Answer:
7 feet, 3 feet, 7.5 feetStep-by-step explanation:
Let the sides be a, b and c.
GivenThe length of one side of a triangle is 2 feet less than three times the length of its second side
a = 3b - 2The length of the third side is 3/4 of the sum of the lengths of the first two sides:
c = (3/4)(a + b)The perimeter of the triangle is 17.5 feet:
a + b + c = 17.5SolutionSubstitute a with b in the second equation:
c = (3/4)(3b - 2 + b) = (3/4)(4b - 2) = (3/4)(4b) - (3/4)(2) = 3b - 1.5Now substitute a and c with b in the third equation and solve for b:
a + b + c = 17.53b - 2 + b + 3b - 1.5 = 17.57b - 3.5 = 17.57b = 17.5 + 3.57b = 21b = 3Find the value of a:
a = 3b - 2 = 3*3 - 2 = 9 - 2 = 7Find the value of c:
c = 3b - 1.5 = 3*3 - 1.5 = 9 - 1.5 = 7.5The sides of the triangle are:
a = 7 feet, b = 3 feet, c = 7.5 feetFind the future value if $25000 is invested at 20% for 15 years compounded continuously
The future value of the amount is $458,083.92.
What is the future value?The formula for calculating future value when there is continuous compounding is :
A x e^r x N
Where:
A= amount e = 2.7182818 N = number of years r = interest rate25,000 x e^0.20 x 15 = $458,083.92
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Solve the system using substitution.
5x + 3y = -4
y - 2x = 6
([?], [? ])
Answer:
(-2,2)
Step-by-step explanation:
i tried sorry if it's wrong
Willey’s Grill & Restaurant Corporation wholesales ovens and ranges to restaurants throughout the Southwest. Willey’s Grill & Restaurant, which had 335,000 shares of common stock outstanding, declared a 2-for-1 stock split. This information has been collected in the Microsoft Excel Online file. Open the spreadsheet, perform the required analysis, and input your answers in the questions below.
a. The number of shares outstanding after the stock split by Willey’s Grill & Restaurant Corporation is 670,000 (335,000 x 2).
b. If the common stock had a market price of $510 per share before the stock split, the approximate market price per share after the split is $255 ($510/2).
What is a stock split?A stock split is a way that a corporation increases the number of its shares by increasing or decreasing the number of shares outstanding.
When a stock split increases the number of outstanding shares, the market price reduces proportionately.
Question Completion:a. What will be the number of shares outstanding after the split? Round your answer to the nearest whole number.
b. If the common stock had a market price of $510 per share before the stock split, what would be an approximate market price per share after the split? Round your answer to the nearest dollar.
Thus, the stock split increased the outstanding shares to 670,000 while reducing the market price to $255.
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Marcelina uses a blend of white corn and yellow corn to make tortilla chips at her restaurant. She needs to buy 50kg, end text of corn in total for her next order. White corn costs $0.30 per kilogram, yellow corn costs $.015 per kilogram, and she wants to spend 12 dollars total.
What does point F represent in this context?
Answer:
Marcelina spends less that the intended amount of money and buy less than enough corn
The point F tells us that Marcelina spends the intended amount of money and she buys more than the intended amount of corn.
How to solve the graphFrom the question the entire kg of maize that she buys from what we see in the graph is
50 + 20 = 70kg
But the amount that she was supposed to buy is 50 kg
Hence the point F shows that she spent the money the way she wanted to use in to buy the product.,
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if ABC is a isosceles triangle with sides(2a+b)cm, (6a+2b+1)cm and a base of 4a cm, if it's perimeter is 28 find sides
Answer:
a=1.5, b= 3
Step-by-step explanation:
sum of all sides will give us perimeter. Therefore
2a+b+6a+2b+1+4a=28
12a+3b+1=28
12a+3b=27....eqn1
since t two sides of this isosceles triangle are equal:
2a+b=4a
2a=b
[tex]a=\frac{b}{2}[/tex]
substituting [tex]\frac{b}{2}[/tex][tex].[/tex] in the first equation whenever we encounter a we will have:
[tex]12(\frac{b}{2})+3b=27\\[/tex]
6b+3b=27
9b=27
b=3
therefore [tex]a= \frac{b}{2} =\frac{3}{2} =1\frac{1}{2} or 1.5[/tex]
Compare the graph of g(x) = -0.3x² + 7 with the graph of p(x) = x².
