All x-values that are within the scope of both f and g make up the domain of the (f g)(x).
How can the domain of f/g)(x be determined?All x-values that are within the scope of both f and g make up the domain of the (f g)(x). In this case, f has a domain of "x | x 0" and g has a domain of "all real numbers," thus (f g)(x) also has a domain of "x | x 0" since these values of x fall within both f and g's domains.Utilizing graphs is another method for determining the domain and range of functions. The collection of possible input values is referred to as the domain, and the domain of a graph is made up of all the input values displayed on the x-axis. The collection of potential output values that make up the range is represented by the y-axis.To learn more about Domain refer to:
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What is the probability of rolling a number less than or equal to 8 with the
sum of two dice, given that at least one of the dice must show a 6?
Step-by-step explanation:
this is a combined event.
a probability is always the ratio
desired outcomes / totally possible outcomes
in our experiment with rolling 2 dice we have totally 6×6 = 36 different outcomes.
from these 36 total possibilities we desire the following (sum is less than or equal to 8, one die has to show a 6) :
6 1
6 2
1 6
2 6
that is 4 desired outcomes.
and the probabilty for this is
4/36 = 1/9 = 0.111111111...
in mathematical terms we could say :
one possible scenario is one die is 6, and the other is either 1 or 2.
the probability for one die to show 6 is
1/6
the probability for one die to show 1 or 2 is
2/6.
the probabilty for the combination is then
1/6 × 2/6 = 2/36 = 1/18
how many different combinations can we have ?
well, with 2 dice we have 2 such possible combinations.
therefore, the probability is
2 × 1/18 = 2/18 = 1/9.
Question 7 of 9
A
B
Answer here
D
Given that ZBCD and ZBCA form a straight angle, if ZBAC measures 20
degrees and ZABC measures 45 degrees, then the measure of ZBCD is
degrees.
solve 2y=3x+12 for y
Answer: y=[tex]\frac{3x}{2}[/tex]+6
Step-by-step explanation:
isolate y by dividing both sides by 2
y=3x+12/2
y=[tex]\frac{3x+12}{2}[/tex]
y=[tex]\frac{3x}{2}[/tex]+6
What is the missing statement in the proof? Scroll down to see the entire proof. a ∠BDC≅∠ADB b ∠BCA≅∠DCB c ∠BAC≅∠BAD d ∠DBC≅∠BAC
The missing statement in the proof specified is given by: Option B: ∠BCA≅∠DCB
What is the reflexive property of congruence?
According to the reflexive feature of congruence, the geometric quantity under consideration—be it an angle, a line segment, a form, etc.—is congruent to itself.
The third step's key concept in this instance is the "Reflexive Property of Congruence."
The fourth phase comes to the conclusion that the ΔABC~ΔBDC criteria for similarity are met.
Since it was established in the second stage that ∠ABC≅∠BDC
Therefore, in order for the fourth step to apply angle-angle similarity, which requires the angles of two considered triangles to be congruent, we must demonstrate another angle pair of triangles ΔABC and ΔBDC being congruent in the third step.
Internal angle C is the common angle in the triangles ABC and BDC.
As a result, it is consistent with itself due to its reflexive characteristic.
We obtain the following notation for internal angle C from triangle ABC:
Angle ACB and BCA(both will work)
Triangle BDC gives us the following notation for internal angle C:
Angle DCB and BCD(Both will function.)
They fit together. The second choice takes angle C into account, so:
∠BCA≅∠DCB (second option) is the missing statement.
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function is g(x)=|x+3| can you find the domain and range?
Answer:
domain: all real numbers
range: [ 3 , infinity]
what is 258 divided by 4?
The quotient of 258 divided by 4 is calculated to be 64.5
Division is considered to be one of the four basic operations of arithmetic. The other three are multiplication, addition, and subtraction, and they can be defined as how numbers are combined to make new numbers.
Out of the four basic operations, division is the process of splitting an amount or a number into equal parts.
In division, the numerator is the top number of a fraction, and on the other hand, the denominator is the bottom number of a fraction and the result of division is called the quotient
In this case, 258 is the numerator and 4 is the denominator;
258 / 4 = 64.5
Therefore the quotient of this division is calculated to be 64.5
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Rewrite using a single positive exponent.
5^7/5^4
Answer:
5³
Step-by-step explanation:
5^7 : 5^4
5^(7-4)
5^3
Renata is purchasing a condominium for $ 125,000. She wants to put down a down payment of 20%. Select all the true statements.
