Answer: The required x-co-ordinate of the point of intersection of two lines is [tex]-\dfrac{2bm}{m^2+1}.[/tex]
Step-by-step explanation: Given that two perpendicular lines have opposite y-intercept and the equation of one of the lines is
[tex]y=mx+b.[/tex]
We are to express the x-coordinate of the intersection point of the lines in terms of m and b.
Let the slope and y-intercept of the other line be s and c respectively.
Since the product of the slopes of two perpendicular lines is -1 and -b is the opposite of b, so we have
[tex]ms=-1~~~\Rightarrow s=-\dfrac{1}{m}[/tex]
and c = -b.
That is, the equation of the other line is
[tex]y=sx+c\\\\\Rightarrow y=-\dfrac{1}{m}-b.[/tex]
Comparing the equations of both the lines, we get
[tex]mx+b=-\dfrac{1}{m}x-b\\\\\\\Rightarrow mx+\dfrac{1}{m}x=-2b\\\\\\\Rightarrow \dfrac{m^2+1}{m}x=-2b\\\\\\\Rightarrow x=-\dfrac{2bm}{m^2+1}.[/tex]
Thus, the required x-co-ordinate of the point of intersection of two lines is [tex]-\dfrac{2bm}{m^2+1}.[/tex]
Final answer:
The x-coordinate of the intersection point for two perpendicular lines with opposite y-intercepts can be expressed as x = -2b/(m + 1/m), where m is the slope and b is the y-intercept of one of the lines.
Explanation:
When we have two perpendicular lines with opposite y-intercepts, and one of the lines has the equation y = mx + b, we can find the x-coordinate of the intersection point by expressing the other line's equation.
Since the other line is perpendicular, its slope will be the negative reciprocal of m. Therefore, if the first line's slope is m, the second line's slope will be -1/m. Also, if the y-intercept of the first line is b, the y-intercept of the second line will be -b, given that they are opposite.
The equation of the second line will then be y = (-1/m)x - b. To find the intersection point, we set the y-values of both equations equal to each other:
mx + b = (-1/m)x - b
Now, let's solve for x:
mx + (1/m)x = -2b
x(m + 1/m) = -2b
x = -2b/(m + 1/m)
Thus, the x-coordinate of the intersection point in terms of m and b is x = -2b/(m + 1/m).
What is the process of converting a fraction to a decimal?
8/12 in 0.667 (right?), but i dont actually know the process to figuring it out.
thank youu
You would use long division to determine that 8/12 = 0.667 approximately. The steps would go something like this
-------------
Step 1) Write 8 under a long division bar and place 12 to the left and outside the bar
Step 2) Change the 8 into 8.0000 (you can have as many trailing zeros as you want)
Step 3) While 12 doesn't go into 8, 12 does go into 80. So you can think of 8.0 as 80. Ask yourself "how many times does 12 go into 80?". The answer is 6 times because 6*12 = 72 (note how 7*12 = 84 is too big). So we write a "6" above the 8.0000 such that it is above the first "0". So far we have 0.6 as the quotient.
Step 4) Multiply that 6 by 12 to get 72. We write the 72 under the 80 and then subtract to get 80 - 72 = 8. Pull down the second 0 from "8.0000" to change that 8 into 80.
Step 5) We effectively repeat steps 3 and 4 again. So thats why we get another 6 as the next part of the quotient, and why the 6's repeat forever. The only reason why there's a 7 at the end is because of rounding.
-----------------------
The long division is shown in the image attachment below. The gridlines are there to show how things line up though when you write out the long division steps (to show to your teacher) you wont use any grid lines.
side note: the fraction 8/12 reduces to 2/3. You divide both top and bottom by the GCF 4. So 8/4 = 2 and 12/4 = 3.
How do you get 7.5/8
Answer:
0.9375
Step-by-step explanation:
Do the long division and you'll find the answer.
