The volume of a rectangular prism is 48x^3+56x^2+16 X. Answer the following questions 1. What are the dimensions of the prism?2. If x=2, use the polynomial 48x^3+56x^2+16x to find the prism. 3. If x=2, use the factors found in part a to calculate each dimension. 4. Using the dimensions found in part c, calculate the volume.
Answers
(1) From the statement, we know that:
• the volume of a rectangular prism is:
[tex]V(x)=48x^3+56x^2+16x\text{.}[/tex]• and it also can be computed by:
[tex]V=a\cdot b\cdot c.[/tex]Where a, b and c are the lengths of the sides.
By factoring in the expression of V(x), we get:
[tex]\begin{gathered} V(x)=(48x^3+56x^2+16x) \\ =(8x)\cdot(6x^2+7x+2) \\ =(8x)\cdot(3x+2)\cdot(2x+1)\text{.} \end{gathered}[/tex]Comparing this result with the expression above, we see that the sides of the prism are:
[tex]\begin{gathered} a=8x, \\ b=3x+2, \\ c=2x+1. \end{gathered}[/tex](2) Relacing the value x = 2 in the polynomial V(x), we get:
[tex]V(2)=48\cdot2^3+56\cdot2^2+16\cdot2=640.[/tex](3) Replacing the value x = 2 in the expressions for the dimensions found in point (2), we get:
[tex]\begin{gathered} a=8\cdot2=16, \\ b=3\cdot2+2=8, \\ c=2\cdot2+1=5. \end{gathered}[/tex](4) Replacing the dimensions from point (3) in the expression for the volume, we get:
[tex]V=16\cdot8\cdot5=640.[/tex]Which is the same result that we get in point (2), as it should be.
Answer
(1) Dimensions
[tex]\begin{gathered} a=8x, \\ b=3x+2, \\ c=2x+1. \end{gathered}[/tex](2) Volume when x = 2:
[tex]V(2)=640.[/tex](3) Dimensions when x = 2:
[tex]\begin{gathered} a=16, \\ b=8, \\ c=5. \end{gathered}[/tex](4) Volume when x = 2:
[tex]V=16\cdot8\cdot5=640.[/tex]Related Questions
Baldwin's family is going to get a hamster, but first they need to buy the supplies. The pet store has 2 cages and 5 kinds of wood shavings to cover the floor of the cage. There are 2 kinds of water bottles. How many different ways can Baldwin's family set up a home for their hamster?
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To solve this problem it is necessary to use the multiplication counting rule, that states that if there are two events a and b, the total number of outcomes will be a times b (a*b).
In this case, the total number of outcomes will be the product of the number of options for each event (cages, kinds of wood shavings and water bottles).
[tex]2\cdot5\cdot2=20[/tex]Baldwin's family can set up a home for their hamster in 20 different ways.
NEED THIS DONE ASAP!!!!
Answers
Answer:
0,4,2
Step-by-step explanation:
all you have to do is divide the y value with the x value and I got these solutions
The amount of light I (in milliwatts per square centimeter) of a 50-watt light bulb at a distwnce of d centimeters is described by thrme equation I = 8910d^-2 a. simplify the right side of the equation b. Substitute 80 for d in the equation you found in a.) and solve for I. What does your result mean in this situation?
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The amount of light of the bulb is represented by I
[tex]I=8910d^{-2}[/tex]a) Simplify the right side of the expression
[tex]I\text{ = }\frac{8910}{d^2}[/tex]b) Substitute 80 for d in the equation.
d = 80
[tex]\begin{gathered} I\text{ = }\frac{8910}{80^2} \\ I\text{ = }\frac{8910}{6400} \\ I\text{ = }1.39 \end{gathered}[/tex]The amount of light I of the 50-watt light bulb is 1.39 milliwatts per square centimeter
While writing a paper on making decisions in groups, Todd researched the number of members on a variety of committees. Number of members Number of committees 19 8 27 2 57 2 74 1 79 2 83 5 X is the number of members that a randomly chosen committee has. What is the expected value of X? Write your answer as a decimal.
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Step 1: Draw the expected value table
[tex]p(x)\text{ = }\frac{committee\text{ selected}}{\text{total number of committe}e}[/tex]Expected value = sum of x.p(x)
Expected value = 48.35
Hi, can you help me with this question, problem 7, attached picture. Thank you.
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∠2 and ∠4 are a linear pair, so they are supplementary, regardless of lines l and m are parallel or not.