The proportion that represents the down payment is 20100=125,000
20
100
=
125
,
000
x
.
The down payment is $ 25,000
25
,
000
.
The proportion that represents the down payment is 20100=125,000
20
100
=
x
125
,
000
.
The down payment is $ 50,000
50
,
000
.
The down payment is 15
1
5
of the cost of the house.
Answer:
The following statements are true:
The proportion that represents the down payment is 20/100 = x/125,000.
The down payment is $25,000.
The down payment is 15 of the cost of the house.
The other statements are not true. The proportion that represents the down payment is not 20100=125,000, as this is not a valid proportion. Additionally, the down payment is not $50,000, as this is more than the cost of the condominium.
Step-by-step explanation:
in your own words, explain what the concept of density means—without using the formula.
The concept of density explains how tightly packed together the matter in an object is
How to explain the concept of density?The density of a material is a measure of how tightly packed matter is together.
The concept of density compares the quantity of matter the material has to its volume. A material with much matter in a certain volume has a high density.
An object with high density is considered highly dense or heavy and an object with low density is considered less dense. For example, the air is about 830 times less dense than water.
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Find m∠VYW plsssssss helppppppppp
Is there any other information provided in the given question? For instance, is there data showing what each of the given letters represent?
A combat jet is flying at 750 mph, at 40,000 feet above the ground. It needs to descend all the way to 100 meters above the ground in order to engage the target, which is an oil depot several miles ahead. It descends along a diagonal, maintaining a constant speed of 750 mph along the diagonal. Ground radar at the oil depot shows the jet approaching along the horizontal at 738 mph. How fast is the jet descending vertically, in mph?
After solving the equation the jet is descending vertically at 12 mph.
What is Pythagorean Theorem?The Pythagorean Theorem is a mathematical statement that states that in a right triangle the sum of the squares of the lengths of the two sides that make up the right angle are equal to the square of the length of the hypotenuse. This theorem is named after the ancient Greek mathematician, Pythagoras. It can be expressed as the equation: a2 + b2 = c2, where 'a' and 'b' are the lengths of the two sides adjacent to the right angle and 'c' is the length of the hypotenuse. The Pythagorean Theorem is a fundamental tool used in geometry and trigonometry, and it is used to calculate the lengths of the sides of a right triangle. It can also be used to calculate the distances between two points or to calculate the area of a triangle.
The jet is descending at 12 mph. This can be calculated by using the Pythagorean Theorem. Since the jet is approaching the oil depot horizontally at 738 mph and descending diagonally at a constant speed of 750 mph, we can use the Pythagorean Theorem to calculate the vertical component of the jet's descent.
750^2 = 738^2 + x^2
x = √(750^2 - 738^2)
x = 12 mph
Therefore, the jet is descending vertically at 12 mph.
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we can make statistical inferences aobout populations even if we do not know the true population distribution bcause
Distribution will be possible even if we do not know the true population distribution.
What is a frequency distribution?
The gathered data is arranged in tables based on frequency distribution. The information could consist of test results, local weather information, volleyball match results, student grades, etc. Data must be presented meaningfully for understanding after data gathering. A frequency distribution graph is a different approach to displaying data that has been represented graphically.
Here, the A5 sample size is infinite then the sample distribution becomes normal. So as per the C.L.T n→∞ so distribution is normal.
Hence, Distribution will be possible even if we do not know the true population distribution.
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Select the graph of the solution. Click until the correct graph appears. 2x - 1 > 5
open circle on 3
open circle on 3 pointing > right >
open circle on 3 pointing < left
Answer:
lick on 3 pointing left under the wear
Mr. Fishman decides to build a rectangular pen for his pet llama. He will use the side of his house for one wall of the pen; therefore, he will not need any fencing on that side. On the other three sides, he will use special fencing that costs
$12
per foot. He wants the entire enclosed area to be 2400 square feet. Lastly, he needs four corner posts that cost
$5
each. a) [5 pts] Let
x
represent the length of the side of the pen perpendicular to the house and let
y
represent the length of the side pf the pen parallel to the house. Write a function
C(x)
for the cost of the materials for the Ilama pen in terms of
x
. Simplify your answer. Note: Diagram not to scale. b) [4 pts] Since the Cost Function is not quadratic, you will need to use your graphing calculator to determine the dimensions of the pasture that minimize cost. Set your viewing window on your calculator from
[0,150]
on the
x
-axis and
[0,5000]
on the
y
-axis and provide a labeled sketch of your graph. Then use the CALC capability on the calculator to find the minimum point. Round your answers for both the
x
and
y
dimensions to two decimal places and express your final answer using appropriate units. c) [3 pts] What is the minimum cost for Mr. Fishman to build this Ilama pen? Round your answer to the nearest cent and express your answer with appropriate units.