Angeline made a dish with 5 1/3 cups of pasta. One serving is 0.2 cups. How many servings of pasta were in the dish Angeline made?
Answer:
26 2/3
Step-by-step explanation:
We could turn both of these numbers into decimals, but I find fractions easier.
so 5 1/3 as an improper fraction is 16/3
0.2 as a fraction is 2/10 or 1/5
so now we have the equation 16/3 ÷ 1/5 or 16/3 x 5/1
this give me 80/3, then when simplified gives me 26 2/3.
A soccer game is 90 minutes long. 36 minutes have passed. What percentage of the game has passed?
Answer:
40%
Step-by-step explanation:
Percentage Passed = 36 ÷ 90 × 100 = 40%
The volume of a rectangular prism is 72 cubic centimeters. The prism is 2 centimeters wide and 4 centimeters high. What is the length of the prism?
Answer:
The length of prism is 9 centimetres.
Step-by-step explanation:
Given:
The volume of a rectangular prism is 72 cubic centimeters.
The prism is 2 centimeters wide and 4 centimeters high.
Now, to find the length of prism:
Let the length be [tex]l[/tex].
Width = 2 centimeters.
Height = 4 centimeters.
Volume = 72 cubic centimeters.
So, putting the formula to get the length of prism:
[tex]Volume = length\times width\times height[/tex]
[tex]72=l\times 2\times 4[/tex]
[tex]72=l\times 8[/tex]
Dividing both sides by 8 we get:
[tex]9=l[/tex]
Length = 9 centimetres.
Therefore, the length of prism is 9 centimetres.
What expression is equivalent to x-5/(x-5)(x-4)
Answer:
[tex]\large\boxed{\dfrac{1}{x-4}\ \text{for}\ x\neq5\ \wedge\ x\neq4}[/tex]
Step-by-step explanation:
[tex]\dfrac{x-5}{(x-5)(x-4)}\\\\\text{Domain:}\ (x-5\neq0\ \wedge\ x-4\neq0)\Rightarrow(x\neq5\ \wedge\ x\neq4)\\\\\dfrac{x-5}{(x-5)(x-4)}\qquad\text{cancel}\ (x-5)\\\\=\dfrac{1}{x-4}[/tex]
Lines c and d are parallel. The equation of line c is y=−3x−5. What is the equation of line d?
Answer: y = -3x - 5
Step-by-step explanation: since the two lines are parallel, it implies that
m1 = m2. From the equation given,
m1 = -3 , m2 = -3
Substantial for m2 in y = mx + c
We now have
y = -3x - 5 since the new line is not passing through any coordinate.
Answer:
The answer is y=-3x-2
I hope it's not too late;)
Step-by-step explanation:
3. READING Alejandro read 4 pages in a book in 6
minutes. How long would you expect him to take to
read 6 pages at this rate?
Alejandro read 4 pages in 6 minutes, then it would take him 9 minutes to read 6 pages
Solution:
Given that Alejandro read 4 pages in a book in 6 minutes
Therefore,
4 pages = 6 minutes
Let us find the time taken to read 1 page
time taken to read 1 page = time taken to read 4 pages divided by 4
[tex]\text{ time taken to read 1 page } = \frac{6 minutes}{4} = \frac{3}{2} minutes[/tex]
Thus to read 6 pages at this rate, multiply time taken to read 1 page by 6
[tex]\rightarrow \frac{3}{2} \times 6 = 9[/tex]
Therefore it would take 9 minutes to read 6 pages
which expression represents 4 times as much as 12
Answer:
4x+12
Step-by-step explanation:
An expression is a mathematical sentence that doesn't end in an equal sign. This question is asking for you to find four times as much as 12, but doesn't explain what number or phrase you need to find it for. We would replace the unknown number with the variable X.
I may be confusing this with another subject, so sorry if I am! Hope this helps!