∠1 and ∠8 are alternate exterior angles since they are on the outer side of each line (l and m) but on the opposite side of the transversal line. If l and m are parallel lines, then the alternate exterior angles are equal, so ∠1=∠8
∠6 and ∠3 are alternate interior angles. If lines m and l are parallel, then these angles are congruent. The only case in which these equal angles will add up to 180º is if they are right angles.
∠7 and ∠6 are vertically opposite angles since they are opposite to each other when the two lines cross formin an X-shape. These angles are always equal.
The only fact that will allow you to prove that l and m are parallel is m∠1=m∠8
The sum of 5 and three times a number is 23
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We are asked to formulate an algebraic expression for the statement "The sum of 5 and three times a number is 23", first a "sum" is mentioned and it says that the result is 23, let's say that we are adding two numbers a and b and we get 23, like this:
a + b = 23
From the statement we know that one of the numbers that is being added is 5, then by using 5 instead of a, we get:
5 + b = 23
And we are told that the other number that is being added is being multipliend by 3, by replacing 3x (three times a number "x") for b we get:
5 + 3x = 23
Then, the answer is 5 + 3x = 23
find the exact values of the three trigonomic functions of the angle 0 (sin 0, cos 0 and tan 0) in the figure.Use the Pythagorean theorem to find the third side of the triangle
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In the given triangle we have, a = 7, b = 24
In right angle triangle we have, Perpendicular = 7 , base=24
Apply Pythagoras to find the Hypotenuse of the triangle:
Puthagoras Theorem:
In the right angle triangle, The sum of square of perpendicular and base is equal to the square of the hypotenuse.
Hypotenuse²= Base² + Perpendicular²
In the given figure we hvae to evaluate the hypotenuse
[tex]\begin{gathered} \text{Hypotenuse}^2=perpendicular^2+base^2 \\ \text{Hypotenuse}^2=7^2+24^2 \\ \text{Hypotenuse}^2=49+576 \\ \text{Hypotenuse}^2=625 \\ \text{Hypotenuse}=\text{ 25} \end{gathered}[/tex]Hypotenuse =25
All sides are : 25, 7, 24
The ratio of Perpendicular to the base is the tangent.
[tex]\begin{gathered} \text{Tan}\theta=\frac{Perpendicular}{Base} \\ \text{Tan}\theta=\frac{a}{b} \\ \text{Tan}\theta=\frac{7}{24} \\ \text{Tan}\theta=0.291 \\ \theta=\tan ^{-1}(0.291) \\ \theta=16.2^o \end{gathered}[/tex]So, we get
[tex]\begin{gathered} \text{For, Sin}\theta=\frac{Perpendicular}{Hypotenuse} \\ \text{ Sin}\theta=\frac{7}{25} \\ \\ \text{for Cos}\theta=\frac{Base}{Hypotenuse} \\ \text{Cos}\theta=\frac{24}{25} \end{gathered}[/tex]thus, the trignometric ratio of angle is :
[tex]\text{ Sin}\theta=\frac{7}{25},\text{ Cos}\theta=\frac{24}{25},\text{ Tan}\theta=\frac{7}{24}[/tex]ANswer:
A)
[tex]\text{ Sin}\theta=\frac{7}{25},\text{ Cos}\theta=\frac{24}{25},\text{ Tan}\theta=\frac{7}{24}[/tex]You randomly draw twice from this deck of cards(9) (6) (5) (5) (7) (7) (2)What is the probability of drawing a 7, then drawing a 2, replacing the first card? Write as fraction.
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Given:
Total number of cards : 7 cards
No. of 7 cards = 2
No. of 2 cards = 1
[tex]\text{ Probability = Probability of getting 7 x Probability of getting 2}[/tex][tex]\text{ = }\frac{\text{ 2}}{\text{ 7}}\text{ x }\frac{\text{ 1}}{\text{ 7}}\text{ = }\frac{\text{ 2}}{\text{ 49}}[/tex]Therefore, the probability of drawing 7, then drawing 2 is 2/49.
The answer is 2/49.
Complete the table with the missing values.Explain what it means to say that the pairs of numbers are equivalent ratios.
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ANSWER
EXPLANATION
To complete the table, we have to see what is the ratio between the given pairs,
[tex]\frac{b}{a}=\frac{15}{3}=\frac{50}{10}=\frac{30}{6}=5[/tex]This means that b is 5 times a, so the value of b when a = 1 is 5. And, the value of a when b = 80 is 16,
If the pairs of numbers are equivalent ratios, it means that the division between the numbers, for example, b/a, is constant - i.e. it always gives the same result. In this case, the ratio is always 5.
