In the diagram below, \overline{EF}
EF
is parallel to \overline{BC}
BC
. If DC=27DC=27, EF=27.5EF=27.5, and BC=49.5BC=49.5, find the length of \overline{DF}
DF
. Figures are not necessarily drawn to scale.
)
Answers
Applying the AA similarity theorem, the length of DF is: 15 units.
What is the AA Similarity Theorem?When two angles in one triangle are congruent to the corresponding two angles in another triangle, both are said to be similar to one another. This means that, the corresponding sides of these two triangles would be proportional or have the same ratio.
In the image attached bellow, triangle DEF and triangle DBC are similar to each other because they have two pairs of corresponding angles that are congruent which are:
<DFE ≅ <DCB [right angles are equal]
<EDF ≅ <BDC [reflexive property of congruence]
Therefore, their corresponding sides will have equal ratios such as:
DC/DF = BC/EF
Substitute:
27/DF = 49.5/27.5
Cross multiply:
DF = (27.5 * 27) / 49.5
DF = 15
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Related Questions
Popular wisdom is that eating presweetened cereal tends to increase the number of dental caries (cavities) in children A sample of children was (with parental consent) entered into a study and followed for several years. Each child was classified as a sweetened-cereal lover or a unsweetened cereal lover. At the end of the study, the amount of tooth damage was measured. Here are the summary data: 10 15 Mean 6.41 5.20 Std Dev 5.0 15.0 225 Sweetened Unsweetened Assuming the necessary conditions for inference are met, which of the following an approximate 95% confidence interval for the difference in the mean tooth damage?
O (6.41 -5.20) + 2.145 3+
O (6.41 - 6:20) +1.96 Vi+
O (6.41 -- 5.20) +1,96/13 + 38
O (6.41 -5.20) +2.202/33+
O (6.41 - 5:20): 2.2821+13 228 X
Answers
An approximate 95% confidence interval for the difference in the mean tooth damage is (6.41 - 5.20) ± 2.26[tex]\sqrt{\frac{25}{10}+\frac{225}{15} }[/tex]
For sweetened samples
The value of n = 10
The mean of the samples = 6.14
The standard deviation of the sample = 5.0
For unsweetened samples
The value of n = 15
The mean of the sample = 5.20
The standard deviation of the sample = 15.0
We have to assume that necessary conditions for inference are met
The confidence interval = 95%
The equation will be
(6.41 - 5.20) ± 2.26[tex]\sqrt{\frac{25}{10}+\frac{225}{15} }[/tex]
Therefore, the difference in the mean tooth damage is (6.41 - 5.20) ± 2.26[tex]\sqrt{\frac{25}{10}+\frac{225}{15} }[/tex]
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write a system of linear equations in which (2, -1) is a solution of equation 1 but not a solution of equation 2, and (5, 5) is a solution of the system.
Answers
The system of linear equation of the line are y=2x-5 when (2,-1) is a solution of the first equation but not for the second equation.
Given that,
We have to find the system of linear equation in which (2,-1) is a solution of the first equation but not for the second equation and (5,5) is a solution for system of equation.
We know that,
To find the system of equation we have a formula that is
y=mx-mx₁+y₂
Here m is slope
m=-1-5/2-5
m=-6/-3
m=2
The equation is
y=2x-2(5)+5
y=2x-10+5
y=2x-5
Therefore, The system of linear equation of the line are y=2x-5 when (2,-1) is a solution of the first equation but not for the second equation.
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Solve pls brainliest
Answers
Answer:
d = 9.43398113206
Step-by-step explanation:
d=√((x2 – x1)² + (y2 – y1)²)
d = √((-4-1)² + (5+3)²)
d = √((-5)² + (8)²)
d = √(25 + 64)
d = √89
d = 9.43398113206
Step by step
Using the two points, graph the two points and count the units of length. Draw your triangle.