The expression which represents 4 times as much as 12 is, [tex]4x=12[/tex]
Let us consider that number is x.
Since, 4 times of number is equal to 12.
So that, [tex]4*x=12\\\\4x=12[/tex]
Hence, the expression which represents 4 times as much as 12 is, [tex]4x=12[/tex]
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Expand
Your answer should be a polynomial in standard form
(
+1)(2-6)
Answer:
x^2 - 6x + x - 6
= x^2 - 5x - 6
Change to slope-intercept form. Then find the y-intercept, first point, and second point.
-x+3y>6
(choices in photo)
Answer:
1) [tex]y> \frac{x}{3}+2[/tex] 2) (0,2) 3) The first and second Points must have x coordinate <-6, or y-coordinate y >2 e.g. (-7,2), (-6,3)
Step-by-step explanation:
1) To Rewrite it as Slope-intercept form, is to isolate the y on the left side and on the right side the rest of the inequality.
[tex]-x+3y>6\Rightarrow 3y>x+6 \Rightarrow y> \frac{x+6}{3}\Rightarrow y> \frac{x}{3}+2[/tex]
2) Since this is a linear inequality the y intercept is given by "b" parameter.[tex]y> mx+b \Rightarrow y> \frac{x}{3}+2 \Rightarrow b=2[/tex]
So the y-intercept is y > 2, coordinate point (0,2). In the graph, we have a dashed line over 2.
3) Since there no choices, the points that satisfy this inequality lie within the green area. We know that the points for this inequality must satisfy x < -6 or y> 2:
Testing for (-7,2) for x<-6 ⇒-7 <-6
[tex]-x+3*y>6\\-(-7)+3*2>6\\7+6>6\\13>6\:\\True\\[/tex]
Testing for (-6,3) for y>2 ⇒3>2
[tex]-x+3*y>6\\-(-6)+3*3>6\\6+9>6\\15>6\:True\\[/tex]
Robin can clean
72
7272 rooms in
6
66 days.
How many rooms can Robin clean in
9
99 days?
Robin can clean 108 rooms in 9 days, given she cleans 72 rooms in 6 days; this is determined by calculating her daily cleaning rate and extending it over the 9-day period.
The student asks how many rooms Robin can clean in 9 days if she can clean 72 rooms in 6 days. To solve this problem, we first need to determine the rate at which Robin cleans rooms per day, which is done by dividing the total number of rooms by the total number of days. Once we have the daily rate, we can then multiply this rate by 9 to find out how many rooms can be cleaned in 9 days.
First, we calculate the daily rate:
Rate = Total Rooms / Total Days
Rate = 72 rooms / 6 days
Rate = 12 rooms/day.
Now, we multiply the daily rate by 9 days:
Rooms in 9 days = Rate times Number of Days
Rooms in 9 days = 12 rooms/day times 9 days
Rooms in 9 days = 108 rooms.
Therefore, Robin can clean 108 rooms in 9 days.
Can someone help me with this problem? I just can't seem to get the answer.
Answer:
[tex]\frac{9}{4}[/tex] or [tex]2\frac{1}{4}[/tex] or [tex]2.25[/tex]
Step-by-step explanation:
1 cm = 25000cm = 1/4 km. 9*1/4=9/4=2 1/4= 2.25
MR Arsalan bought a plot of land for rs 108 000 and sold it for a profit of 20% at what price did he sold at
Answer:
The plot of land was sold for rs 129,600
Step-by-step explanation:
The cost of the plot of land = rs 108,000
Profit calculation ( 20% x 108,000) = rs 21,600
Price land was sold = (108,000 + 21,600) = 129,600
what is the degree of differential equation? ?
Answer:
Degree = 1
Step-by-step explanation:
Given:
The differential equation is given as:
[tex]\frac{d^2y}{dx^2}+(\frac{dy}{dx})^2+6y=0[/tex]
The given differential equation is of the order 2 as the derivative is done 2 times as evident from the first term of the differential equation.
