac{9x}{\sqrt[2]{9{x}^2-1}} = \frac{dy}{dx}[/tex]
Hence the solution is consistent.
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There are 17 goats in the barn 8 goats goes outside how many goats are still in the barn
Total number of goats still in Barn is 9 goats
Given that;
Total number of goats in Barn = 17
Number of goat go outside = 8
Find:
Total number of goats still in Barn
Computation:
Total number of goats still in Barn = Total number of goats in Barn - Number of goat go outside
Total number of goats still in Barn = 17 - 8
Total number of goats still in Barn = 9
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Abbreviate 375,000 using scientific notation is what eactly?
Find the ordered pairs for the x- and y-intercepts of the equation 3x − 2y = 18 and select the appropriate option below.
The x-intercept is (0, 6); the y-intercept is (–9, 0).
The x-intercept is (6, 0); the y-intercept is (0, –9).
The x-intercept is (3, 0); the y-intercept is (0, –2).
The x-intercept is (0, 3); the y-intercept is (–2, 0).
Answer:
The x-intercept is (6, 0); the y-intercept is (0, –9).
Step-by-step explanation:
Is each line parallel, perpendicular, or neither parallel nor perpendicular to a line whose slope is -3/4?
Who is vladimir lenin?
Which of the following graphs represents the function f(x) = x4 - 2x3 - 3x2 + 4x + 1?
Answer: Graph D will be correct graph for the given function.
Explanation:
Given function [tex]f(x) = x^4-2x^3-3x^2+4x+1[/tex]
Since it is a bi-quadratic equation thus it must have 4 roots and (0,1) is one of its point.
Moreover, the degree of the function is even thus the end behavior of the function is[tex]f(x)\to+\infty[/tex], as [tex]x\to-\infty[/tex] and [tex]f(x)\to+\infty[/tex] as [tex]x\to+\infty[/tex]
In graph A, function has four root but it does not have the end behavior same as function f(x).( because in this graph [tex]f(x)\to-\infty[/tex], as [tex]x\to-\infty[/tex] and [tex]f(x)\to-\infty[/tex], as [tex]x\to+\infty[/tex].) so, it can not be the graph of given function.
In graph B, neither it has four root nor it has the end behavior same as function f(x).(because in this graph [tex]f(x)\to+\infty[/tex] as [tex]x\to-\infty[/tex] and [tex]f(x)\to-\infty[/tex] as [tex]x\to+\infty[/tex].) so, it can not be the graph of given function.
In graph C, neither it has four root nor it has the same end behavior as function f(x).(because in this graph [tex]f(x)\to-\infty[/tex] as [tex]x\to-\infty[/tex] and [tex]f(x)\to+\infty[/tex] as[tex]x\to\infty[/tex].) so, it also can not be the graph of given function.
In graph D it has four root as well as it has the same end behavior as the given function. Also it passes through the point (0,1).
Thus, graph D is the graph of given function.
Which of the following points does not lie on the line whose equation is x+ y = 7?
(-10, 17)
(13/3, 8/3)
(-1, -6)
What type of number is
- square root 81?
Select all that apply.
Whole number
Integer
Rational
Irrational
The square root of 81 is 9. Hence, it falls under the categories of Whole number, Integer, and Rational number since it can be a positive whole number without any fractional part and can be written as a ratio.
Explanation:The square root of 81 is 9. Now, considering the types of numbers, 9 is a Whole number, because it is not a fraction, and it does not have a decimal point. It is also an Integer, which are numbers that can be written without a fractional component, including both positive and negative whole numbers plus zero. Lastly, 9 is a Rational number, a number that can be written as a fraction. So, the square root of 81 is all three: Whole number, Integer, and Rational number. It is not an Irrational number, which is a number that cannot be expressed as a ratio of two integers.
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Final answer:
The square root of 81 is 9, which makes it a whole number, an integer, and a rational number. It is not an irrational number.