a) The function C(x) for the cost of the materials for the pen in terms of x
is : c(x) = 24x+(28800/x)+20
b) the dimensions of the pasture that minimize cost is:
x = 24.62 and y = 69.28
c) minimum cost for Mr. Fishman to build this Ilama pen is $1682.77
a) Perimeter of fence = 2x+y
∴ cost of fencing = 12(2x+y)
also cost of poses corner house = 4×5 = 20
∴ Total cost of pen = 24x+!2y+20
But area of the pen = 2400
xy = 2400
y = 2400/x
Putting 1 in total cost c(x)
= 24x+{(12×2400)/x}+20
c(x)= 24x+{28800/x}+20
b) we need to minimize c(x)
using the graphical calculator we obtain
That x = 34.61
x = 34.61 & y = 2400/34.61 = 69.28
c) Minimum value of c(x) = c(34×64) = boxed 1682.77
Minimum cost for Mr. Fishman to build this Ilama pen is $1682.77
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find the flux through the boundary of the rectangle for fluid flowing along the vector field .
For fluid moving in a vector field, the flux through the rectangle's perimeter is 750.52.
Flux = ∫ F. X ds
Flux = ∫∫ delta. F da (by divergence theorem)
delta. F = d/dx(x³+4) + d/dy(ycos(3x))
= 3x² + cos(3x)
So flux = ∫∫ 3x² + cos(3x) dy dx
= ∫ (3x² + cos(3x)) (y)⁶₀ dx
= 6 ∫ 3x² + cos(3x) dx
After evaluating,
= 6(125 + sin(15)/3)
= 750.52
Therefore, the flux through the boundary of the rectangle for fluid flowing along the vector field is 750.52
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2.115) There are 10 votes in the box,
and only one
of them has a symbol.
=V”Now people A chooses one vote
first, without replacement; people B
one
chooses one. please determine the
probability of B chooses the ....
Answer:
1/9
Step-by-step explanation:
The probability of A choosing is 1/10
then the probability of B choosing( without replacement)= 1/9
Use quadratic regression and a graphing calculator to find the quadratic function that best fits the data set. Then use the model to forecast the value of the function at the indicated point.(Round your coeffiecients to two decimal places.) Years Abortions Since 1985(per 1000 live births) t3 10 15 16 17 422 389 350 324 325 319 What was the abortion rate in 2005? (Round your answer to the nearest whole number) abortions
The abortion rate in 2005 was 309 per 1000 live births.
Given:
y = Ax²+Bx+C
from calculator : A = 0.13
B = -8.34
C = 423.50
y = 0.13x² - 8.34x + 423.50
Abortion rate in 2005:
x=20
y = 0.13x(20)² - 8.34×20 + 423.50
y = 308.7
y = 309 per 1000 live births.
Hence we get the required answer.
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I need help with 2nd question please
Answer:26.78
Step-by-step explanation:
350x.062=21.70
350x.0145=5.08
21.70+5.08= 26.78
A culture of the bacterium Salmonella enteritidis initially contains 50 cells. When introduced into a nutrient broth, the culture grows at a rate proportional to its size. After 1.5 hours, the population has increased to 875. (a) Find an expression for the number of bacteria after t hours. (Round your numeric values to four decimal places.) P(t) = (b) Find the number of bacteria after 7 hours. (Round your answer to the nearest whole number.) P(7) = bacteria (c) Find the rate of growth (in bacteria per hour) after 7 hours. (Round your answer to the nearest whole number.) P'(7)= bacteria per hour (d) After how many hours will the population reach 250,000? (Round your answer to one decimal place.) t = hr
a) An expression for the number of bacteria after t hours - [tex]50e^{0.8t}[/tex]
b) The number of bacteria after 7 hours, P(7)=13521.320
c) The rate of growth (in bacteria per hour) after 7, P(t)=1081.705
d) The population will reach 250,000 after - 4.6hours.
