The shortest distance [tex]x + y = 4[/tex] that is closest to the origin is [tex]\boxed{2\sqrt 2 {\text{ units}}}.[/tex]
Further explanation:
The formula for distance between the two points can be expressed as follows,
[tex]\boxed{{\text{Distance}} = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} }[/tex]
Given:
The line is [tex]x + y = 4.[/tex]
Explanation:
The coordinate of the origin is [tex]\left( {0,0} \right).[/tex]
The first point is [tex]\left( {x,y} \right)[/tex] and the second point is [tex]\left( {0,0} \right).[/tex]
The distance between the two points can be calculated as follows,
[tex]\begin{aligned}{\text{Distance}}&= \sqrt {{{\left( {x - 0} \right)}^2} + {{\left( {y - 0} \right)}^2}}\\&= \sqrt {{x^2} + {{\left( {4 - x} \right)}^2}}\\&= \sqrt {{x^2} + {x^2} - 8x + 16}\\&= \sqrt {2{x^2} - 8x + 16}\\\end{aligned}[/tex]
Differentiate the above equation with respect to [tex]x[/tex].
Substitute the first derivative equal to zero.
[tex]\begin{aligned}\frac{d}{{dx}}\left( {{\text{Distance}}} \right) &= 0\\\frac{{\left( {2x - 4} \right)}}{{\sqrt {2{x^2} - 8x + 16} }} &= 0\\2x - 4 &= 0\\2x &= 4\\x&= 2\\\end{aligned}[/tex]
Substitute [tex]x = 2[/tex] in equation [tex]x + y = 4[/tex] to obtain the value of [tex]y[/tex].
[tex]\begin{aligned}2 + y &= 4\\y&= 4 - 2\\y&= 2\\\end{aligned}[/tex]
The point is [tex]\left( {2,2} \right).[/tex]
The shortest distance can be obtained as follows,
[tex]\begin{aligned}{\text{Distance}} &= \sqrt {{2^2} + {2^2}}\\&= \sqrt {4 + 4}\\&= 2\sqrt 2\\\end{aligned}[/tex]
The shortest distance [tex]x + y = 4[/tex] that is closest to the origin is [tex]\boxed{2\sqrt 2 {\text{ units}}}.[/tex]
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Application of derivatives
Keywords: Derivative, shortest distance, curve, origin, attains, maximum, value of x, function, differentiate, minimum value, closest point, line, y+x = 4.
The midpoint of a segment is (2,-5) and one of the end points is (3,6). Where is the other endpoint?
Given a polynomial f(x), if (x + 3) is a factor, what else must be true?
What applies here is one of the laws of the factorization of polynomials, called the factor theorem and it states that:
For a polynomial f(x) , if for any value a, f(a) =0 then (x-a) is factor of f(x)
Example:
Consider the polynomial
[tex]f(x) = x^{3} - 3x^{2} - 8x+4[/tex]
For a =3,
[tex]f(a) = (3)^{3} - 3(3)^{2} - 8(3)+24[/tex]
= 27-27-24+24 = 0
f (3)= 0
which means (x-3) is a factor of f(x)
Applying the above rule to the question:
if (x + 3) is a factor of a polynomial f(x), then f(-3) = 0
note that (x+3) can also be written as (x- (-3)).
You have a 32-foot fence around a square garden. You paint 1/3 of one side of the fence. What fraction of the fence did you paint?
if h(x)=-1/2x+3, find h(-29)
Which situations can be represented by a linear function?
Select each correct answer.
Every day, the number of bacteria in the dish is 3 times what it was the previous day.
Every year, Luisa puts $10 into her savings account.
A country's population increases by 0.8% each year.
The barrel leaks 0.5 L of water each day.
The barrel leaks 0.5 L of water each day.
Every year, Luisa puts $10 into her savings account.
Linear functions represent situations with a constant rate of change. Saving $10 every year and a barrel leaking 0.5 L of water each day are examples of linear relationships due to this constant change.
Situations that can be represented by a linear function are those where the rate of change remains constant. Let's analyze the given situations according to this condition:
Every year, Luisa puts $10 into her savings account. This describes a constant rate of change, with $10 being added each year, which can be represented by a linear function.The barrel leaks 0.5 L of water each day. Here, the change in water level is constant (0.5 L per day), also indicative of a linear relationship.These two scenarios fit the criteria for being linear because they involve a constant rate of change over time. On the other hand, scenarios like the exponential growth of bacteria or percentage-based population growth are not linear since the rate of change is not constant; these would be represented by exponential functions.
The drawing of a building, shown below, has a scale of 1 in to 30 ft. what is the actual height, in ft, of the building
how do you solve this?
how much is 2/7 of 1 and 3/4
Answer:
0.5 or half of it (1/2)
How many steps does it take to complete 0.1 km?