Explanation:
The type of number the square root of 81 is can be determined simply by calculating the square root. The square root of 81 is 9, since 9 multiplied by 9 equals 81. Now, let's classify this number:
Whole number: Yes, because whole numbers include all the non-negative numbers (0, 1, 2, 3...), and 9 falls into this category.
Integer: Yes, because integers are all whole numbers, both positive and negative, including zero (-3, -2, -1, 0, 1, 2, 3...), so 9 is also an integer.
Rational number: Yes, because rational numbers are ratios of integers (like 2/1 or -3/4), and 9 can be expressed as 9/1.
Irrational number: No, because irrational numbers cannot be expressed as ratios of integers. Examples of irrational numbers include √2 or π.
Therefore, the square root of 81 is a whole number, an integer, and a rational number, but it is not irrational.
In a scatter plot or diagram, what is it called when all the points lie approximately along a straight line?
Suppose that the area of a square is eight times its perimeter. find the length of a side of the square.
Final answer:
To solve the problem, an equation is set where the area of the square (a²) is equal to eight times its perimeter (8(4a)). Solving for a, we find the side length of the square to be 32 inches.
Explanation:
The student is presented with a problem where the area of a square is stated to be eight times its perimeter. Let the side length of the square be a inches. The area of a square is given by a², and the perimeter of a square is given by 4a. According to the problem, we have a² = 8(4a).
Firstly, we set up the equation based on the given information:
a² = 8(4a)
Solving for a:
a² = 32a
a² - 32a = 0
a(a - 32) = 0
Therefore, the side length a can be 0 or 32. Since the side length cannot be 0, the length of a side of the square is 32 inches.
Use the rule 4x to find the missing number on the table.
X Y
0 0
2 8
5 20
7
10 40
Answer: The missing number would be 28.
Step-by-step explanation:
Use the rule
4
to find the missing number on the table.
x y
0 0
2 8
5 20
7 28
10 40
7*4=28.
in the rectangle below, what is x?
Kesara Santiago purchased a riding mower for $2,746.00. She received a $49.95 rebate from the manufacturer and a $38.95 rebate from the store. What is the final price?
Answer: $2657.1
Step-by-step explanation:
Given: The price of a riding mower = $2,746.00
The amount she received by manufacturer as rebate= $49.95
The amount she received by the store as rebate = $38.95
The total amount she received as rebate =[tex]\$49.95+\$38.95=\$88.9[/tex]
Therefore, the final price of the riding mower is given by :-
[tex]\text{Final Price}=\$2746-\$88.95=\$2657.1[/tex]
4. A video game sells at Arnolds for $14.99. Arnold's marks the game up at 40% of the selling price. What is the cost of the game to Arnold?
A. $6.00 B. $9.10 C. $6.50 D. $8.99
Answer:
8.99
Step-by-step explanation:
In America, a person uses about 100 gallons of water each day, which is equal to 1,600 water-filled drinking glasses. How many water-filled drinking glasses would 281 Americans fill in a weeks time?
a. 449,600 water-filled glasses
b. 28,100 water-filled glasses
c. 3,147,200 water-filled glasses
d. 196,700 water-filled glasses
Answer:
c. 3,147,200 water-filled glasses
Step-by-step explanation:
The amount of water consumed by an American per day= 100 gallons
So, the amount of water consumed by an American per week = 7×100 = 700 gallons
In 100 gallons of water 1,600 drinking glasses can be filled.
In 700 gallons of water 7×1,600= 11,200 drinking glasses can be filled.
The amount of water consumed by 281 American per week = 700×281 = 196,700 gallons
As in 700 gallons of water 11,200 glasses can be filled.
So, in 1 gallons of water [tex]\dfrac{11,200}{700}=16[/tex] glasses can be filled.
So, in 196,700 gallons of water [tex]16\times 196,700=3,147,200[/tex] glasses can be filled.
