The negative solution of the absolute value function is x = - 1.
What is the negative solution of an absolute value set?In this problem we need to solve for x in an absolute value function, whose procedure is done by the use of algebra properties:
Step 1 - Initial condition:
|2 · x - 4| = 6
Step 2 - By definition of absolute value:
2 · x - 4 = 6 or - 2 · x + 4 = 6
Step 3 - By compatibility with addition, existence of additive inverse, associative, commutative and modulative properties:
2 · x = 10 or - 2 · x = 2
Step 4 - By compatibility with multiplication, existence of multiplicative inverse, associative, commutative and modulative properties we get this result:
x = 5 or x = - 1
The negative solution of the function is x = - 1.
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f(x) = (x ^ 2 + 2x + 7) ^ 3 then
Answer
[tex]f^{\prime}(x)=6(x+1)(x^{2}+2x+7)^{2}[/tex][tex]f^{\prime}(1)=1200[/tex]Explanation
Given
[tex]f\mleft(x\mright)=(x^2+2x+7)^3[/tex]To find the derivative, we have to apply the chain rule:
[tex][u(x)^n]^{\prime}=n\cdot u(x)^{n-1}\cdot u^{\prime}(x)[/tex]Considering that in our case,
[tex]u(x)=x^2+2x+7[/tex][tex]u^{\prime}(x)=2x+2+0[/tex]and n = 3, then:
[tex]=3\cdot(x^2+2x+7)^{3-1}\cdot(2x+2)[/tex]Simplifying:
[tex]f^{\prime}(x)=3\cdot2(x+1)(x^2+2x+7)^2[/tex][tex]f^{\prime}(x)=6(x+1)(x^2+2x+7)^2[/tex]Finally, we have to replace 1 in each x in f'(x) to find f'(1):
[tex]f^{\prime}(1)=6((1)+1)((1)^2+2(1)+7)^2[/tex][tex]f^{\prime}(1)=6(1+1)(1+2+7)^2[/tex][tex]f^{\prime}(1)=6(2)(10)^2[/tex][tex]f^{\prime}(1)=6(2)(100)[/tex][tex]f^{\prime}(1)=12(100)[/tex][tex]f^{\prime}(1)=1200[/tex]Refer to your equation for the line that models the association between latitude and temperature of the cities: Yours y = -12 + 120 Computer calculated y = -1.07 + 119 What does the slope mean in the context of this situation?
The slope in the equations represent the change in temperature by the change in lattitude. This means that for each unit change in the latitude the temperature will decrease by an amount given by the slope.
An inverted pyramid is being filled with water at a constant rate of 30 cubic centimeters per second . The pyramíd , at the top, has the shape of a square with sides of length 7cm and the height is 14 cm.
ANSWER :
The answer is 30 cm/sec
EXPLANATION :
We have an inverted square pyramid with a square side of 7 cm and a height is 14 cm.
We need to find the area of the square at 2 cm from below.
Using similar triangles, we will express the side view as 2D :
We need to find the side of the square at 2 cm level.
The ratio of the sides of the smaller triangle and bigger triangle must be the same :
[tex]\begin{gathered} \frac{\text{ smaller}}{\text{ bigger}}=\frac{x}{7}=\frac{2}{14} \\ \\ \text{ Solve for x :} \\ \text{ Cross multiply :} \\ 14x=7(2) \\ 14x=14 \\ x=\frac{14}{14}=1 \end{gathered}[/tex]So the value of x is 1, then the side of the square at 2 cm level is 1 cm
The area of that square is :
A = 1 x 1 = 1 cm^2
The inverted pyramid is filled with water at a constant rate of Q = 30 cm^3 per second.
And we are asked to find the rate when the water level is 2 cm or when the area of the square is 1 cm^2 from the result we calculated above.