Your question mentions Pythagorean Theorem so we will use that to solve.
a^2 + b^2 = c^2
Our triangle sides a= 5 and b=8
5^2 + 8^2 = c^2
25 + 64 = c^2
89 = c^2
Take square root of both sides
9.43 = c
Your diagonal length is
how many ways are there to make an r-arrangement of pennies, nickels, dimes, and quarters with at least one penny and an odd number of quarters? (coins of the same denomination are identical
Answers
The exponential generating function is,
G(x) = (x¹/1! + x²/2! + ...)(x¹/1! + x³/3! + x⁵/5! +...)(x°/0! + x¹/1! + x²/2! + ...)²
where the first factor describes the pennies, the second describes the quarters, and the last describes nickles and dimes.
We want the coefficient of [tex]x^{r}[/tex]/r!.
Using exponential identities we get,
G(x) = ([tex]e^{x}[/tex] - 1)([tex]\frac{e^{x}-e^{-x} }{2}[/tex]) ([tex]e^{x}[/tex])² = [tex]\frac{1}{2}[/tex] ([tex]e^{4x} - e^{2x} - e^{3x} + e^{x}[/tex]),
and the coefficient is
[tex]\frac{1}{2} (4^{r} - 2^{r}- 3^{r} + 1 )[/tex]
The generating function is
G(x) = (x¹/1! + x²/2! + ...)⁴(x°/0! + x¹/1! + x²/2!+...)²² = ([tex]e^{x}-1[/tex])⁴[tex]e^{22x}[/tex],
and we again want the coefficient of [tex]x^{10}[/tex]/10!.
Multiplying we get,
G(x) = ([tex]e^{4x}-4e^{3x}+2e^{2x}+1[/tex])[tex]e^{22x}[/tex] = [tex]e^{26x}-4e^{25x}+2e^{24x}+e^{22x}[/tex]
so the coefficient of [tex]x^{10}[/tex]/10! is
26¹⁰ - 4 × (25)¹⁰ + 2 × (24)¹⁰ + 22¹⁰
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suppose that the number of miles that a car can run before its battery wears out is exponentially distributed with an average value of 11k miles. (a) what is the probability that the battery lasts between 12k and 14k miles?
Answers
The probability that the battery lasts between 12,000 and 14,000 miles is approximately 0.084.
The number of miles that a car can run before its battery wears out follows an exponential distribution with an average value of 11,000 miles. The probability that the battery lasts between 12,000 and 14,000 miles can be calculated as follows:
P(12,000 < x < 14,000) = P(x < 14,000) - P(x < 12,000)
To calculate the probabilities on the right-hand side of the equation, we can use the formula for the cumulative distribution function (CDF) of an exponential distribution:
F(x) = 1 - e^(-x/λ)
where λ is the mean value of the distribution. In this case, λ = 11,000 miles.
Plugging in the values and solving, we get:
P(12,000 < x < 14,000) = F(14,000) - F(12,000)
= (1 - e^(-14,000/11,000)) - (1 - e^(-12,000/11,000))
= e^(-12,000/11,000) - e^(-14,000/11,000)
≈ 0.242 - 0.158
≈ 0.084
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Three ballet dancers are positioned on stage. Oliver Is 6.3 feet straight behind Dana and 6.3
feet directly left of Ashley. When the music begins, Oliver twirls to Ashley's position, then
leaps to Dana's position, and finally walks back to his original position. How far did Oliver
travel? If necessary, round to the nearest tenth.
Answers
Oliver travelled the distance of 21.5 ft
What is perimeter?
The complete length of a shape's edge serves as its perimeter in geometric terms. Adding the lengths of all the sides and edges that surround a form yields its perimeter. It is calculated using linear length units such centimeters, meters, inches, and feet.
As shown in the figure, we need to find the perimeter of the triangle ABC.