The degree of a differential equation is the exponent of the term which is the order of the differential equation. The terms which represents the differential equation must satisfy the following points:
They must be free from fractional terms.Shouldn't have derivatives in any fraction.The highest order term shouldn't be exponential, logarithmic or trigonometric function.The above differential equation doesn't involve any of the above conditions. The exponent to which the first term is raised is 1.
Therefore, the degree of the given differential equation is 1.
A total of $11,000 is invested in two accounts. Part was invested at 4% and the rest was invested at 7%. If the investments earn $680 peer year, how much was invested at each rate? I=prt
Answer:
The amount invested at 4% was $3,000 and the amount invested at 7% was $8,000
Step-by-step explanation:
we know that
The simple interest formula is equal to
[tex]I=P(rt)[/tex]
where
I is the Final Interest Value
P is the Principal amount of money to be invested
r is the rate of interest
t is Number of Time Periods
in this problem we have
[tex]t=1\ year\\ P_1=x\\P_2=11,000-x\\I=\$680\\r_1=4\%=4/100=0.04\\r_2=7\%=7/100=0.07[/tex]
substitute in the formula above
[tex]I=P_1(r_1t)+P_2(r_2t)[/tex]
[tex]680=0.04x+(11,000-x)0.07[/tex]
solve for x
[tex]680=0.04x+770-0.07x[/tex]
[tex]0.07x-0.04x=770-680[/tex]
[tex]0.03x=90[/tex]
[tex]x=\$3,000[/tex]
so
[tex]\$11,000-x=\$11,000-\$3,000=\$8,000[/tex]
therefore
The amount invested at 4% was $3,000 and the amount invested at 7% was $8,000
Solve |12x + 1| = 10
(9/2, 11/2)
(-9/2, 9/2)
(-112.9/2)
|12x + 1| = 10
Remove the absolute value term and make two equations:
12x +1 = 10
12x +1 = -10
Now solve for both x values:
12x +1 = 10
Subtract 1 from both sides:
12x = 9
Divide both sides by 12:
x = 9/12
12x +1 = -10
Subtract 1 from both sides:
12x = -11
Divide both sides by 12:
x = -11/12
The answer would be (-11/12, 912)
If f (x) = 3 x squared and g (x) = 4 x cubed + 1, what is the degree of (f circle g) (x)?If f (x) = 3 x squared and g (x) = 4 x cubed + 1, what is the degree of (f circle g) (x)?
options are
2
3
5
6
Answer:
degree = 6
Step-by-step explanation:
Given [tex]f(x)=3x^2[/tex], and [tex]g(x)=4^3+1[/tex], we can find the composition of functions: [tex]fog(x)[/tex] by applying the definition of composition and performing the needed algebra.
Recall that the composition of functions is defined as: [tex]fog(x)=f(g(x))[/tex], where we use as input for the function f(x) the actual expression in terms of "x" of the function g(x):
[tex]f(g(x))=f(4x^3+1)\\f(g(x))=3(4x^3+1)^2\\f(g(x))=3\,(4x^3+1)\,(4x^3+1)\\f(g(x))=3\,[16x^6+4x^3+4x^3+1]\\f(g(x))=3\,[16x^6+8x^3+1]\\f(g(x))=48x^6+24x^3+3[/tex]
Therefore, the degree of this expression is "6" (the highest power at which the variable "x" appears)
Answer:
the answer is 6
Step-by-step explanation:
don't have one
How many feet are in 100 inches? Write your answer two ways: in feet and inches and in feet only
Answer: 8 ft and 4 in, 8 1/3 ft
Step-by-step explanation:
100/12=50/6=25/3=8.3333333...
1/3 of a foot is 1/3x12 which is 12/3 which is 4 in.
Final answer:
100 inches is equal to 8 feet and 4 inches or 8.33 feet.