Recall the formula of rate :
[tex]\begin{gathered} Q=AV \\ \text{ where :} \\ Q\text{ = constant rate in }\frac{cm^3}{sec} \\ \\ A\text{ = Area of the section in }cm^2 \\ \\ V\text{ = Velocity or rate in }\frac{cm}{sec} \end{gathered}[/tex]We have the following :
[tex]\begin{gathered} Q=30\text{ }\frac{cm^3}{sec} \\ \\ A=1\text{ }cm^2 \end{gathered}[/tex]Using the formula above, the rate is :
[tex]\begin{gathered} Q=AV \\ V=\frac{Q}{A} \\ \\ V=\frac{30\text{ }\frac{cm^{\cancel{3}}}{sec}}{1\text{ }\cancel{cm^2}} \\ \\ V=30\text{ }\frac{cm}{sec} \end{gathered}[/tex]Set A is the set of all whole numbers to 20. Set B is the set of all odd integers between 8 and 18. How many numbers do the two sets have in common?
ANSWER
5 numbers they have in common
EXPLANATION
Set A has all whole numbers to 20:
[tex]A\colon1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20[/tex]Set B has only odd integer between 8 and 18:
[tex]B\colon9,11,13,15,17[/tex]We can see that set B is inside set A, because it has whole numbers that are less than 20, so the amount of numbers they have in common is all of set B: 5 numbers.
Answer this fraction based Question I will make you btainliest & provide you 50 points
Answer:
i) 2/3, ii) 2/9,iii) 4/27,iv) Rs. 40000.Step-by-step explanation:
i)A person gives 1/3 of his wealth to his wife, then he is left with:
1 - 1/3 = 2/3 of the total amountii)Then he gives 1/3 of the remainder to his son, the son gets:
2/3*1/3 = 2/9 of the total amountiii)The remaining portion is:
2/3 - 2/9 = 6/9 - 2/9 = 4/9 of the total amountEach daughter gets 1/3 of it as there are three daughters:
4/9 * 1/3 = 4/27 of the total amountiv)If the total amount is x, the son gets 2/9x and a daughter gets 4/27x and the difference of the two is Rs 20000:
2/9x - 4/27x = 200006/27x - 4/27x = 200002/27x = 20000x = 20000*27/2x = 270000This is the total amount.
The amount obtained by a daughter is:
4/27*270000 = 40000If quadrilateral WXYZ is transformed using the rule T(-1.2), in whatdirections should WXYZ be translated?O 1 unit down, 2 units rightO 1 unit left, 2 units upO 1 unit up, 2 units leftO 1 unit right, 2 units up
helppppppppppppppppppppppppppppp
Answer:
(see attached image)
Step-by-step explanation:
Imagine that there is a line drawn at y=x, when a problem wants you to "show the inverse of a function", imagine that y=x acts as a mirror and you have to make the "reflection" of your given function across that mirror.
Given the function [tex]y=(m^2-1)x^2+2(m-1)x+2[/tex] , find the values of parameter m for which the function is always positive.
Answer: [tex](-\infty, -1) \cup (1, \infty)[/tex]
Step-by-step explanation:
The function is always positive when it has a positive leading coefficient (since that means the graph will open up), and when the discriminant is negative (meaning the graph will never cross the x-axis).
Condition I. Leading coefficient is positive
[tex]m^2 -1 > 0 \implies m < -1 \text{ or } m > 1[/tex]
Condition II. Discriminant is negative
[tex](2(m-1))^2 -4(m^2 -1)(2) < 0\\\\4(m^2 -2m+1)-8(m^2 -1) < 0\\\\4m^2 -8m+4-8m^2 +8 < 0\\\\-4m^2 -8m+12 < 0\\\\m^2 +2m-3 > 0\\\\(m+3)(m-1) > 0\\\\m < -3 \text{ or } m > 1[/tex]
Taking the intersection of these intervals, we get [tex]m < -1[/tex] or [tex]m > 1[/tex].
I'll send a pic of the problem
Weare given a graph that relates the number of strawberries to the number of containers in pairs (x, y)
being x the number of containers, and y the number of strawberries.
The points of the graph read:
(3. 57)
(5, 95)
(7, 133)
(9, 171)
and we are asked to find the proportionality between those values.