Perimeter = g + h +f ............(1)
g=f = 6.3 ft
Finding h using pythogoras theorem formula:
h² = g² + f²
= 6.3² + 6.3²
= 79.38
h =√79.38 = 8.909 ≈ 8.9 ft
So, (1) => Perimeter = 6.3 + 6.3 + 8.9 = 21.5 ft
Thus, Oliver travelled the distance of 21.5 ft
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What is the center and radius of the circle defined by the equation x² y² 49?
Answers
The equation x² + y² = 49 can be written in standard form as (x-0)² + (y-0)² = 49, which is the equation for a circle with a center at (0, 0) and a radius of 7.
The equation for a circle is (x-h)² + (y-k)² = r², where (h, k) represents the center of the circle and r represents the radius.
In order to find the center (h, k) and radius (r) of the circle, we need to rearrange the equation to put it in the standard form. To do this, we subtract 49 from each side of the equation, which will give us x² + y² - 49 = 0. Then, we can factor the equation to get (x² + y²) - 7(7) = 0. This can be further simplified to (x + 7)(x - 7) + (y + 7)(y - 7) = 0.
Since there are no solutions for (x + 7) or (x - 7) when set equal to 0, we can conclude that the equation for the circle is (x-0)² + (y-0)² = 49, which is the equation for a circle with a center at (0, 0) and a radius of 7.
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An elevator can carry 25 adults or 30 children at one time. During the course of a day, the elevator carries a full passenger load 36 times. If all the passengers were children, how many more people would the elevator carry than if all the passengers were adults?
Answers
180 people would the elevator carry than if all the passengers were adults.
What is the meaning of times?
The mathematical symbol is used to represent the multiplication operation and the consequent product. It is often referred to as the times sign or the dimension sign.
Given an elevator can carry 25 adults or 30 children at one time.
One day the elevator carries a full passenger load 36 times.
If the elevator carries all adults, then it carries (25 × 36) = 900 adults.
If the elevator carries all children, then it carries (30 × 36) = 1080 adults.
If all the passengers were children, then the elevator carry more than (1080 - 900) = 180 people if all the passengers were adults.
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How do you integrate 1/ln(x)?
Answers
The integration of 1/ln(x) is not possible as We permit y = lnx, we then recognize that x = ey.
By differentiating each aspects of this equation with admire to y we get:dx/dy = ey, because the exponential characteristic differentiates to itself whilst differentiated with admire to its power.
In mathematics, the logarithmic quintessential characteristic or quintessential logarithm li(x) is a unique characteristic. It is applicable in issues of physics and has variety theoretic significance. In particular, in keeping with the high variety theorem, it's far a excellent approximation to the high-counting characteristic, that is described because the variety of high numbers much less than or same to a given value x.
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Graph f(x) = √√x+2 over the domain -2 ≤x≤7.
Answers
The graph of the function square root of the square root of x + 2 in the given domain is given by the image shown at the end of the answer.
How to graph the function?The function for this problem is defined as follows:
f(x) = √√x+2
The double square root means that in reality, we have the fourth root of the function, hence the function is then defined as follows:
f(x) = fourth root(x + 2).
The fourth root function is defined for values of x for which the inner term is zero and greater, hence:
x + 2 >= 0
x >= -2.
Then the graph is made over a domain that goes until x = 7, and is given by the image presented at the end of the answer.
(the domain is chosen because it is the domain given by the problem).
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an equation is written to represent the relationship between the temperature in alaska during a snow storm, y, as it relates to the time in hours, x, since the storm started. if the storm lasted five hours and the temperature saw a drop from 0o to - 20o, which quadrant(s) of a coordinate grid should be used to display this data?
Answers
The first and fourth quadrant of a coordinate grid should be used to display the given data .
In the question ,
it is given that ,
from the time the storm has started the temperature change is from 0° to - 20° .
we know that time cannot be in negative , and temperature can be both negative and positive .So only quadrant 1 and 4 can be used for representation .