The sum of two numbers is -6 andtheir difference is 12. What is thevalue of the smallest of the twonumbers?
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Formulating a system of equations, we have:
x+y=-6 Equation 1
x-y=12 Equation 2
Solving the system of equations, we have:
y = -6 - x (Subtracting x from both sides of the Equation 1)
x - (-6 - x)=12 (Replacing y=-6-x in the Equation 2)
x + 6 + x = 12 (Using the sign rules)
2x + 6 = 12 (Adding like terms)
2x = 6 (Subtracting 6 from both sides of the Equation)
x=3 (Dividing on both sides by 2)
Replacing x=3 in y=-6-x , we have:
y = -6 - 3
y= -9
Given that the numbers are 3 and -9, the smallest of them is -9.
The answer is -9.
- Lucy's hair grew foot over the past year. She cut 1 foot off for summer. a. One of the fractions should be written as negative number. Which one is it and why? b. Find the sum of the two numbers to determine the total change in the length of Lucy's hair over the past year.
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Lucy's hair grew 1 foot over the past year, and she cut 1 foot off for summer. So, the negative fraction is the amount that she cut off, which is:
What information is missing to use SAS
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Given data:
The given figure.
For SAS two sides and included angle must be equal.
The missing angle is,
[tex]\angle ABD\cong\angle CBD[/tex]Thus, the third option is correct.
Please help it’s my homework. You have to graph the line.
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Thr equation of a line in Slope-Intercept form is:
[tex]y=mx+b[/tex]Where "m" is the slope and "b" is the y-intercept.
You have this equation:
[tex]2x+y=8[/tex]Solve for "y" in order to write it in Slope-Intercept form:
[tex]y=-2x+8[/tex]You can see that the y-intercept is:
[tex]b=8[/tex]Since "x" is zero when the line cuts the y-axis, you have this point:
[tex](0,8)[/tex]You know that "y" is zero when the line cuts the x-axis. So to find the x-intercept, substitute that value of "y" into the equation and solve for "x":
[tex]\begin{gathered} 0=-2x+8 \\ \frac{-8}{-2}=x \\ \\ x=4 \end{gathered}[/tex]So you also have this point:
[tex](4,0)[/tex]Knowing these points, you can plot them of the Coordinate plane and graph the line.
The graph of the line is:
Which x and y scales are most appropriate for the graph?
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As the graph increases 1 vertical unit per 4 horizontal units the scale in x (horizontal) needs to be -10 to 10.
As the y-intercept is 8 you don't need to have a negative part of y-axis to graph the line.
y-scale from 0 to 15.
Then, as you can see in the next graph you can use scales:
x from -10 to 10
y from 0 to 15
Use the drop down to fill in the blank for the sentence
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1 isosceles
2. legs (isosceles trapezoid has same size legs)
3. congruent (The opposite angles at the endpoints of the cross diagonal are congruent)
4. perpendicular (The diagonals are perpendicular.)
The cost of a dozen apricots is 360 cents. What is the cost of an apricots in cents, if c represents the cost of an apricot?
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To answer the problem we need to divide the total price by the amount of apricots, since the total price given is for a dozen this means that the price for one apricot is:
[tex]c=\frac{360}{12}[/tex]ind the area of the shaded region.12 cmSelect the correct response:shaded area = (72 – 97)cm?, ~ 43.7cm2shaded area = (72 – 4.57 )cm?, ~ 57.9cm2shaded area = (36 – 97 )cm?, ~ 7.7cm2shaded area = (72 – 187 )cm2, 15.5cm2
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Ok, so the diameter of the circles is 6 because is the half of the rectangle,
the radius is half the diameter so is 3
the height of the rectangle is equal to the diameter of the circle so is 6
the shaded area is the total area of the rectangle minus the area of the two circles
then
[tex]\begin{gathered} (12\times6-2(3^2\pi))cm^2 \\ (72-18\pi)cm^2\approx15.5^{}cm^2 \end{gathered}[/tex]Expand: (3a - 4b)^8. Write a brief explanation of your process for finding the expansion of this binomial.
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To expand the binomial
[tex](3a-4b)^8[/tex]We can use the binomial theorem.