We then calculate the slope that joins the points, using for example the first two pairs:
slope = (y2 - y1) / (x2 - x1)
in our case:
slope = (95 - 57) / (5 - 3) = 38 / 2 = 19
we check this same type of calculation with another pair of points to see if it holds true as well:
slope = (171 - 133) / (9 - 7) = 38 / 2 = 19
So we can answer that the proportionality is 19 strawberries per container.
With aging body fat increases in muscle mass declines the graph to the right shows the percent body fat in a group of adult women and men as they age from 25 to 75 years age is represented along the X-axis and percent body fat is represented along the Y-axis use interval notation to give the domain and range for the graph of the function for women
Step 1
The domain and range of a function is the set of all possible inputs and outputs of a function respectively. The domain is found along the x-axis, the range on the other hand is found along the y-axis.
Find the domain of the graph of the function of women using interval notation.
[tex]\text{Domain:\lbrack}25,75\rbrack[/tex]Step 2
Find the range of the graph of the function of women using interval notation.
[tex]\text{Range:}\lbrack32,40\rbrack[/tex]Therefore, the domain and range in interval notation for the women respectively are;
[tex]\begin{gathered} \text{Domain:\lbrack}25,75\rbrack \\ \text{Range:}\lbrack32,40\rbrack \end{gathered}[/tex] The graph of a toy car's speed y
over time x is a parabola that
shows a minimum speed of 2 m/s
after 3 seconds. After 5 seconds,
the car's speed is 3 m/s. What is
the equation in vertex form of the
parabola?
The equation in vertex form of the parabola is y=-1/30(x+23/2)²+529/120
Y axis represends the toy car's speed
X axis represents time
y=ax²+bx+c
c=0
y=ax²+bx
2=9a+3b multiplied with -5
-10 = -45a -15b........equation 1
3=25a+5b multiplied with 3
9 = 75a + 15b............equation 2
adding equation 1 and 2
9-10=75a-45a+15b-15b
30a=-1
a=-1/(30)
2=9×(-1/30)+3b
3b=2+3/10=23/10
b=23/30
y=-1/30 x²+23/30=-1/30(x²+23x+(23/2)²)+1/30 ×(529/4)
y=-1/30(x+23/2)²+529/120
Therefore, the equation in vertex form of the parabola is y=-1/30(x+23/2)²+529/120
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Question 3
If your rectangular yard is 8 feet wide and requires 160 pieces of sod that are cut into 1 foot squares. how
long is it?
The length of the rectangle yard is 2 feet.
What is a rectangle?A rectangle in Euclidean plane geometry is a quadrilateral with four right angles. It can also be explained in terms of an equiangular quadrilateral—a term that refers to a quadrilateral whose angles are all equal—or a parallelogram with a right angle. A square is an irregular shape with four equal sides.So, the length f the rectangular yard:
Width is 8 feet.Requires 160 pieces of sod.Then 160ft² is the area of the rectangular yard.
Now, calculate the length as follows:
A = l × w160 = l × 80l = 160/80l = 2 feetTherefore, the length of the rectangle yard is 2 feet.
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Rebecca must complete 15 hours of volunteer work. She does 3 hours each day.
For the linear equation that represents y, the hours Rebecca still has to work after x days, what does the y-intercept represent?
The y-intercept represents the hours Rebecca must work.
How to represent linear equation?Linear equation can be represented in slope intercept from, point slope form and standard form.
Therefore, in slope intercept form it can be represented as follows:
Hence,
y = mx + b
where
m = slopeb = y-interceptShe must complete 15 hours of volunteer work. She does 3 hours each day. Let's represent Rebecca situation in linear form.
where,
y = hours Rebecca still has to work
x = the number of days
Therefore,
y = 15 - 3x
The y-intercept is 15 which implies the number of hours she must complete.
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The number of hours Rebecca must work is represented by the y-intercept in the linear equation.
What is the linear equation?An equation is said to be linear if the power output of the variable is consistently one.
The linear equation is y = mx + c, where m denotes the slope and c is its intercept.
Given that she is required to put in 15 hours of volunteer work. Each day, she works three hours.