Calculate dy/dx
You need not expand your answer.
y=3x^2-9x+13/2x+4 ...?
How do you solve
x-3/(x-4) + 4= 3x/3
If the purchase price for a house is $445,500, what is the monthly payment if you put 5% down for a 30 year loan with a fixed rate of 6.25%?
Answer:
The monthly payment is $2603.17
Step-by-step explanation:
The purchase price is = $445500
5% is down payment = [tex]0.05\times445500=22275[/tex]
Loan amount is = [tex]445500-22275=423225[/tex]
The EMI formula is = [tex]\frac{p*r*(1+r)^{n} }{(1+r)^{n}-1 }[/tex]
p = 423225
r = 6.25/12/100=0.0052
n = 30*12 = 360
Putting the values in the formula we get:
[tex]\frac{423225*0.0052*(1.0052)^{360} }{(1.0052)^{360}-1 }[/tex]
= $2603.17
The monthly payment is $2603.17 approx
Only $1.87 to buy a present for Jim. Her Jim. Many a happy hour she had spent planning for something nice for him. Something fine and rare and sterling—something just a little bit near to being worthy of the honor of being owned by Jim.What inferences about Della can be made based on this excerpt Check all that apply
Della does not have much money to spend on Jim’s gift.
Della wants to buy Jim a watch chain for his watch.
Della loves her husband, Jim, very much
Della wants to buy Jim a really nice gift
Della will need to sell he
Answer:
The following are correct inferences:
1. Della does not have much money to spend on Jim’s gift.
Della is financially poor and does not have enough to gift Jim. This can be understood from the line "Only $1.87 to buy a present for Jim."
2. Della loves her husband, Jim, very much.
Della has been saving for months to gift her husband something rare and unique that his husband is worth for.
"Something fine and rare and sterling—something just a little bit near to being worthy of the honor of being owned by Jim." these lines shows her love and affection for her husband.
3. Della wants to buy Jim a really nice gift.
"Many a happy hour she had spent planning for something nice for him. Something fine and rare and sterling—" These lines show that Della needs to buy something for Jim.
Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation d = 144 – 16t2, where t is the number of seconds it takes the car to travel down to each point on the ride. For which interval of time is Greg’s car moving in the air?
What is 2.5 in expanded and word form?
A rain gutter is made from sheets of aluminum that are 20 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross sectional area to allow the greatest amount of water to flow
To find the depth of the gutter that will maximize its cross sectional area, we can use optimization. We need to set up the area equation, differentiate it, find critical points, and determine the maximum area.
Explanation:To find the depth of the gutter that will maximize its cross sectional area, we need to use the concept of optimization. Let's assume the depth of the gutter is x inches. The width of the gutter would be 20 - 2x inches, as the edges are turned up to form right angles.
The cross-sectional area of the gutter would be x(20 - 2x) = 20x - 2x^2 square inches. To maximize this area, we can find the derivative of the area function with respect to x, set it equal to 0, and solve for x.
After finding the critical points, we can analyze which value of x gives the maximum area. Lastly, we can plug this value back into the area function to find the maximum area.
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Consider the basis B of R2 consisting of vectors
5
-6
and
-2
-2
Find x in R2 whose coordinate vector relative to the basis B is
[x]B = [-6]
[ 2 ]
The idea is to find a linear combination a_1(5, -6) + a_2(-2, -2) = (-6, 2) It boils down to a system of equations: Take the augmented matrix:
[5−6−2−2−62] Reduced form: [1001−8111311] -(8/11)*(5, -6) + (13/11)*(-2, -2) = (-6, 2)Answer: The required vector x is
[tex]x=\begin{bmatrix}-\dfrac{8}{11}\\ \dfrac{13}{11}\end{bmatrix}[/tex].
Step-by-step explanation: Given that a basis B of R² consists of vectors (5, -6) and (-2, -2).