Answer:1250
Step-by-step explanation:
Which is NOT a way to state the meaning of the expression h + 7?
A-7 more than a number
B-A number added 7 times
C-A number increased by 7
D-7 is added to a number
use the table below to find (FoG)(1)
x 0 1 2 3 4 5 6 7
f(x) 5 7 9 11 13 15 17 19
g(x) 3 6 9 12 15 18 21 24
Answer:
value of [tex](f o g)(1)[/tex] is, 17
Step-by-step explanation:
We have to find the [tex](f o g)(1)[/tex]
Using the given tables;
[tex](f o g)(1) = f(g(1))[/tex] ......[1]
At x = 1
g(1) = 6
Substitute this in [1] we have;
[tex](f o g)(1) = f(6)[/tex]
At x = 6
f(6) = 17
then;
[tex](f o g)(1) = 17[/tex]
Therefore, the value of [tex](f o g)(1)[/tex] is, 17
A line contains the points (−26, −37) and (−32, −61) .
What is the slope of the line in simplified form?
Answer:
the answer should be 4, you left the - off of the 61
Step-by-step explanation:
f a computer costs $600.00 and the sales tax rate is 7.5 percent, what is the total cost of the computer?
As per linear equation, the total cost of the computer is $645.00.
What is a linear equation?"A linear equation is an equation in which the highest power of the variable is always 1."
Given, the cost of computer is $600.00.
The sales tax rate is 7.5%.
Therefore, the total cost of the computer is
= $[tex][600.00 + \frac{(7.5)(600.00)}{100}][/tex]
= $[tex][600.00 + 45.00][/tex]
= $[tex]645.00[/tex]
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A store had 100 t-shirts. Each month, 30% of the t-shirts were sold and 25 new t-shirts arrived in shipments. Which recursive function best represents the number of t-shirts in the store, given that f(0) = 100?
f(n) = f(n - 1) • 0.3 + 25, n > 0
f(n) = 100 - f(n - 1) • 0.3 + 25, n > 0
f(n) = f(n - 1) • 0.7 + 25, n > 0
f(n) = 100 - f(n - 1) • 0.7 + 25, n > 0
Divide. Write each quotient in simplest form.
2/3 ÷ 2 = _____ Show step-by-step solution.
in one week there are 10,080 minutes. what is this number in scientific notation?
Blake has more than five friends.
Let f represent the number of Blake's friends.
Which inequality describes the number of Blake's friends?
f < 5
f > 5
f ≤ 5
f ≥ 5
Answer: The correct option is (B) [tex]f>5.[/tex]
Step-by-step explanation: Given that Blake has more than five friends.
We are to select the inequality that describes the number of Blake's friends.
The number of Blake's friends is represented by f.
According to the given information, we have
the inequality that describes the number of Blake's friends is given by
[tex]f>5.[/tex]
Thus, (B) is the correct option.
Is 5 a prime composite or neither
Answer:
Prime
Step-by-step explanation:
Its factors are:1 and itself.
ALL prime numbers have the same 2 factors: 1 and itself.
If p(x) = x2 – 1 and q(x)=5(x-1), which expression is equivalent to (p – q)(x)?
5(x – 1) – x2 – 1
(5x – 1) – (x2 – 1)
(x2 – 1) – 5(x – 1)
(x2 – 1) – 5x – 1
Ryan's gas tank is 1/10 full. After he buys 11 gallons of gas, it is 3/5 full. How many gallons can Ryan's tank hold?
What method can researchers employ in order to counter bias in their sampling?
a. snowball sampling
b. weighting
c. convenience sample
d. non-random surveying
Researchers can use b) weighting to counter bias in their sampling, which adjusts sample representation to match a desired population. While convenience sampling offers easy data collection, it lacks generalizability, and weighting is necessary to enhance the validity of findings.
To counter bias in their sampling, researchers can employ a method known as b) weighting. This technique adjusts the representation of samples to match a desired population, countering potential biases that may arise from non-random sampling methods such as convenience sampling. It is important to distinguish between non-random sampling methods that are potentially biased and strategies that are used to ensure data accuracy and representativeness.
Convenience sampling, also referred to as availability sampling or haphazard sampling, is a nonprobability sampling strategy where researchers collect data from individuals or elements that are most easily accessible. This approach is useful in exploratory research or student projects where probability sampling is too costly or difficult but does not offer the rigor needed to make generalizations to larger populations. To overcome these limitations and enhance the validity of their findings, researchers may adjust their data using weighting to better reflect the population being studied.
Write 796.384 in expanded form
Find the exact value of sin(11pi/8).
Choices are in the attachment.