As per the given situation,
If x represents the number of days and y represents the number of hours she must work
So the linear representation shows Rebecca's situation will be:
y = 15 - 3x
Therefore, the number of hours Rebecca must work is represented by the y-intercept in the linear equation.
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Freya counted then number of cars that came to a complete stop at stop sign. of the 25 cars, 13 cars came to a complete stop. if Freya observes the next 75 cars that reach the stop sign, how many cars can she expect to come to a complete stop?
The expected value can be calculated with the formula
[tex]E(x)=x\cdot p(x)[/tex]Where p represents the probability, and x represents the new event.
Basically, we just have to find the probability of the 13 cars
[tex]p(x)=\frac{13}{25}=0.52[/tex]Then, we multiply by the numbers of cars x = 75.
[tex]E(75)=75\cdot0.52=39[/tex]Hence, the right answer is 39. The expected value is 39.if Em=11, Am=16,CF=27, What are the lengths of the following sides
We will have the following:
First, we calculate AE as follows:
[tex]AE=\sqrt[]{AM^2-EM^2}[/tex]Now, we replace values and solve it:
[tex]AE=\sqrt[]{16^2-11^2}\Rightarrow AE=3\sqrt[]{15}[/tex]From theorem AE = EC therefore EC = esqrt(15); so, the following is true:
[tex]AC=AE+EC\Rightarrow AC=2AE\Rightarrow AC=2(3\sqrt[]{15})\Rightarrow AC=6\sqrt[]{15}[/tex]Knowing this, we then determine FA as follows:
[tex]FA=\sqrt[]{AC^2-CF^2}[/tex]We then determine BE, DM & CM as follows:
Sergio believes he is five years younger than double the age of Chloe and Chloe believes she is five yearsolder than half of Sergio's age. Are they both right?
Let S represent Sergio's age
Let C represent Chloe's age
Sergio believes he is five years younger than double the age of Chloe. This would be expressed as
S = 2C - 5
Chloe believes she is five years older than half of Sergio's age. This means that
C = 5 + S/2
If we multiply the second equation by 2, it becomes
2C = 10 + S
This means that both equations are not the same. Therefore, they are not right
4x^{3}=3y+2x^{3}y^{3}
Answer: This would be the answer to your question!!
Step-by-step explanation:
Find LM if LN = 137mm.
If you are not knowledgeable in college algebra please let me know so I can move on more quickly. Thanks in advance!
Given polynomial is
[tex]3x^5-4x^4-5x^3-8x+25[/tex]We have to check whether the polynomial x-2 is a factor.
If x-2 is a factor then x = 2 is a root of the given polynomial.
Substitute x = 2 in the given polynomial,
[tex]\begin{gathered} 3.2^5-4.2^4-5.2^3-8.2+25=96^{}-64-40-16+25 \\ =121-120=1 \end{gathered}[/tex]Hence 2 is not a root of given polynomial.
And so x - 2 is not a factor.
The exponential function that represents an experiment to track the growth of agroup bacterial cells is f(x) = 2200(1.03)*, where f(x) is the number of cells and x isthe time in minutes.• Sketch this scenario, including variables, title, axes and appropriate scales.• How many bacterial cells were there to begin the experiment?• What is the percentage growth of the bacterial cells per minute?• How many bacterial cells are there after one-half hour? Round to the nearestthousand.• How long will it take for there to be 7500 bacterial cells? Round your answerto the nearest whole minute?
For this problem we are going to be working with the function:
[tex]f(x)=2200(1.03)^x[/tex]where x is the time in minutes and f(x) represents the number of bacteria at any given time x.
Part 1.
To sketch the graph we need to determine some points of it; to get them we give values to x and plug them in the expression for the funtion.
If x=0 we have that:
[tex]\begin{gathered} f(0)=2200(1.03)^0 \\ f(0)=2200 \end{gathered}[/tex]Then we have the point (0,2200).
If x=10 we have that:
[tex]\begin{gathered} f(10)=2200(1.03)^{10} \\ f(10)=2956.616 \end{gathered}[/tex]Then we have the point (10,2956.616).