a)The exponential growth model is [tex]P = P_{0} e^{rt}[/tex]
Where,
[tex]P_{0}[/tex] = Initial growth
r = growth rate
t = time period
It is given that initially cells, [tex]P_{0}[/tex] = 50
time period, t = 1.5 hours
Population after t = 1.5 hours is , P =875
Substitute the given values in the growth exponential model:
875 = [tex]50 e^{r * 1.5}[/tex]
⇒ [tex]\frac{875}{50}[/tex] = [tex]e^{r * 1.5}[/tex]
⇒ 17.5 = [tex]e^{r * 1.5}[/tex]
⇒ log (17.5 ) = log ([tex]e^{r * 1.5}[/tex] )
⇒ log (17.5 ) = 1.5r
⇒ 1.24 = 1.5r
⇒ r = [tex]\frac{1.24}{1.5}[/tex]
r = 0.8%
Thus, P(t) = [tex]50e^{0.8t}[/tex]
b) Number of bacteria after 7 hours is calculated as follows:
P(7) = [tex]50 e^{0.8 * 7}[/tex]
P(7 ) = 13521.320
c) The rate of growth (in bacteria per hour) after 7 hours:
P(t) = [tex]50e^{0.8t}[/tex]
Differentiate w.r.t :
P(t) = 50 × [tex]0.8e^{0.8t}[/tex]
= 50 × [tex]0.8e^{0.8 * 7}[/tex]
= 1081.705
d) The population at time t is P(t) = 250000
250000 = [tex]50e^{0.8t}[/tex]
⇒ [tex]\frac{250000}{50}[/tex] =[tex]e^{0.8t}[/tex]
⇒ 5000 = [tex]e^{0.8t}[/tex]
⇒ 0.8t = 3.7
⇒ t = 4.6
a) An expression for the number of bacteria after t hours - [tex]50e^{0.8t}[/tex]
b)The number of bacteria after 7 hours, P(7) = 13521.320
c) The rate of growth (in bacteria per hour) after 7, P(t) = 1081.705
d) The population will reach 250,000 after - 4.6hours.
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What is the absolute value of 3 by 5?; What's the absolute value of 3?; What is the negative absolute value of 3?; What is the absolute value of 5 on the number line?
The absolute value of 3 by 5 is 3 by 5, absolute value of 3 is 3, negative absolute value of 3 is 3, absolute value of 5 on the number line is 5.
The absolute value if a number is the distance of the number from zero, in a number line. Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value |x| of a real number x is the non-negative value of x without regard to its sign. Absolute value is just the distance from zero in either the positive or negative direction.
Hence the absolute value of 3/5 is 3/5,
absolute value of 3 is 3.
Absolute value never be negative so negative absolute value of 3 is 3,
the absolute value of 5 on the number line is 5.
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For an independent-measures experiment comparing two treatment conditions with a sample of n = 10 in each treatment, the F-ratio would have df equal to ________.A. 1,18B. 18C. 1,19D. 19
The F - ratio is equal to 1, 18.
Given information,
For an independent-measures experiment comparing two treatment conditions with a sample of n = 10 in each treatment, the F - ratio would have df equal to what?
Let,
p = 2
n = 10
df of factor = p - 1 = 2 - 1
= 1
df of error = (np - p) = 10 * 2 - 2
= 18
Therefore, the F - ratio is equal to 1, 18.
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Each board is 3 ¾ feet and Max makes four shelves. What is the length of the piece remaining after four shelves have been cut from the 16 foot board?
The length of the remaining piece after four shelves have been cut from the 16 feet board is 1 foot.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that each board is 3 ¾ feet and Max makes four shelves. The length of the remaining board will be calculated as,
Length = 16 - ( 4 x 3 ¾ )
Length = 16 - ( 4 x15/4 )
Length = 16 - 15
Length = 1 foot
Therefore, the length of the remaining piece after four shelves have been cut from the 16 feet board is 1 foot.
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A set of experimental runs was made to determine a way of predicting cooking time y at various values of oven width x, and flue temperature x2. The coded data were recorded as follows: y X1 x2 6.4 1.32 1.15 15.05 2.69 3.4 18.75 3.56 4.1 30.25 4.41 8.75 44.85 5.35 14.82 48.94 6.2 15.15 51.55 7.12 15.32 61.5 8.87 18.18 100.44 9.8 35.19 111.42 10.65 40.40 Estimate the multiple linear regression equation.