Explanation:
To solve this problem, you can convert 100 inches to feet using the unit equivalence:
1 foot = 12 inches
Multiply the number of inches by the unit equivalence:
100 inches ÷ 12 inches per foot = 8 feet and 4 inches
g(x) = x2 + x - 2
Find g(x - 3)
Answer:
x^2-5x+4
Step-by-step explanation:
(x-3)^2+(x-3)-2
x^2-3x-3x+9+x-3-2
x^2-6x+9+x-5
x^2-5x+9-5
x^2-5x+4
HELP I WILL GIVE BRAINLIEST THANKYOU PLEASE YOU DONT HAVE TO ANSEER ALL
Answer:
none of the above
Step-by-step explanation:
PLEASE MARK BRAINLIEST!
Answer:
Sorry for the incorrect answer everyone!
Step-by-step explanation:
My old answer: Perpendicular bisector
THE CORRECT ANSWER: None of the aboveReasons:
Median doesn't relate to this problemAltitude doesn't relate to this problemA perpendicular bisector cuts through a line segment, making an angle of 90. This triangle does not show thatThe answer is none of the above, because none of the other answers make sense.I know I made a mistake before, but this is better late than never! I hope this helps for the future! *^-^*
- sincerelynini
What is the domain and range of the relation shown in
the table?
Y
5
9
-1
5
X
10
15
19
32
A. domain: {−1, 5, 9}{−1, 5, 9} range: {10, 15, 19, 32}{10, 15, 19, 32}
B. domain: {10, 15, 19, 32}{10, 15, 19, 32} range: {−1, 5, 9}{−1, 5, 9}
C. domain: {10, 15, 19, 32}{10, 15, 19, 32} range: {y|y∈R}{y|y∈ℝ}
D. domain: {x|x∈R}{x|x∈ℝ} range: {−1, 5, 9}
Answer:
Domain : {10, 15, 19, 32}
Range: {-1, 5, 9}
Step-by-step explanation:
As the table of the relation is given as follows:
X Y
10 5
15 9
19 -1
32 5
From this table, we can define the relation by combining the ordered pairs from the table. Each and every order pair consists of x-coordinate and corresponding y-coordinate of any corresponding point.
So, the relation from the table can be made as follows:
relation : {(10, 5), (15, 9), (19, -1), (32, 5)}
Domain of a relation consists of all the x-coordinates (first elements) of order pairs.
Range of a relation consists of all the y-coordinates (second elements) of ordered pairs.
So, domain and range of relation will be as follows:
Domain : {10, 15, 19, 32}
Range: {-1, 5, 9}
Note: If there is any duplicate element in any x or y-coordinate of any ordered pair, it will be written only once when we determine domain and range. Here, in this example, 5 is duplicate, so, it will be mentioned only one time when we determine the range of this relation.
Keywords: domain, relation, range
Learn more about domain and range of a relation from brainly.com/question/11422136
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The domain and range of the relation given in the table are the sets of x-values and y-values respectively. In this case, the domain is {-1, 5, 9} and the range is {10, 15, 19, 32}, which corresponds to Option A.
To determine the domain and range of the relation shown in a table, we must look at the set of ordered pairs provided. The domain consists of all the first elements in these pairs, which represent the x-values, while the range consists of all the second elements, which are the y-values. Looking at the table, we can list the x and y values separately to clearly identify the domain and range.
If the table of the relation presents the ordered pairs as (X, Y), interpreting the question that the domain should be {-1, 5, 9} and the range should be {10, 15, 19, 32}, then the correct option would be:
Option A.
Domain: \{-1, 5, 9\} and Range: \{10, 15, 19, 32\}.
It is essential to note that a function is a special relation where each element of the domain maps to a unique element in the range. In this case, as each x-value maps to a distinct y-value, it suggests the relation is indeed a function.