If x=20 we have that:
[tex]\begin{gathered} f(20)=2200(1.03)^{20} \\ f(20)=3973.445 \end{gathered}[/tex]Then we have the point (20,3973.445).
If x=30 we have that:
[tex]\begin{gathered} f(30)=2200(1.03)^{30} \\ f(30)=5339.977 \end{gathered}[/tex]Then we have the point (30,5339.977).
If x=40 we have that:
[tex]\begin{gathered} f(40)=2200(1.03)^{40} \\ f(40)=7176.483 \end{gathered}[/tex]Then we have the point (40,7176.483).
If x=50 we have that:
[tex]\begin{gathered} f(50)=2200(1.03)^{50} \\ f(50)=9644.593 \end{gathered}[/tex]Then we have the point (50,9644.593).
Then we have the points (0,2200), (10,2956.616), (20,3973.445), (30,5339.977), (40,7176.483) and (50,9644.593). Plotting this points on the plane and joining them with a smooth line we have that the grah of the function is:
Part 2.
To determine how many bacteria were at the beginnning of the experiment we plug x=0 in the function describing the population, we did this in the previous question; therefore we conclude that there were 2200 bacteria at the beginning of the experiment.
Part 3.
We notice that the function fgiven has the form:
[tex]f(x)=a(1+r)^x[/tex]where a=2200 and r=0.03; for this type of function the growth rate in decimal form is given by r. Therefore we conclude that the percentage growth in this function is 3%.
Part 4.
To determine how many bacteria were in the experiment after one half hout we plug x=30 in the function give; we did this in part 1 of the proble.Therefore we conclude that after one half hour there were approximately 5340 bacteria cells. (for this part we roun to the neares whole number)
Part 5.
To determine how long it takes to have 7500 cells we plug f(x)=7500 in the expression given and solve the resulting equation for x:
[tex]\begin{gathered} 2200(1.03)^x=7500 \\ 1.03^x=\frac{7500}{2200} \\ 1.03^x=\frac{75}{22} \end{gathered}[/tex]To remove the base we need to remember that:
[tex]b^y=x\Leftrightarrow y=\log _bx[/tex]Then we have:
[tex]\begin{gathered} 1.03^x=\frac{75}{22} \\ x=\log _{1.03}(\frac{75}{22}) \end{gathered}[/tex]Now we use the change of base property for logarithms:
[tex]\log _bx=\frac{\ln x}{\ln b}[/tex]Then we have:
[tex]\begin{gathered} x=\log _{1.03}(\frac{75}{22}) \\ x=\frac{\ln (\frac{75}{22})}{\ln 1.03} \\ x=41.491 \end{gathered}[/tex]Therefore it takes 41 minutes to have 7500 cells.
Multiply the expressions.
-0.6y(4.5 - 2.8y) =
answer 1
-2.86
-2.7
1.68
3.9
--------- y² +
answer 2
-2.86
-2.7
1.68
3.9
Answer:
1.68y²+ 2.7y is the answer
hope it helps
The entire company went in together to buy lottery tickets. Inside the safe are two different types of lottery tickets. The Mega Million Tickets cost $5 each and the Scratch Off Tickets cost $2 each. They bought 60 tickets totaling $246. what are my x and y variables?x=y=
we have the following system
[tex]\begin{gathered} \begin{cases}x+y=60 \\ 5x+2y=246\end{cases} \\ \end{gathered}[/tex]where x is the number of mega million ticktets and y the number of scratch off tickets, so we have that y=60-x and we get that
[tex]\begin{gathered} 5x+2(60-x)=246 \\ 5x-2x+120=246 \\ 3x=126 \\ x=\frac{126}{3}=42 \end{gathered}[/tex]so they bought 42 mega million tickets and 18 scratch off
Hiwhat is 18×18[tex]18 \times 18[/tex]
The answer for 18 x 18 is 324.
Ride 'em Rodeo is a traveling rodeo show. Last night, there were 5 people wearing
boots at the rodeo for every 2 people who were not wearing boots.