The fitted regression equation is :y = 25.228X1 + 0.861X2 + 1283
Estimate the multiple linear regression equation ?Given the summary output :
Intercept 1283.000_ 352.000 _ 3.65
X1 _____ 25.228 ____8.63
X2 _____0.861 _____0.372
General form of a multiple regression model:
Y = B1X1+B2X2 +E
The fitted regression equation is :
y = 25.228X1 + 0.861X2 + 1283
Intercept and slope corresponding to the standard error :
Intercept = 352
Slope 1 = 8.61 ; slope 2 = 0.372
'B' coefficient represents the change in the mean response, Y, per unit increase in a version predictor variable when all the other predictors are held constant.
B1 represents the change in the mean response, U, per unit increase in x1 when x2, x3, ..., xk are held constant
B2 represents the change in the mean response, U, per unit increase in x2 when x1, x3, ..., xk are held constant
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T/F a 2.70 kgkg bucket is attached to a disk-shaped pulley of radius 0.131 mm and mass 0.692 kgkg . if the bucket is allowed to fall,
If the bucket is allowed to fall, then its linear acceleration is 8.687 m/s^2.
From the given question,
We have to solve, if the bucket is allowed to fall, then its linear acceleration
Mass of Pulley(M)= 0.692 kg
Radius of pulley(r)= 0.131 m
Mass of bucket(m)= 2.70 kg
Net torque on pulley,
[tex]\Sigma[/tex] r = rT = Iα
T = Iα/r
T= tension on string(connecting pulley and bucket)
I= moment of inertia of pulley
α= angular acceleration of pulley
linear acceleration of bucket(a),
a = rα
α = a/r
Pulley is disc shaped so
I = 1/2 Mr^2
Now, T= Iα/r = (1/2 Mr^2) (a/r)/r
T = 1/2 Ma
Now considering the motion of bucket
ma = mg-T
ma = mg-1/2 Ma
ma+1/2 Ma = mg
a = m/(m+1/2 M) g
Now putting the value
a = (2.70)×9.8/ (2.70+1/2 (0.692))
After solving
a = 8.687 m/s^2
If the bucket is allowed to fall, then its linear acceleration is 8.687 m/s^2.
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The right question is;
A 2.70 kg bucket is attached to a disk-shaped pulley of radius 0.131 mm and mass 0.692 kg . If the bucket is allowed to fall, What is its linear acceleration.
I Need help please with this Math problem.
Solve -3(x + 3) = -6 (x - 3)
help me please.!
Answer:
x = 9
Step-by-step explanation:
–3( x + 3 ) = –6 ( x–3)
–3x –9 = –6x + 18
–3x + 6x = 18 + 9
3x = 27
x = 27/3
x= 9
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Since Scotty started doing his workout plan Janet has been inspired to set herself and go to do more exercises and walk a little more each day she has decided to walk 10 m more every day on day 20 she walked 800 M how many meters will she walk on today 21 on day 60
They walk 810 m on 21 day and 1,200 m on 60th day.
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
She has decided to walk 10 m more every day on day 20 she walked 800 m.
Now,
Since, She has decided to walk 10 m more every day on day 20 she walked 800 m.
Hence, The amount of meters she walk everyday can be given as a linear function, and, the rate of variation is a constant (10 m/day):
⇒ 10 m / day
This constant (10 m/day) is the slope of the linear function.
And, We know that;
One point of the linear function (D(20) = 800) and the slope, so we can calculate the y-intercept:
⇒ D (t) = 10t + b
⇒ 800 = 10 × 20 + b
⇒ 800 - 200 = b
⇒ b = 600
So, The equation become;
⇒ D (t) = 10t + 600
Hence, For day t = 21;
⇒ D (21) = 10 x 21 + 600
⇒ D (21) = 810 m
And, For day t = 60;
⇒ D (60) = 10 x 60 + 600
⇒ D (60) = 1,200 m
Thus, They walk 810 m on 21 day and 1,200 m on 60th day.
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how do you Draw a line representing the "rise" and a line representing the "run" of the line. State the slope of the line in simplest form.
The slope of the line is 9/13 according to the given line.
What is a slope?The same number is given for every two different points on the same line when the ratio is stated as a quotient ("rise over run"). On a falling line, there is a negative "rise." In a diagram that depicts a roof or a road as a description or a design, the line may be practical as established by a road surveyor.