Which function has a removable discontinuity?
a. g(x)=(2x-1)/(x)
b. p(x)=(x+2)/(x²-x-2)
c. f(x)=(5x)/(x-x²)
d. h(x)=(x²-x+2)/(x+1) ...?
The function [tex]\( f(x) = \frac{5x}{x-x^2} \)[/tex] has a removable discontinuity at ( x = 0 ) after canceling the common factor [tex]\( x \).[/tex]
A removable discontinuity occurs at a point where a function is not defined due to a factor in the denominator that could be canceled out with a factor in the numerator.
Let's analyze each function to determine if any of them have a removable discontinuity.
[tex]a. \( g(x) = \frac{2x-1}{x} \)[/tex]
- The denominator ( x ) is zero at ( x = 0 ).
- The numerator ( 2x-1 ) is non-zero at ( x = 0 ).
- Since there is no common factor in the numerator and denominator that could cancel out, the discontinuity at ( x = 0 ) is not removable.
[tex]b. \( p(x) = \frac{x+2}{x^2-x-2} \)[/tex]
- The denominator [tex]\( x^2-x-2 \) factors as \( (x-2)(x+1) \).[/tex]
- The function becomes [tex]\( p(x) = \frac{x+2}{(x-2)(x+1)} \).[/tex]
- The denominator is zero at ( x = 2 ) and ( x = -1 ).
- The numerator ( x+2 ) is zero at ( x = -2 ).
- There is no common factor between the numerator and the denominator that could be canceled out. So, the discontinuities at [tex]\( x = 2 \) and \( x = -1 \)[/tex] are not removable.
[tex]c. \( f(x) = \frac{5x}{x-x^2} \)[/tex]
- The denominator [tex]\( x-x^2 \) factors as \( x(1-x) \).[/tex]
- The function becomes [tex]\( f(x) = \frac{5x}{x(1-x)} = \frac{5x}{x-x^2} \).[/tex]
- The denominator is zero at x = 0 and x = 1 .
- The numerator ( 5x ) is zero at ( x = 0 ).
- There is a common factor of ( x ) in the numerator and denominator which could be canceled out, making the discontinuity at ( x = 0 ) removable.
- After canceling the common factor ( x ), the function becomes [tex]\( \frac{5}{1-x} \),[/tex] which is defined at ( x = 0 ).
[tex]d. \( h(x) = \frac{x^2 - x + 2}{x + 1} \)[/tex]
- The denominator ( x + 1 ) is zero at ( x = -1 ).
- The numerator [tex]\( x^2 - x + 2 \) is not zero at \( x = -1 \).[/tex]
- There is no common factor in the numerator and denominator that could be canceled out, so the discontinuity at ( x = -1 ) is not removable.
Conclusion
The function [tex]\( f(x) = \frac{5x}{x-x^2} \)[/tex] has a removable discontinuity at ( x = 0 ), because the discontinuity can be removed by canceling the common factor ( x ) in the numerator and the denominator.
So, the correct answer is:
[tex]c. \( f(x) = \frac{5x}{x-x^2} \)[/tex]
What is the domain and range for the function below?
1/square root x+4
How to solve this please help
Which equation can be used to solve the problem?
How many 12-packs of juice boxes contain a total of 84 juice boxes?
A.
12 + b = 84
B.
12 ÷ b = 84
C.
12b = 84
D.
b ÷ 12 = 84
Your medical bill is $2345. Your health insurance covers 70% after a $120 deductible. What amount of the bill will you pay?
Answer:
It is $787.50.
find the amount that results from each. $50 invested at 6% compounded monthly after a period of 3 years ...?
Final answer:
Using the compound interest formula, the total amount after three years for $50 invested at 6% compounded monthly is approximately $59.70.
Explanation:
To calculate the future value of $50 invested at 6% compounded monthly after a period of three years, we use the formula for compound interest:
FV = P × (1 + (r/n))³⁼ ×⁴
where:
FV is the future value of the investment,
P is the principal amount ($50),
r is the annual nominal interest rate (6% or 0.06),
n is the number of times the interest is compounded per year (12, for monthly compounding),
t is the time the money is invested for, in years (3 years).
Using these values, we calculate as follows:
FV = $50 × (1 + (0.06/12))³·×´ = $50 × (1 + 0.005)·¹·´
Now we calculate this raised to the power of 36 (3 years times 12 months):
FV = $50 × (1.005)³···¶ = $50 × 1.194052
Thus, FV ≈ $59.70
The total amount after three years would be approximately $59.70, assuming the interest is compounded monthly at a rate of 6%.
The ratio of the number of cats to the number of dogs in an animal shelter was 2:3. After 120 cats were adopted, the ratio of the number of cats to the number of dogs became 3:7. Find the total number of cats and dogs in the animal shelter in the end?