If there were 125 people wearing boots at the rodeo last night, how many people were
there altogether?
The total number of people that were there altogether at the radio show is 175 people.
How to calculate the value?From the information, there were 5 people wearing boots at the rodeo for every 2 people who were not wearing boots.
It was also illustrated that there were 125 people wearing boots at the rodeo last night, those that aren't wearing boots will be illustrated by x.
2/5 = x/125
Collect like terms
5x = 125 × 2
5x = 250
Divide
x = 250/5
x = 50
Those not wearing boots = 50
Total number of people will be:
= Those wearing boots + Those not wearing
= 125 + 50
= 175
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Answer:
175
Step-by-step explanation:
Juliet has a choice between receiving a monthly salary of $1750 from a company or a base salary of $1600 and a 3% commission on the amount of furniture she sells during the month. For what amount of sales will the two choice be equal?The two salary choices will be equal when the amount of sales is [$ ]
For an amount of sales of $5,000, the two salary choice will be equal
Let the amount of sales be $x
The 3% she will receive will be;
[tex]\frac{3}{100}\times x\text{ = 0.03x}[/tex]We add this to the base salary and equate to the former monthy salary
We have this as;
[tex]\begin{gathered} 1750\text{ =1600 + 0.03x} \\ 1750-1600\text{ = 0.03x} \\ 150\text{ = 0.03x} \\ x\text{ = }\frac{150}{0.03} \\ x\text{ = \$5000} \end{gathered}[/tex]Solve the following compound inequalities. Use both a line graph and interval notation to write each solution set.
t+1-5 ort+1> 5
The value of the inequality expression given as t + 1 < -5 or t + 1 > 5 is (-oo, -6) u (4, oo)
How to determine the solution to the inequality?The inequality expression is given as
t + 1 < -5 or t + 1 > 5
Collect the like terms in the above expressions
So, we have
t < -5 - 1 or t > 5 - 1
Evaluate the like terms in the above expressions
So, we have
t < -6 or t > 4
Hence, the solution to the inequality is t < -6 or t > 4
Rewrite as an interval notation
(-oo, -6) u (4, oo)
See attachment of the number line
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Find (and classify) the critical points of the following function and determine if they are local max, local min, or neither: f(x) =2x^3 + 3x^2 - 120x
As given by the question
There are given that the function:
[tex]f(x)=2x^3+3x^2-120x[/tex]Now,
To find the critical point, differentiate the given function with respect to x and put the result of function equal to zero
So,
[tex]\begin{gathered} f(x)=2x^3+3x^2-120x \\ f^{\prime}(x)=6x^2+6x-120 \end{gathered}[/tex]Then,
[tex]\begin{gathered} f^{\prime}(x)=0 \\ 6x^2+6x-120=0 \\ x^2+x-20=0 \\ x^2+5x-4x-20=0 \\ x(x+5)-4(x+5) \\ (x-4)(x+5) \\ x=4,\text{ -5} \end{gathered}[/tex]Now,
To find the y-coordinate, we need to substitute the above value, x = 4, -5, into the function f(x)
So,
First put x = 4 into the given function:
[tex]\begin{gathered} f(x)=2x^3+3x^2-120x \\ f(4)=2(4)^3+3(4)^2-120(4) \\ =128+48-480 \\ =-304 \end{gathered}[/tex]And,
Put x = -5 into the function f(x):
[tex]\begin{gathered} f(x)=2x^3+3x^2-120x \\ f(-5)=2(-5)^3+3(-5)^2-120(-5) \\ =-250+75+600 \\ =425 \end{gathered}[/tex]Hence, the critical point is, (4, -304) and (-5, 425).
Now,
To find the local maxima and local minima, we need to find the second derivative of the given function:;
So,
[tex]\begin{gathered} f^{\prime}(x)=6x^2+6x-120 \\ f^{\doubleprime}(x)=12x+6 \end{gathered}[/tex]Now,
The put the value from critical point into the above function to check whether it is maxima or minima.