-x+8=6 what is the answer for x
Answer:
x=-2
Step-by-step explanation:
-2+8=6
subtract 8 from six and you get your answer since it is a simpler linear equation. hoped this helped a little bit
Answer:2
Step-by-step explanation:
-X+8=6
It takes 2 wooden sticks and 1.5 square feet of paper to make a kite, and it takes 12 wooden sticks and 8 square
feet of paper to make a lamp.
Min-Young wants to make kites and lamps using at least 87 wooden sticks and more than 63 square feet of
paper.
Let K denote the number of kites she makes and L the number of lamps she makes.
Write an inequality that represents the condition based on the number of wooden sticks. Write an inequality that represents the condition based on the number of square feet of paper.
Answer:
2K + 12L ≥ 87 and 1.5K + 8L > 63
Step-by-step explanation:
It takes 2 wooden sticks and 1.5 square feet of paper to make a kite, and it takes 12 wooden sticks and 8 square feet of paper to make a lamp.
If K denotes the number of kites that Min-Young made and L denotes the number of lamps she made then
2K + 12L is the number of wooden sticks that she used which is greater than equal to 87
i.e. 2K + 12L ≥ 87 ........ (1)
Again, 1.5K + 8L is the number of square feet of paper that she used which is greater than 63 sq. feet
i.e. 1.5K + 8L > 63 ........... (2)
Hence, inequalities (1) and (2) are the required answer. (Answer)
Answer:
1) 2K + 12L ≥ 87
2) 1.5K + 8L > 63
Step-by-step explanation:
i have 150 i share one third how much did i share
Answer:
50
Step-by-step explanation:
150 divided by 3 is 50 and since you shared only one third the answer will be 50
Answer:50
Step-by-step explanation:
One third equals 1/3 of 150
1/3 ×150 =50
Using the expressions above, find the cost to the family of each company's phone plan if:
a. Four people want a phone line, four people want unlimited texting, and the family needs two interna
Company A
Company B
Company C
Which cell phone company should John's family use? Why?
Answer:
company B
Step-by-step explanation:
the company is good!
Final answer:
To find the cost to the family of each company's phone plan, we calculated the total cost for four phone lines and four unlimited texting plans, as well as the cost for two international calls. Based on the calculations, John's family should use Company A because it has the lowest total cost of $100.50.
Explanation:
To find the cost to the family of each company's phone plan, we need to calculate the total cost for four phone lines and four unlimited texting plans, as well as the cost for two international calls. Let's calculate the cost for each company:
Company A:
Cost for four phone lines = 4 * $20 = $80
Cost for four unlimited texting plans = 4 * $5 = $20
Cost for two international calls = 2 * $0.25 = $0.50
Total cost for the family with Company A = $80 + $20 + $0.50 = $100.50
Company B:
Cost for four phone lines = 4 * $15 = $60
Cost for four unlimited texting plans = 4 * $10 = $40
Cost for two international calls = 2 * $0.50 = $1
Total cost for the family with Company B = $60 + $40 + $1 = $101
Company C:
Cost for four phone lines = 4 * $25 = $100
Cost for four unlimited texting plans = 4 * $8 = $32
Cost for two international calls = 2 * $0.15 = $0.30
Total cost for the family with Company C = $100 + $32 + $0.30 = $132.30
Therefore, John's family should use Company A because it has the lowest total cost of $100.50.
The top band in the state charges C(x)=55x+15,000, where x is the total number of attendees at the concert. The venue charges $70 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?
The venue will break even after selling 1000 tickets.
The value will be $70,000 at that point.
Step-by-step explanation:
Given,
Charges of band;
C(x) = 55x+15000
Per ticket charges by Venue = $70
V(x) = 70x
For breaking even,
C(x) = V(x)
[tex]55x+15000=70x\\15000=70x-55x\\15000=15x\\15x=15000[/tex]
Dividing both sides by 15
[tex]\frac{15x}{15}=\frac{15000}{15}\\x=1000[/tex]
The venue will break even after selling 1000 tickets.