We are to find the vector x in R² whose co-ordinate vector relative to the basis B is [tex]\begin{bmatrix}-6\\ 2\end{bmatrix}[/tex].
Let us consider that a, b are scalars such that
[tex]a(5,-6)+b(-2,-2)=(-6,2)\\\\\Rightarrow (5a-2b,-6a-2b)=(-6,2)\\\\\Rightarrow 5a-2b=-6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\-6a-2b=2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
Subtracting equation (ii) from equation (i), we get
[tex](5a-2b)-(-6a-2b)=-6-2\\\\\Rightarrow 11a=-8\\\\\Rightarrow a=-\dfrac{8}{11}[/tex]
and from equation (i), we get
[tex]5\times\left(-\dfrac{8}{11}\right)-2b=-6\\\\\\\Rightarrow 2b=-\dfrac{40}{11}+6\\\\\\\Rightarrow 2b=\dfrac{26}{11}\\\\\\\Rightarrow b=\dfrac{13}{11}.[/tex]
Thus, the required vector x is
[tex]x=\begin{bmatrix}-\dfrac{8}{11}\\ \dfrac{13}{11}\end{bmatrix}[/tex].
Expand and simplify 4 (2x-1) +3 (2x+5)
Also,
Find the first four terms and the 10th term
5-2n....
_ _ _ _......_
Which equation could be used to solve this problem?
Kate weighs 3 times as much as Jen. Their total weight is 140 pounds. How much does each girl weigh?
A) 140 = 3x
B) 3x + x = 140
C) 3x - x = 140
D) x - 140 = 3x
Answer:
B
Step-by-step explanation:
Segment BDSegment BD is a perpendicular bisector of segment ACsegment AC . AD=8x−15AD=8x−15 and DC=57DC=57 .
What is the value of x?
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A standard showerhead in Marty's house dispenses 7 gallons of water per minute. Marty changed this showerhead to an energy-saving one. The equation shows the amount of water dispensed, y, as a function of the number of minutes, x, for the new showerhead.
y = 3x
How much water does Marty save each day with the change in showerheads if he uses the shower for 8 minutes each day?
4 gallons
12 gallons
32 gallons
53 gallons
Answer:
the answer would be 32
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explain why you can use subtraction to solve a division problem.
Final answer:
Subtraction can be used to solve division problems as division can be thought of as repeated subtraction. In scientific notation, dividing numbers involves subtracting exponents, underscoring the relationship between subtraction and division, especially with exponent rules.
Explanation:
Subtraction can be used to solve a division problem because division essentially involves determining how many times a number (the divisor) can be subtracted from another number (the dividend) until nothing is left or a remainder is obtained. This concept is similar to repeated subtraction, and understanding this helps clarify why we subtract exponents when dividing numbers in scientific notation. For example, when dividing $10^6$ by $10^3$, you are essentially finding out how many times $10^3$ can be subtracted from $10^6$, which can be intuitively understood as reducing the exponent by 3 (hence $10^{(6-3)}$ or $10^3$).
When working with scientific notation, the operation of division is handled by dividing the coefficients (the numbers in front of the exponents), and then subtracting exponent of the divisor from the exponent of the dividend. This subtraction of exponents is based on the properties of powers where dividing two numbers with the same base results in a power with the base raised to the difference of the exponents. Division and subtraction are intimately related in mathematics, particularly in dealing with exponent rules which require a deep conceptual understanding, not just a mechanical execution of operations.
y=3(1.2)x growth or decay
A farmer sold 54 pumpkins the first weekend of a fall festival and 42 pumpkins the second weekend. Which expression uses the distributive property to show equivalent expressions for this situation?
Answer:
D.) 54 + 42 = 6(9+7)
Step-by-step explanation:
This shows the distributive property as, when the 6 is distributed to the 9 you get 54 (pumpkins), and when distributed to the 7 you get 42 (pumpkins).