The graph of g(x) is wider.
The graph of q(x) opens downward and the graph of p(x) op
Both have the axis of symmetry x = 0.
The vertex of q(x) is (0, 7); the vertex of p(x) is (0, 0).
O The graph of q(x) is narrower.
The graph of q(x) opens downward and the graph of p(x) op
Both have the axis of symmetry x = 1.
The vertex of q(x) is (1, 7); the vertex of p(x) is (1, 0).
The graph of q(x) is wider.
Both graphs open upward.
Both have the axis of symmetry x = 1.
The vertex of q(x) is (1, 7); the vertex of p(x) is (1, 0).
The graph of q(x) is narrower.
Both graphs open downward.
Both have the axis of symmetry x = 0.
The vartov nf n/v) is in 7) the vartav nf n/v) ie in my
Answer:
Step-by-step explanation:
Below is the graphs of the given g(x) and p(x)
g(x) is green
p(x) is blue
The graph of g(x) is wider.
The graph of g(x) opens downward
Both have the axis of symmetry x = 0
The vertex of g(x) is (0, 7); the vertex of p(x) is (0, 0).
The graph of g(x) opens downward and the graph of p(x) opens upward
The graph of g(x) is wider.
Find all real zeros for the function y= -5x - 7
A. -5
B. -5, -7
C. 7/5
D. - 7/5
The sum of two numbers is 32. One number is 3 times as large as the other. What are the numbers?
Answer: 8, 24
Step-by-step explanation:
Since the sum of two numbers is 32 and one number is 3 times as large as the other, the first number can be defined as x and the second number is 3x.
Divide 32 by (3+1) = 4 and you get 8, which is x.
The second number is 3(8) = 24.
Classify the graph as a linear function, nonlinear function, or relation (non- function). -i0- O A. Nonlinear function O B. Linear function C. Relation
Answer:
I need to see the graph
No explanation
The function ()=2f(x)=x2is transformed to
()=0.4(+1)2f(x)=0.4(x+1)2
Which statement describes the effect(s) of the transformation on the graph of the original function?
The parabola is narrower and shifted 1 unit to the right.
The parabola is narrower and shifted 1 unit to the left.
The parabola is wider and shifted 1 unit to the right.
The parabola is wider and shifted 1 unit to the left.
Answer:
StepWhat is the measure of z? Z X y 4 9 z = [ ? ]VI-by-step explanation:
I forgot how to find the x coordinates for the intersection already :( I will be very much appreciated if I could get a step by step clear solution for both answers!
(Question b)
In the graph,
Amplitude = 1.5 units
Period = 180°
The x-intercepts are 40°, 140°, 220°, and 320°
Determining x-interceptsFrom the question, we are to determine the amplitude, period and the x-intercepts of the given graph
Amplitude is the distance from rest to crest.
In the graph,
Amplitude = 1.5 units
Period is how long from crest to crest OR trough to trough
In the graph,
Period = 180°
x-intercepts refers to the points where the graph cuts the x-axis
In the graph,
The x-intercepts are 40°, 140°, 220°, and 320°
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Two bikers meet at a park. Biker A needs to stop at the store that is 12 miles east of the park. Biker B heads southeast at a 61° angle at the same time for 24 miles. Once biker A leaves the store he heads southwest at an angle of 89° for 21 miles. Do NOT use the law of cosines, use your knowledge from the content of this course.
a. Use your knowledge of triangles to figure out if the two bikers will be able to meet up if each biker travels the distance given.
b. If they do not meet up, how much farther would one of the bikers have to travel to meet the other?
c. What is the measure of the angle between the bikers?
d. What is the relationship between the measure of the angles and the paths the bikers took?
e. Classify the triangle the paths created.
f. How many miles did they travel together?
The bikers will meet at point N, all the questions have been answered below.
What is a Triangle?A triangle is a polygon with three sides, angles, and vertices.
The triangle MNO is formed on the basis of the data given in the question
If the triangle satisfies the sine-rule then the biker meets the other biker at point N
According to the sine rule
In a Triangle ABC of side length a,b and c respectively
a/ sin A = b/ sin B = c/sin C
NM/sin NOM =21/ sin 61°
NM/sin NOM = 24.01
NO/sin OMN =24/sin 89°
NO/sin OMN = 24.003
∠ONM = ( 180-61-89) = 30°
OM / sin ONM = 12/ sin 30 = 24
The ratio of the side and angles of the triangle are almost equal, therefore they satisfy the sine ruleand so, the bikers will meet at point N.