The steepness, incline, or grade of a line can be approximated by the slope's absolute value. The slope of a line in the plane including the x and y axes is typically symbolized by the letter m and is defined as the difference in the y coordinate by the equivalent difference in the x coordinate between two different places on the line. The following equation encapsulates this:
Slope=rise/run
Rise=y₂-y₁
=-1-(-10)
=9
Run=x₂-x₁
=9-(-4)
=13
Therefore, slope=9/13
Hence, the slope of the line is 9/13.
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Which of the following is not polynomial function a.f(x)=2x-10 b.g(x)=4x²-3x+8 c.p(x)=x³-7 d.s(x)=|3x-4|-9
Option d is correct, s(x)=|3x-4|-9 this is not a polynomial.
Which is a polynomial?An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial (No division operation by a variable).
What are the 5 polynomials?Monomial. Monomial refers to a polynomial with one non-zero term.
Binomial. A binomial is a type of polynomial that has two non-zero terms.
Quadrinomial polynomial. Trinomial.
a polynomial with five factors.
Quadratic polynomial. Linear polynomial.
polynomial cube.
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the directions on a sewing pattern; which equation can be used to find the length of the hypotenuse; elsa took out a 2-year loan
On solving the provided question, we got to know that -Pythagoras' Theorem - [tex]a^2 + b^2 = c^2 =[/tex][tex]5^2 + 8^2 = c^2[/tex]
What is the Pythagoras Theorem?The square of the hypotenuse of a right triangle (90 degrees) is equal to the sum of the squares of the other two sides, according to the Pythagorean theorem. Keep in mind that BC2 = AB2 + AC2 in the triangle ABC below. where BC is the hypotenuse, AB is the base, and AC is the height (height). Keep in mind that the hypotenuse of a right triangle is its longest side.
Pythagoras' Theorem -
[tex]a^2 + b^2 = c^2[/tex]
where the other two sides' lengths are a and b, and c is the length of the hypotenuse.
so
[tex]5^2 + 8^2 = c^2\\c^2 = 25 + 64\\c^2 = 89\\c = \sqrt{89}[/tex]
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five points $a$, $b$, $c$, $d$, and $o$ lie on a flat field. $a$ is directly north of $o$, $b$ is directly west of $o$, $c$ is directly south of $o$, and $d$ is directly east of $o$. the distance between $c$ and $d$ is 140 m. a hot-air balloon is positioned in the air at $h$ directly above $o$. the balloon is held in place by four ropes $ha$, $hb$, $hc$, and $hd$. rope $hc$ has length 150 m and rope $hd$ has length 130 m.to reduce the total length of rope used, rope $hc$ and rope $hd$ are to be replaced by a single rope $hp$ where $p$ is a point on the straight line between $c$ and $d$. (the balloon remains at the same position $h$ above $o$ as described above.) determine the greatest length of rope that can be saved.
The amount of rope that can be saved is 114.5 m.
Let us consider
OH = X
OD = Y
OC = Z
We can see that the rope and the hot air balloon and the points on the ground make a right-angled triangle. Hence by Pythagoras' Theorem
X² + Y² = 130²
X² + Z² = 150²
Y² + Z² = 140²
Y² - Z² = 130² - 150²
or, 2Y² = 130² - 150² +140²
or, Y= √7000 [1]
From equation [1] we get
X² + 7000 = 130²
or, X = √9900 [2]
Again using [1] we get
7000 + Z² = 140²
or, Z = √12600
From [1] and [3] we can now find OP by
Z² + 70² = (OP)²
or, 12600 + 70² = 17500
OP= √17500
Since HP is the new rope, we will see that
X² + 17500 = HP²
or, HP² = 9900 + 17500
or, HP² = 27400
or, HP = √27400
= 165.5
The amount of rope that has been saved is
HC + HD - HP
= 130+150 - 165.5
= 114.5 m
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Correct Question
Five points A, B, C, D, and O lie on a flat field. A is directly north of O, B is directly west of O, C is directly south of O, and D is directly east of O. The distance between C and D is 140 m. A hot-air balloon is positioned in the air at H directly above O. The balloon is held in place by four ropes HA, HB, HC, and HD. Rope HC has a length of 150 m and rope HD has a length of 130 m.
To reduce the total length of rope used, rope HC and rope HD are to be replaced by a single rope HP where P is a point on the straight line between C and D. (The balloon remains at the same position H above O as described above.) Determine the greatest length of rope that can be saved.
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