Y = temperature
X = time
Quadrant 1 = (+x , +y) this can be used because x can only be positive and y is also positive.
Quadrant II = (-x , +y) this cannot be used because x cannot be negative
Quadrant III = (-x , -y) this cannot be used because x cannot be negative
Quadrant IV = (+x , -y) this can be used because x can only be positive and y can be negative.
Therefore , only 1 and 4 quadrants can be used .
The given question is incomplete , the complete question is
An equation is written to represent the relationship between the temperature in Alaska during a snow storm, y, as it relates to the time in hours, x, since the storm started. if the storm lasted five hours and the temperature saw a drop from 0° to - 20°, which quadrant(s) of a coordinate grid should be used to display this data ?
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dollar general sends out a $5 off coupon to its email subscribers. what percentage of customers between 20 and 30 years of age who lived within 5 miles of the store used the coupon?
Answers
The classification model is used to define the percentage of customers between 20 and 30 years of age who lived within 5 miles of the store used the coupon.
Percentage:
In math, number or ratio that can be expressed as a fraction of 100 refers the percentage.
Given,
Here we have a dollar general sends out a $5 off coupon to its email subscribers.
And we need to find the percentage of customers between 20 and 30 years of age who lived within 5 miles of the store used the coupon.
Then the model used to define the this type of situation is refers as the classification model, because in this model, we have classify the given group of people based on the constraint.
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The volume of a cone with height 29 cm is 1758 CM3, find the radius of the conw
Answers
Answer:13.77 cm.
Step-by-step explanation:
r = sqrt((3V)/(pih))
= sqrt((31758)/(pi29))
= sqrt((5274)/(pi*29))
= sqrt(182.16)/(pi/29)
= 13.77 cm
with one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is accepted if every item in the sample is okay. the abc electronics company has just manufactured 1200 write-rewrite cds, and 150 are defective. if 3 of these cds are randomly selected for testing, what is the probability that the entire batch will be accepted?
Answers
The probability that the entire batch will be accepted.
What is Acceptance Sampling?
Acceptance sampling is a statistical measure used in quality control. It allows a company to determine the quality of a batch of products by selecting a specified number for testing.
abc Electronics Company manufactured 1200 write-rewrite cds, and 150 are defective. Since 150 are defective, the number of write-rewrite cds that would be accepted is (1200 - 150 = 1050).
Since the items at selected without replacement, the number of write-rewrite cds will continue to decrease anytime a write-rewrite cds is taken out. For example if they are 1050 accepted write-rewrite cds if one is taken out they would then be 1049 accepted write-rewrite cds, and if another one is taken out they would be 1049 accepted write-rewrite cds and so on.
If 3 of these cds are randomly selected for testing without replacement,
The probability that the entire batch will be accepted
= (1050/1200) × (1049/1200) × (1048/1200) × (1047/1200)
= 0.8750 × 0.8742 × 0.8733 × 0.8725
=0.5828
The probability that the entire batch will be accepted is 0.5828 .
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A submarine dives 20 m each minute for 15 minutes. What is the total change in depth after 15 minutes?
Answers
Answer:
Step-by-step explanation:300 mStep-by-step explanation:20 meters per minute. 15 minutes it dives. 20x15 is 300
invert a multiply to evaluate the division expression 3/4 ÷ 1/3 help
Answers
To multiply the division expression 3/4 ÷ 1/3 is 9/4. Division expressions are mathematical expressions using division
What is meant by mathematical expression ? When numbers and variables are properly combined using operations like addition, subtraction, multiplication, division, exponentiation, and other as-yet-unlearned operations and functions, a set of mathematical symbols known as an expression is produced. For instance, "4 added to 2" will have the mathematical formula 2+4. Addition, subtraction, multiplication, and division are all considered operations. There are a lot more operations that can be employed in a mathematical equation. You must substitute a number for each variable and carry out the arithmetic operations in order to evaluate an algebraic expression. Since 6 + 6 equals 12, the variable x in the example above is equal to 6.3/4 : 1/3
= 3/4 · 3/1
= 3·3/4.1
= 9/4
Hence this proved.