As they are meeting the distance between them will be zero.The measure of the angle between the bikers is as calculated above = 30°.The largest angle is opposite to the longer path.The triangle formed is an acute angle triangle.Total miles they travelled = 24+12+21 = 57 miles.To know more about Triangle
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Factor the trinomial below.
x2 - 14x + 45
Answer:
(x-9) (x-5)
x = 9 x = 5
Step-by-step explanation:
find out which 2 numbers when multiplied give 45 and
when added give -14
they are -9 and -5
(x-9) (x-5)
x = 9 x = 5
Answer:
[tex]x_{1}=9, x_2=5[/tex]
Step-by-step explanation:
Recall a quadratic formula for quadratic equation [tex]ax^2+bx+c=0[/tex]: [tex]x_{1/2}=\frac{-b\pm \sqrt{b^2 -4ac}}{2a}[/tex]
Notice the constant in the given equation [tex]x^2-14x+45[/tex]:
[tex]a=1, b=-14, c=45[/tex]
Substitute the values into the formula:
[tex]x_{1/2}=\frac{-(-14)\pm \sqrt{(-14)^2 -4\cdot 1\cdot 45}}{2\cdot 1}[/tex]
Calculate the values:
[tex]x_{1/2}=\frac{14\pm \sqrt{196 -180}}{2}=\frac{14\pm\sqrt{16}}{2}=\frac{14\pm 4}{2}[/tex]
It follows that there are two solutions:
[tex]x_1=\frac{14+4}{2}=\frac{18}{2}=9[/tex] and [tex]x_2=\frac{14-4}{2}=\frac{10}{2}=5[/tex].
Can anybody tell me how to solve this?
The real risk free rate is 2%. The inflation premium for the next three years is 4.0%. The maturity risk premium for three year bonds is 3.7%. Liquidity premium is 3% and default risk premium is 3.5%. What is the nominal rate on a three year Treasury Bond?
answer in % without the symbol
The nominal rate on a three year Treasury Bond, which is the risk-free rate plus the inflation premium, is 6%.
What are nominal rates?The nominal interest rate (also known as the money interest rate) is the percentage of interest that the borrower pays to the lender for a loan.
The nominal interest rate includes the real rate and the inflation premium.
Data and Calculations:Risk-free rate = 2%
Inflation premium = 4.0%
Maturity risk premium = 3.7%
Liquidity premium = 3%
Default risk premium = 3.5%
Nominal rate for Treasury Bond = Risk-free rate plus Inflation premium
= 6% (2% + 4%)
Thus, the nominal rate on a three year Treasury Bond is 6%.
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Two trains leave a station at the same time. One train is heading south at a rate that is 1.5 times faster than the other train, which is heading north. After 5 hours, the trains are 750 miles apart. At what speed, in mph, are the two trains getting farther away from each other?
Answer:
150 mph
Step-by-step explanation:
The relationship between time, speed, and distance is ...
speed = distance/time
ApplicationThe trains have separated 750 miles in 5 hours, so their rate of separation is ...
(750 mi)/(5 h) = 150 mi/h
The trains are separating at a speed of 150 mph.
In the dining room how many circles are there for each square HELP
Whih number produces a rational number multiplied by 0.5
The question is incomplete. Please find the full content below.
Which number produces a rational number multiplied by 0.5
(C) 2.22 produces a rational number multiplied by 0.5.
If a number can be expressed by p/q form where p and q are whole numbers and q does not equal to 0, is called a Rational number.
If a number cannot be expressed as above then that number is called Irrational Number.
In simple term, we can say that the never-ending decimal numbers and square roots of non perfect square numbers are Irrational Numbers.
Product of two rational numbers is always a rational and the product of one rational and one irrational is always an irrational number.
Here given that the number is 0.5 which is clearly a rational number.
In order to get a rational number, we have to multiply another rational number with that.
In the given choices,
[tex]\pi=3.14......[/tex]
is a never-ending decimal number that is irrational.
7 is not a perfect square so
[tex]\sqrt{7}[/tex]
is an irrational number.
2.020020002.... is also a never-ending decimal number hence it is irrational.
Only 2.22 is rational here.