According to the binomial theorem, our binomial can be rewritten as:
[tex](3a-4b)^8=\sum_{n\mathop{=}0}^8\begin{pmatrix}8 \\ n\end{pmatrix}(3a)^{8-n}(-4b)^n[/tex]Then, expanding this sum, we have:
[tex]\begin{gathered} (3a-4b)^8=\frac{8!}{0!(8-0)!}(3a)^8(-4b)^0+\frac{8!}{1!(8-1)!}(3a)^7(-4b)^1+... \\ +\frac{8!}{7!(8-7)!}(3a)^1(-4b)^7+\frac{8!}{8!(8-8)!}(3a)^0(-4b)^8 \\ \\ =6561a^8-69984a^7b-870912a^5b^3+1451520a^4b^4-1548288a^3b^5 \\ +1032192a^2b^6-393216ab^7+65536b^8 \end{gathered}[/tex]A small company is creating a new product to sell to buyers. They have estimated that it will costthem $8 to produce each item and they will have start-up costs of $242000. This leads to thefollowing expression, which gives the total cost, in dollars, to produce q of these new products:8q + 242000How much will cost them to produce 10500 items?
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Recall that to evaluate an expression at a given value, we substitute the variable by the given value.
Substituting q=10500 in the given expression we get:
[tex]8\cdot10500+242000.[/tex]Simplifying the above sum we get:
[tex]84000+242000=326000.[/tex]Answer: It will cost them 326000 dollars to produce 10500 items.
A person in the audience of a talk show has a 15% chance of winning free movie tickets. What is the chance that a person in the audience may not win free movie ticketsa.85 percentb.79 percent c. 35 percentd.30 percent
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Since it's mentioned that a person in the audience of a talk show has a 15% chance of winning free movie tickets, the rest of the percentage must be the chance that he may not win.
We get,
[tex]\text{Chance to win + Chance to lose = 100\%}[/tex][tex]\text{ 15\% + Chance to lose = 100\%}[/tex][tex]\text{ Chance to lose = 100\% - 15\%}[/tex][tex]\text{ Chance to lose = 85\%}[/tex]Therefore, the chance that the audience may not win is 85%.
A woman drives her car 260 miles in 4 hours. At this rate how far will she travel in 6 hours?She will travel ? miles.
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The speed formula is given by:
[tex]v=\frac{d}{t}[/tex]Where d is the distance traveled in t hours. From the problem, we know that:
[tex]v=\frac{260}{4}=65\text{ mi/h}[/tex]Using the speed formula, we can get the distance formula:
[tex]d=v\cdot t[/tex]Now, at this rate (v = 65 mi/h), the distance traveled in 6 hours is:
[tex]d=65\cdot6=390\text{ mi}[/tex]She will travel 390 miles in 6 hours.
The cost of a pound of nails increased from $2.18 to $2.37. What is the percent of increase to the nearest whole-number percent?
Answers
Answer:
9%
Explanation:
To calculate the percentage of the increase, use the formula below:
[tex]\%\text{ of change}=\frac{\text{Final Value-Initial Value}}{\text{Initial Value}}\times100[/tex]Substitute the given values:
[tex]\begin{gathered} \text{Percent of change=}\frac{2.37-2.18}{2.18}\times100 \\ =\frac{0.19}{2.18}\times100 \\ =8.72 \\ \approx9\% \end{gathered}[/tex]The percent of the increase is 9% (to the nearest whole number percent).
Patricia built 11 models avery 3 hours. At that rate, how long, in hours, will it take to build 5 models?
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start by calculating the rate
[tex]\frac{11\text{models}}{3\text{hours}}[/tex]write a relationship with the numer of model required
[tex]\begin{gathered} 11mod\Rightarrow3h \\ 5\mod \Rightarrow x \end{gathered}[/tex]solve for x the relationship
[tex]\begin{gathered} x=5\cdot\frac{3}{11} \\ x=\frac{15}{11}h \end{gathered}[/tex]it will take 15/11 hours to buld 5 models.
A coefficient of determination, r = 0.75, indicates that A: 75% of the data lie on the regression lineB: the data have a strong positive correlationC: the slope of the regression line is 0.75D: 75% of the variance in y is a result of the variance in x
Answers
Remember that the coefficient of determination is an indicative of how good of a fit a model is for a set of data.
-1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation and 0 means that there's no correlation at all.
Given a coefficient of determination, r = 0.75, we can conclude that there's a strong positive correlation.