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How do you simplify 27^(3/4)?
Answers
By simplifying the above equation i.e 27^(3/4) we get, 9 x (3)^(1/4).
Simplification is the process of substituting a mathematical expression with a simpler (typically shorter) counterpart. A fraction is simplified to an irreducible fraction. In computer algebra, expressions are simplified.
In order to simplify the given equation we need to factorize the bases. You can see that in the following steps :
(a^b)^c=a^(bc)
a^(b+c)=a^b x a^c
By using this above equation, we can factorize the bases.
So, factorization would be equal to :
27 ^(3/4) = ((3^3) )^ (3/4)
= 3^(9/4)
= 3^ (2+1/4)
= 3^2 x 3^(1/4)
= 9 x 3^(1/4)
That’s how we simplified the given equation.
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The values of f(x) and g(x) are shown for selected values of x in the table below. x f(x) g(x) 0 0 1 1 3 2 2 6 4 3 9 8 4 12 16 Of these two functions, one is exponential and the other is linear. Determine which of the functions is linear, and explain why this must be the case.
Answers
The linear function is f(x) and the exponential function is g(x).
What is an exponential function?The exponential function is a mathematical function that is represented by e^x. Unless otherwise specified, the term refers to a positive-valued function of a real variable, though it can be extended to complex numbers or generalized to other mathematical objects such as matrices or Lie algebras.
Linear functions have a straight line as their graph. The following is the form of a linear function. y = f(x) + a+bx There is one independent variable and one dependent variable in a linear function.
In this case, the function for g(x) is 2^x.
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Write the equation of a line with an undefined slope through the point (-4, 2)
Answers
Answer:
x = - 4
Step-by-step explanation:
A line with an undefined slope is a vertical line parallel to the y- axis with equation
x = c ( c is the value of the x- coordinates the line passes through )
the line passes through (- 4, 2 ) with x- coordinate - 4 , then
x = - 4 ← equation of line
3.8x10^-2 as an ordinary number
Answers
Answer:0.038
Step-by-step explanation:
The ordinary number equivalent of 3.8x10⁻² is expressed as 0.038.
To express the number 3.8x10⁻² as an ordinary number, we need to understand scientific notation. In scientific notation, numbers are written in the form of a coefficient (a decimal number greater than or equal to 1 but less than 10) multiplied by 10 raised to a certain power (positive or negative). This notation is commonly used to represent very large or very small numbers in a concise and standardized way.
Let's break down 3.8x10⁻² step by step:
The coefficient: 3.8
The coefficient is the decimal part of the number, which should be greater than or equal to 1 but less than 10. In this case, the coefficient is 3.8.
The exponent: -2
The exponent indicates the power of 10 to which the coefficient is raised. If the exponent is negative, it means the number is less than 1. In this case, the exponent is -2, which means we are dividing the coefficient by 100 (10 raised to the power of -2 is 1/100).
Now, to express 3.8x10⁻² as an ordinary number, we simply perform the calculation:
3.8 × 10⁻² = 3.8 ÷ 100 = 0.038
So, the ordinary number equivalent of 3.8x10⁻² is 0.038.
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what expression is equivalent to 3/4a + 2/3 - 1/2a - 1/3 + 1/2a
Answers
Answer:
(3/4)a + 1/3
Step-by-step explanation:
3/4a + 2/3 - 1/2a - 1/3 + 1/2a
(3/4)a + (2/3) - (1/2)a - (1/3) + (1/2)a [added () to clarify. Is "3/4a" (3/4)a or 3/(4a)?