So multiplication of 2.22 and 0.5 produces a rational number since the product of two rational numbers is always a rational number.
Hence option (C) is correct.
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What is the range for y=x^2-7x-8 ?
Answer:
[-20.25;+∞).
Step-by-step explanation:
1) according to the properties of the given parabola the minimum is:
[tex]x_0=-\frac{-7}{2}=3.5; \ y_0=-20.25;[/tex]
2) y₀=-20.25 means, the value -20.25 is minimum of the range, the maximum →+∞;
3) finally, the range y≥-20.25.
x + 2y = -4
2x + 3y = 1
Answer:
(14, -9)
Step-by-step explanation:
So you can solve this using substitution. When you're solving a systems of equations, you're looking for when (x, y) is the same for both equations, or in other words where the two linear equations intersect. Because of this we can solve for x or y in one equation and substitute it into the other equation, since the x and y in one equation should equal the x and y in the other equation.
Original equation:
x + 2y = -4
Subtract 2y from both sides
x = -2y - 4
Original equation:
2x + 3y = 1
Substitute -2y -4 as x
2(-2y - 4) + 3y = 1
Distribute the 2
-4y - 8 + 3y = 1
Combine like terms
-y - 8 = 1
Add 8 to both sides
-y = 9
Multiply both sides by -1
y = -9
So now that you know what y is, you can plug it into either equation to solve for x:
x + 2y = -4
Plug in -9 as y
x + 2(-9) = -4
Multiply
x - 18 = -4
Add 18 to both sides
x = 14
So this gives you the solution (14, -9)
2. Employed Women If 60% of all women are employed
outside the home, find the probability that in a sample
of 20 women,
aid sel
Mil
2
a. Exactly 15 are employed
b. At least 10 are employed
c. At most 5 are not employed outside the home
Using the binomial distribution, the probabilities are given as follows:
a) 0.0747 = 7.47%.
b) 0.8725 = 87.25%.
c) 0.1256 = 12.56%.
What is the binomial distribution formula?The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
x is the number of successes.n is the number of trials.p is the probability of a success on a single trial.The values of the parameters are:
n = 20, p = 0.6.
Item a:
The probability is P(X = 15), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 15) = C_{20,15}.(0.6)^{15}.(0.4)^{5} = 0.0747[/tex]
Item b:
The probability is:
[tex]P(X \geq 10) = P(X = 10) + P(X = 11) + P(X = 12) + \cdots + P(X = 20)[/tex]
Finding each value with the equation we used in item a, we have that:
[tex]P(X \geq 10) = 0.8725[/tex]
Item c:
At least 20 - 5 = 15 on the road, hence the probability is:
[tex]P(X \geq 15) = P(X = 15) + P(X = 16) + \cdots + P(X = 20)[/tex]
Then, using the same procedure as item b, we have that:
[tex]P(X \geq 15) = 0.1256[/tex]
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Select one of the factors of 3x² + 4x - 4.
O (x + 4)
O (3x + 2)
O (3x - 2)
O(x - 2)
Answer:
(3x-2)
Step-by-step explanation:
3x² + 4x - 4
look for two factors whose product is (3)(-4)=-12 and sum is +4
The factors will be +6 and -2
substitute these factors inplace of 4x to get:
3x² - 2x + 6x - 4
then factorize:
x(3x-2)+2(3x-2)
finally we get factors: (3x-2)(x+2)
1. Simplify (7y²)
(3y³)
Answer:
21 y power 5
Step-by-step explanation:
This table represents a linear function. x y 0 5 5 15 This graph represents another function.The greater unit rate of the two functions is . The greater y-intercept of the two functions is
The greater unit rate (slope) of the two functions is 2 (of the first function).
The greater y-intercept of the two functions is 6 (of the second function).
The unit rate (slope) of a linear function is the rise (or fall) in the independent variable for a unit change in the dependent variable.
When the linear function goes through the points (x₁, y₁) and (x₂, y₂), then the unit rate (slope) can be calculated as (y₂ - y₁)/(x₂ - x₁).
The first function goes through the points (0, 5) and (5, 15).
Thus unit rate (slope) of the first function = (15 - 5)/(5 - 0) = 2.
The second function goes through the points (0, 6) and (-4, 0).
Thus unit rate (slope) of the second function = (6 - 0)/(0 - (-4)) = 1.5.
Therefore, the greater unit rate (slope) of the two functions is 2 (of the first function).