Therefore, the answer is: Option B
Determine whether the given ordered pair is a solution to the system of equations. 6x−2y =36 −3x+3y =12 and (11,15)
Answers
Given the system of equations:
• 6x - 2y = 36
,• -3x + 3y = 12
The solution:
(x, y) ==> (11, 15)
Let's determine whether the given ordered pair is a solution to the system of equations.
Let's solve the system using the addition and elimination method.
Multiply both equations so the coefficients of one variable are opposite.
Multiply equation 2 by 2:
6x - 2y = 36
2(-3x + 3y) = 2(12)
• 6x - 2y = 36
,• -6x + 6y = 24
Add both equations:
6x - 2y = 36
-6x + 6y = 24
____________
0 + 4y = 60
We now have:
4y = 60
Divide both sides by 4:
[tex]\begin{gathered} \frac{4y}{4}=\frac{60}{4} \\ \\ y=15 \end{gathered}[/tex]Plug in 15 for y in either of the equations.
Take the first equation:
[tex]\begin{gathered} 6x-2y=36 \\ \\ 6x-2(15)=36 \\ \\ 6x-30=36 \end{gathered}[/tex]Add 30 to both sides:
[tex]\begin{gathered} 6x-30+30=36+30 \\ \\ 6x=66 \end{gathered}[/tex]Divide both sides by 6:
[tex]\begin{gathered} \frac{6x}{6}=\frac{66}{6} \\ \\ x=11 \end{gathered}[/tex]Therefore, the solution is:
x = 11, y = 15
In point form, the solution is: (11, 15)
Therefore, the given ordered pair is a solution to the system of equations.
ANSWER:
Yes, the ordered pair is a solution to the system of equations.
Solve the equation: 3(x + 4) = 2(x - 6) x = -24x = -28x = -68x = -72
Answers
hello
the equation given is
[tex]3(x+4)=2(x-6)[/tex]step 1
multiply the coeffiecients with the brackets and the proceed to simplfy
[tex]\begin{gathered} 3(x+4)=2(x-6) \\ 3x+12=2x-12 \end{gathered}[/tex]step 2
collect like terms
[tex]\begin{gathered} 3x-2x=-12-12 \\ x=-24 \end{gathered}[/tex]the value of x = -24
F R Wala Which animal ata Rachel's WH Find the equation of a line passing through (5, -6) parallel to: 2x + y = 12
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Two lines are parallel if they have the same slope
The slope of the line y = 2x + y = 12, is found isolating y, as follows:
2x + y = 12
y = 12 - 2x
this means that the slope is -2
The slope-intercept form of a line is:
y = mx + b
where m is the slope and b is the y-intercept
Replacing with m = -2 and the point (5, -6), we get:
-6 = (-2)(5) + b
-6 = -10 + b
-6 + 10 = b
4 = b
Therefore, the equation is:
y = -2x + 4
Given:line HG is the perpendicular bisector of segment jk. Which conclusion is valid
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Given that HG is the perpendicular bisector of JK, it means that
JG = KG
HG is common to triangles JGH and KGH
angle JGH and angle KGH are 90 degrees. They are congruent
Since 2 corresponding sides and an angle of triangles JGH and KGH are congruent, it means that the triangles are congruent. Thus,
HJ = HK
Option C is correct
Determine the level of measurement of the variable.Ranking (First place, 2nd place, etc) Of contestants In a singing competitionA) IntervalB).OrdinalC) nominalD) ratio
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Answer:
B). Ordinal
Explanation:
Last Thursday night, the nurse noted that there were 52 children in the ward. If 75% of them were girls, how many were boys?
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ANSWER
The number of boys in the ward are 13
EXPLANATION
Given that;
The total number of children in the ward is 52
Since 75% of the children are girls, then we can find the number of girls below
[tex]\begin{gathered} \text{ Number of girls in the ward }=\text{ 75\% of children in the ward} \\ \text{ Number of girls in the ward = }\frac{75}{100}\text{ }\times\text{ 52} \\ \text{ Number of girls in the ward }=\text{ 0.75 }\times\text{ 52} \\ \text{ Number of girls in the ward }=\text{ 39} \end{gathered}[/tex]Recall,
Number of children = number of boys + number of girls
we can calculate the number of boys using the relationship above
52 = number of boys + 39
Subtract 39 from both sides
52 - 39 = number of boys + 39 - 39
13 = number of boys
Number of boys in the ward is 13
Hence, the number of boys in the ward are 13