(3/4)a + (2/3) - (1/2)a - (1/3) + (1/2)a
(3/4)a - (1/2)a - (1/3) + (1/2)a + (2/3)
(3/4)a - (1/2)a + (1/2)a - (1/3) + (2/3) [Rearrange: like terms together]
a[(3/4) - (1/2) + (1/2)] - (1/3) + (2/3) [bring out the "a"]
a[(3/4)] + (1/3) [simplify]
(3/4)a + 1/3
2x-1(4x+3)=-7x+2 help me find x please
Answers
Answer:
X=1
Step-by-step explanation:
For step one you have to Distribute
2x-1(4x+3)=-7x+2 --> 2x-4x-3=-7x+2
For step two you have to Combine Like Terms
2x-4x-3=-7x+2 --> -2x-3=-7x+2
For step three you have to Add Three To Both Sides
-2x-3=-7x+2 --> -2x=-7x+5
For step four you have to Add 7x To Both Sides
-2x=-7x+5 --> 5x=5
For step 5 you have to Divide Both Sides By The Same Factor (5)
5x=5 --> x=1
Hope This Helps!
Which is the smallest positive integer that can be written as the product of two cubic integers?
Answers
Answer: 8
Step-by-step explanation:
=(1) ^3×(2) ^3
=1×1×1×2×2×2
=1×8
=8
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Suppose that the average cost, in dollars, of producing a shipment of a certain product is C = 5,000x + 20,000/x, x > 0 where x is the number of machines used in the production process. (a) Find the critical values of this function. (Assume 0 < x < [infinity]. Enter your answers as a comma-separated list.) x = Incorrect: Your answer is incorrect. (b) Over what interval does the average cost decrease? (Enter your answer using interval notation.) (c) Over what interval does the average cost increase? (Enter your answer using interval notation.)
Answers
The average cost function is C(x) = 5000x + 20,000/x, where x is the number of machines used in the production. The critical value is C(x) = 20,000 and it happens when x = 2.
If we have a function f(x), the critical point happens when its first derivative is equal to zero.
f '(x) = 0
In the given problem, the function is:
C(x) = 5000x + 20,000/x
Take the derivative:
C '(x) = 5000 - 20,000/x² = 0
5000 x² = 20,000
x² = 4
x = ±2
Since x is within the interval: 0<x<∞, the solution is x = 2
Substitute x = 2 into the function:
C(2) = 5000 (2) + 20000/2
C(2) = 10000 + 10000 = 20,000
Hence, the critical value is C(x) = 20,000 and it happens when x = 2.
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a popular magazine tests cars and trucks for mileage on the road and around the city. the magazine reports that 50% of the cars tested had mileages below 24.7 mpg on the road, and the other 50% of the cars tested had mileages above 24.7 mpg. which statistic does this 24.7 represent?
Answers
The decimal number 24.7 represents the average mileage of the vehicle.
What are statistics?Statistics is the study of collection, analysis, interpretation, and presentation of data or discipline to collect and summarise the data.
A famous magazine tests vehicles and trucks for mileage out and about and around the city. the magazine reports that half of the vehicles tried had mileages beneath 24.7 mpg out and about, and the other half of the vehicles tried had mileages above 24.7 mpg.
The decimal number 24.7 means the vehicle can cover 24.7 miles with each gallon of gas.
The decimal number 24.7 represents the average mileage of the vehicle.
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How many inches is there in 1/8th of a yard?
Answers
Answer:
4.5
Step-by-step explanation:
You calculate it. look 1 yard is 36 inches. so 1/8 of a yard is 36 divided by 8. 4.5!
A figure has vertices (−13, 13),(26, 52), (39, 39). What would be the new coordinates of the vertices to the nearest tenth if the image were reduced by a scale factor of 0.77 with the origin as the center of dilation?