The y-intercept of a linear function is the y-coordinate of the point at which the graph of the function intersects the y-axis. It is also shown as the point (0, b).
The y-intercept of the first function is 5, as it passes through the point (0, 5).
The y-intercept of the second function is 6, as it passes through the point (0, 6).
Therefore, the greater y-intercept of the two functions is 6 (of the second function).
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Answer:
2 and 6
Source:
Trust me bro
140⁰
D 20⁰
Z 180°
O True
1350
O False
CA
90°
60°
45°
m₂ COX= mzAOX+ m2 COA.
40⁰
The given statement m<COX = m<COA + <AOX is TRUE
What is a protractor?A protractor is a mathematical instrument for measuring angle. From the given protractor, we have that;
The measure of angle COA and angle AOX is equivalent to angle COA. Mathematically;
m<COX = m<COA + <AOX
This shows that the given statement is TRUE
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Given z1 = startroot 2 endroot (cosine (startfraction 3 pi over 4 endfraction) i sine (startfraction 3 pi over 4 endfraction) ) and z 2 = (cosine (startfraction pi over 2 endfraction) i sine (startfraction pi over 2 endfraction) ) , what is the value of z1 – z2?
The subtraction of complex numbers [tex]z_{1} -z_{2}[/tex] is cos(π)+i sin(π).
Given [tex]z_{1} =\sqrt{2}[/tex][cos(3π/4+i sin(3π/4) and [tex]z_{2}[/tex]=cos (π/2) +i sin(π/2)
We have to find the value of [tex]z_{1} -z_{2}[/tex].
A complex number is a number that includes real number as well as a imaginary unit in which [tex]i^{2} =1[/tex]. It looks like a+ bi.
We have to first solve [tex]z_{1} and z_{2}[/tex] and then we will be able to find the difference.
[tex]z_{1} =[/tex][tex]\sqrt{2}[/tex][ cos (3π/4)+i sin (3π/4)]
[tex]=\sqrt{2}[/tex][cos(π-π/4)+ i sin (π-π/4)]
=[tex]\sqrt{2}[/tex] [-cos(π/4)+sin (π/4)]
=[tex]\sqrt{2}[/tex](-1/[tex]\sqrt{2}[/tex]+1/[tex]\sqrt{2}[/tex])
=[tex]\sqrt{2} *0[/tex]
=0
[tex]z_{2} =[/tex]cos(π/2)+i sin (π/2)
=0+i*1
=1
Now putting the values of [tex]z_{1} and z_{2}[/tex],
[tex]z_{1} -z_{2} =0-1[/tex]
=-1
=-1+i*0
=cos (π)+i sin(π)
Hence the value of difference between [tex]z_{1} and z_{2}[/tex] is cos(π)+i sin(π).
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Lenny bought x shares of stock for Sy per share last month. He paid
his broker a flat fee of $20. He sold the stock this month for Sp per
share, and paid his broker a 2% commission. Express Lenny's net
proceeds algebraically.
Lenny's net proceeds algebraically are found as;[tex]\rm |xy + 20 - \frac{98}{100} |[/tex]
What is the equation?A mathematical statement consisting of an equal symbol between two algebraic expressions with the same value is known as an equation.
Let the bought share will be $ x, and $y is the price per share
Paid fees to the broker = $ 20
The total amount spent = $(xy+20)
The price at which each share sells = $ A
Broker commission = 2 %
The total amount obtained after selling the shares;
⇒ ax - 2% of ax
⇒ 98 % of ax
Hence, Lenny's net proceeds algebraically are found as;[tex]\rm |xy + 20 - \frac{98}{100} |[/tex]
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Answer:
Step-by-step explanation:
what were Laura net proceeds
Create both a tree diagram AND sample space of all possible boy/girl births in a family of three kids.
1. How many outcomes are there?
2. What is the probability of all three kids being boys?
3. What is the chance of only the middle kid being a girl?
4. What is the probability of there being at least two boys?
5. What is the chance of there being at most one girl?
Answer: Sample space: {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}
1. 8 outcomes in total
2. 1/8 or 0.125
3. 1/2 or 0.5
4. 1/2 or 0.5
5. 3/8 or 0.375
Step-by-step explanation:
1. As shown by the sample space and the tree diagram, there are 8 total possibilities.
2. There is only a 1/8 or 0.125 chance of there being all boys, which is BBB
3. There are 4 outcomes that allow the middle child to be a girl, and 4/8 = 1/2. There will only be a girl middle child with BGB, GGG, BGG, and GGB.