Putting 1000 in V(x)
[tex]V(1000)=70(1000)\\V(1000)=70000[/tex]
The value will be $70,000 at that point.
Keywords: function, multiplication
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What is the average rate of change of the function f(x)=120(.1)^x from x=0 to x=2
Answer:
- 59.4
Step-by-step explanation:
The given function is [tex]f(x) = 120(0.1)^{x}[/tex]
Therefore, at x = 0, [tex]f(0) = 120(0.1)^{0} = 120[/tex] and
at x = 2, [tex]f(2) = 120(0.1)^{2} = 1.2[/tex]
Hence, the average rate of change of the given function from x = 0 to x = 2 will be
[tex]\frac{f(2) - f(0)}{2 - 0}[/tex]
= [tex]\frac{1.2 - 120}{2 - 0}[/tex]
= - 59.4 (Answer)
Lexi has $80 to spend on an ice cream treat each week. If she spends $4 on ice cream per week, the function C(x)=80−4x
C
(
x
)
=
80
-
4
x
represents how much ice cream money, C, Lexi has left after a certain number of weeks, x. What is the domain in the context of the problem? Why?
Answer:
Domain[tex]=\{1,2,3,4,5,6,7,8,9,10,11,12,,13,14,15,16,17,18,19,20\}[/tex]
In other words Domain[tex]=\{x|x\ is\ an\ integer\ and\ 1\leq x\leq20\}[/tex]
Step-by-step explanation:
[tex]C(x)=80-4x[/tex]
Let after [tex]t[/tex] weeks she has left no money.
[tex]c(t)=0\\80-4t=0\\4t=80\\t=20[/tex]
That means she can have money for ice cream from the week 1 to week 20 and after that she will have no money.
So [tex]x[/tex] can be any number between [tex]1[/tex] and [tex]20[/tex]
Hence Domain[tex]=\{x|x\ is\ an\ integer\ and\ 1\leq x\leq20\}[/tex]
A computer can sort x objects in t seconds, as modeled by the functio
below:
1=0.005x2 + 0.002.x
How many objects are required to keep the computer busy for exactly
seconds?
Round to the nearest whole object.
Question:
A computer can sort x objects in t seconds, as modeled by the function below:
t=0.005x^2+0.002x
How many objects are required to keep the computer busy for exactly 9 seconds?
Round to the nearest whole object.
Answer:
42 objects are required to keep the computer busy for exactly 9 seconds
Solution:
Given function is:
Computer can sort x objects in t seconds, as modeled by the function below:
[tex]t = 0.005x^2+0.002x[/tex]
We have to find number of objects required to keep the computer busy for exactly 9 seconds
Therefore t = 9
Substitute t = 9 in given function
[tex]9 = 0.005x^2+0.002x\\\\0.005x^2+0.002x - 9 = 0[/tex]
Let us solve the above equation by quadratic formula,
[tex]\text {For a quadratic equation } a x^{2}+b x+c=0, \text { where } a \neq 0\\\\x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]
Using the above formula,
[tex]\text{ for } 0.005x^2+0.002x - 9 = 0 , \text{ we have } a = 0.005 ; b = 0.002 ; c = -9[/tex]
Substituting the values of a = 0.005 ; b = 0.002 ; c = -9 in above quadratic formula we get,
[tex]\begin{aligned}&x=\frac{-0.002 \pm \sqrt{0.180004}}{0.01}\\\\&x=\frac{-0.002 \pm 0.4242}{0.01}\\\\&x=\frac{-0.002+0.4242}{0.01} \text { or } x=\frac{-0.002-0.4242}{0.01}\\\\&x=42.22 \text { or } x=-42.62\end{aligned}[/tex]
Ignoring negative value,
x = 42.22 ≈ 42
Thus 42 objects are required to keep the computer busy for exactly 9 seconds