A. −16.9, 16.9), (33.8, 67.6), (50.7, 50.7)
B. (−10, 10), (20, 40), (30, 30)
C. (10, 10), (−20, 40), (−30, 30)
D. (16.9, 16.9), (33.8, 67.6), (50.7, 50.7)
Answers
The new coordinates after a reduced scale factor of 0.77 with the origin will be equal to (−10, 10), (20, 40), (30, 30). Hence, option B is correct.
What is a graph?In math, graph science is the theory of geometric structures called graphs that are used to represent pairwise different objects. Vertices—also known as nodes or points—that are joined by edges make form a network in this sense.
Undirected graphs, where edges connect two vertices equally, and focused therapy, where edges connect two vertices unevenly, are distinguished.
As per the data given in the question,
We will combine the x and y dimensions even by scale factors to shrink the picture by a multiplier of 0.77 with both the origins as that of the center of dilatation because the figure's vertices are (-13, 13), (26, 52), and (39, 39).
So, the first point is (-13, 13)
-13 × 0.77 = -10 and,
13 × 0.77 = 10
So, the new point is (-10, 10).
The second point is (26, 52).
26 × 0.77 = 20
52 × 0.77 = 40
So, the new point is (20, 40)
The third point is (39, 39)
39 × 0.77 = 30
39 × 0.77 = 30
So, the new point is (30, 30).
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f(x) = 2x² + 12x + 16
What is the average rate of change of f(x) on the interval [-3, -2].
Answers
[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]f(x)= 2x^2 + 12x +16 \qquad \begin{cases} x_1=-3\\ x_2=-2 \end{cases}\implies \cfrac{f(-2)-f(-3)}{-2 -( -3)} \\\\\\ \cfrac{[2(-2)^2 + 12(-2) + 16]~~ - ~~[2(-3)^2 + 12(-3) + 16]}{-2+3} \\\\\\ \cfrac{[8-24+16]~~ - ~~[18-36+16]}{1}\implies 0-(-2)\implies \text{\LARGE 2}[/tex]
The sum of two numbers is 20. What is the least possible sum of their squares and what are the numbers?
Answers
The numbers are 10 and 10. And the sum of their squares will be 200.
What are the maxima and minima of a function?For the maximum value of the function if f''(x) < 0 and for the minimum value of the function if f''(x) > 0.
The maximum and minimum value of the function is at f'(x) = 0.
The sum of the two numbers is 20.
Let the first number be 'x'. Then the other number is (20 - x). Then the equation is given as,
y = x² + (20 - x)²
Differentiate the function, then we have
y' = 0
2x + 2(20 - x)(-1) = 0
x - 20 + x = 0
2x = 20
x = 10
Then the other number is given as,
⇒ 20 - 10
⇒ 10
The sum of their squares is given as,
⇒ 10² + 10²
⇒ 100 + 100
⇒ 200
The numbers are 10 and 10. And the sum of their squares will be 200.
More about the maxima and minima links are given below.
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the volume of a rectangular box with a square base remains constant at 1,100 cm3 as the area of the base increases at a rate of 2 cm2/sec. find the rate at which the height of the box is decreasing when each side of the base is 12 cm long. (do not round your answer.)
Answers
If the area of the area of the base increases at rate of 2 cm²/sec then the rate at which the height of the box is decreasing is 0.11 cm/sec .
In the question ,
it is given that ,
the volume of the rectangular box having square base is 1100 cm³ ;
the rate of decrease of the area is (dA/dt) is = 2 cm²/sec ;
we know that ; Volume = Area × Height ;
the area of the square base of side 12 cm is = 12 × 12 = 144 ;
So , height = Volume/Area ;
= 1100/144 ;
Now , V = A × h ;
differentiating the above equation w.r.t. "t" ,
we get ;
dV/dt = A×(dh/dt) + h×(dA/dt)
0 = 144×(dh/dt) + (1100/144)×(2)
dh/dt = -2200/(144 × 144)
= -0.106095
≈ -0.11
Therefore , the rate of change of height is = 0.11 cm/sec .
Learn more about Rate Of Change here
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