4. We can think of at least as meaning no less than two boys, which includes two to three boys. The only outcomes that have either two or three boys is BBB, BBG, BGB, GBB. 4/8 = 1/2.
5. We can think of at most as meaning no more than one girl. The only outcomes that have only one girl are BBG, BGB, and GBB, which is 3/8.
¿What is the area and volume of a cube that has edges 5 m long?
Answer:
Area = 150 m²
Volume = 125 m³
Step-by-step explanation:
• The surface area of a cube is given by the formula:
[tex]Area = 6r^2[/tex]
where r is the length of an edge of the cube = 5 m.
∴ Area = 6(5)²
= 6(25)
= 150 m²
• The volume of a cube is given by the formula:
[tex]Volume = r^3[/tex]
∴ Volume = (5)³
= 125 m³
[tex]\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {\large{\textsf{\textbf{\underline{\underline{Answer :}}}}}}[/tex]
We use the formulas for the area and volume of the cube, assuming that the length [tex] \bold{A=5m}[/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold\red \: \bold \red A \purple = \bold \blue6 \bold \blue a {}^{ \bold \orange2} [/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold \red A \purple= \bold \blue6 \bold \blue \: \: \blue • \: \: \bold \blue5 {}^{ \bold\orange{ 2}} [/tex]
[tex]\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold \red A \purple= \bold \blue 1 \bold \blue5 \bold \blue0 \bold \blue m {}^{ \bold \orange2} [/tex]
Now we will find the volume of the cube :-
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\bold \red V \purple = \bold\blue a {}^{ \bold \orange3} [/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold \red V \purple = \bold \blue 5 {}^{ \bold \orange3} [/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold \red V \purple = \blue{ \bold{125m}} {}^{ \bold \orange3} [/tex]
Answer: The area of the cube is 150 m² and the volume it Equal a 125 m³
I hope I've helped : )
(a) Find the equation of the normal to the hyperbola x = 4t, y = 4/t at the point (8,2).
The slope of the tangent line to the curve at (8, 2) is given by the derivative [tex]\frac{dy}{dx}[/tex] at that point. By the chain rule,
[tex]\dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex]
Differentiate the given parametric equations with respect to [tex]t[/tex] :
[tex]x = 4t \implies \dfrac{dx}{dt} = 4[/tex]
[tex]y = \dfrac4t \implies \dfrac{dy}{dt} = -\dfrac4{t^2}[/tex]
Then
[tex]\dfrac{dy}{dx} = \dfrac{-\frac4{t^2}}4 = -\dfrac1{t^2}[/tex]
We have [tex]x=8[/tex] and [tex]y=2[/tex] when [tex]t=2[/tex], so the slope at the given point is [tex]\frac{dy}{dx} = -\frac14[/tex].
The normal line to the same point is perpendicular to the tangent line, so its slope is +4. Then using the point-slope formula for a line, the normal line has equation
[tex]y - 2 = 4 (x - 8) \implies \boxed{y = 4x - 30}[/tex]
Alternatively, we can eliminate the parameter and express [tex]y[/tex] explicitly in terms of [tex]x[/tex] :
[tex]x = 4t \implies t = \dfrac x4 \implies y = \dfrac4t = \dfrac4{\frac x4} = \dfrac{16}x[/tex]
Then the slope of the tangent line is
[tex]\dfrac{dy}{dx} = -\dfrac{16}{x^2}[/tex]
At [tex]x = 8[/tex], the slope is again [tex]-\frac{16}{64}=-\frac14[/tex], so the normal has slope +4, and so on.
Edith’s age is 1/4 of Ronald’s age. In how many years will Edith’s age be 1/3 of Ronald’s age if Edith is 20 years old?
Answer:
10
Step-by-step explanation:
x = Edith's age = 20 years
y = Ronald's age
x = 20 = y/4
y = 20×4 = 80 years
in z years we have
(x + z) = (y + z)/3
(20 + z) = (80 + z)/3
3(20 + z) = 80 + z
60 + 3z = 80 + z
60 + 2z = 80
2z = 20
z = 10
in 10 years Edith's age will be 1/3 of Ronald's age.
then Edith will be 30, and Ronald will be 90.
30 = 90/3
correct.