So,
First put x = 4 into above function:
[tex]\begin{gathered} f^{\doubleprime}(x)=12x+6 \\ f^{\doubleprime}(4)=12(4)+6 \\ f^{\doubleprime}(4)=48+6 \\ f^{\doubleprime}(4)=54 \\ f^{\doubleprime}(4)>0 \end{gathered}[/tex]And,
Put x = -5 into the above function
[tex]\begin{gathered} f^{\doubleprime}(x)=12x+6 \\ f^{\doubleprime}(-5)=12(-5)+6 \\ f^{\doubleprime}(-5)=-60+6 \\ f^{\doubleprime}(-5)=-54 \\ f^{\doubleprime}(-5)<0 \end{gathered}[/tex]Then,
According to the concept, if f''(x)>0 then it is local minima function and if f''(x)<0, then it is local maxima function
Hence, the given function is local maxima at (-5, 425) and the value is -54 and the given function is local minima at point (4, -304) and the value is 54.
A student sketched some art on an 8-inch x 10-inch piece of paper. She wants to resize it to fit a 4-inch x 6 inchframe (as shown below).What percent of the original sketch was still able to be included in the frame?
So,
The area of art can be found multiplying:
8in * 10in = 80in²
And, the area of the frame, can be also found multiplying the dimentions:
4in * 6in = 24in².
If we divide, we'll obtain a ratio between the area of the frame and the area of the art as follows:
[tex]\frac{24}{80}=0.3[/tex]And, 0.3*100% = 30%.
So,30 percent of the original sketch was still able to be included in the frame.
The product of two consecutive positive even integers is 48. Find the greatest positive integer.
From that statement we can create the following equation,
[tex]n\cdot \left(n+2\right)=48[/tex]solving for n,
[tex]\begin{gathered} n^2+2n=48 \\ n^2+2n-48=0 \\ n_{1,\:2}=\frac{-2\pm \sqrt{2^2-4\cdot \:1\cdot \left(-48\right)}}{2\cdot \:1} \\ n_{1,\:2}=\frac{-2\pm \:14}{2\cdot \:1} \\ n_1=\frac{-2+14}{2\cdot \:1},\:n_2=\frac{-2-14}{2\cdot \:1} \\ n=6,\:n=-8 \end{gathered}[/tex]We can only use the positive number for this problem, therefore n = 6
From the above, the set of numbers is 6 and 6+2=8, since 6*8=48.
Answer: the greatest integer is 8
The area of a rectangle is x2 – 8x + 16. The width of therectangle is x – 4. What is the length of the rectangle?-
To answer this question, we need to remember that the area of a rectangle is given by:
[tex]A_{\text{rectangle}}=l\cdot w[/tex]And we have - from the question - that:
[tex]A_{\text{rectangle}}=x^2-8x+16[/tex]And the width of the rectangle is:
[tex]w=x-4[/tex]If we factor the polynomial that represents the area, we need to find two numbers:
• a * b = 16
,• a + b = -8
And both numbers are:
• a = -4
,• b = -4
Since
• -4 * -4 = 16
,• -4 - 4 = -8
Therefore, we can say that:
[tex]x^2-8x+16=(x-4)(x-4)=(x-4)^2[/tex]Therefore:
[tex]l\cdot w=A_{\text{rectangle}}[/tex][tex]l=\frac{A_{rec\tan gle}}{w}[/tex]Then the length of the rectangle is:
[tex]l=\frac{x^2-8x+16}{x-4}=\frac{(x-4)(x-4)}{x-4}\Rightarrow\frac{x-4}{x-4}=1[/tex][tex]l=\frac{(x-4)}{(x-4)}\cdot(x-4)\Rightarrow l=x-4[/tex]In summary, therefore, the length of the rectangle is x - 4.
[tex]l=x-4[/tex][We can check this result if we multiply both values as follows:
[tex]A_{\text{rectangle}}=l\cdot w=(x-4)\cdot(x-4)=(x-4)^2_{}[/tex]And we already know that the area of the rectangle is:
[tex]x^2-8x+16=(x-4)^2[/tex].]