PLEASE HELP!!! The volume in cubic feet of a box can be expressed as , or as the product of three linear factors with integer coefficients. The width of the box is x-2.
Factor the polynomial to find linear expressions for the height and the length. Show your work.
Answers
x(x^2-6x+8)
x(x^2-2x-4x+8)
x(x(x-2)-4(x-2))
x(x-2)(x-4)
Since we are told that the width is (x-2) the length and height are:
x and (x-4)
To find linear 12/10/2023 for the height and length of the box, factor the given polynomial (x - 2)(...)
Explanation:To find linear expressions for the height and length of the box, we need to factor the given polynomial. The volume of the box can be expressed as (x - 2)(?)(?), where ? represents the linear expressions for height and length. Since we have two unknowns, we need to find two factors that when multiplied together give us the given polynomial.
To factor x^2 - 2x, we need to find two numbers that multiply to give -2 and add up to give -2. The factors are -1 and 2, so we can write the polynomial as (x - 1)(x - 2). Therefore, the linear expressions for the height and length are (x - 1) and (x - 2) respectively.
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Related Questions
The sum of four consecutive even integer numbers is 84. find the four numbers.
Answers
84/4=21
now take the 2 even numbers below 21 and the 2 even numbers above 21
18 +20 + 22 +24 = 84
the numbers are 18, 20, 22 & 24
What is the number 212three in base-two form?
Answers
The correct answer is: B. 10111
To convert the number [tex]\(212_{\text{three}}\)[/tex] to base-two (binary) form, we need to first convert it to base-ten and then convert the base-ten number to base-two.
[tex]\(212_{\text{three}}\)[/tex] in base-ten:
[tex]\[ 212_{\text{three}} = 2 \cdot 3^2 + 1 \cdot 3^1 + 2 \cdot 3^0 = 18 + 3 + 2 = 23_{10} \][/tex]
Now, we convert [tex]\(23_{10}\)[/tex] to base-two:
[tex]\[ 23_{10} = 16 + 4 + 2 + 1 = 2^4 + 2^2 + 2^1 + 2^0 = 10111_{2} \][/tex]
Final answer:
The number 212 in base three (212three) converts to 17 in decimal, which is represented as 10001 in binary notation (base-two).
Explanation:
The number 212 in base three (2123) needs to be converted into base-two form (binary). Converting from base three to base two requires understanding each digit's place value in base three and then converting that to binary. Starting from the rightmost digit to the left, we have:
The units place (30), with a value of 2The threes place (31), also with a value of 2The nines place (32), with a value of 1So, we have:
2123 = (1 × 32) + (2 × 31) + (2 × 30)
2123 = (1 × 9) + (2 × 3) + (2 × 1)
2123 = 9 + 6 + 2 = 17 in decimal.
Now we need to find the binary representation of 17:
17 divided by 2 is 8 with remainder 1 (20)8 divided by 2 is 4 with remainder 0 (21)4 divided by 2 is 2 with remainder 0 (22)2 divided by 2 is 1 with remainder 0 (23)1 divided by 2 is 0 with remainder 1 (24)So, the binary equivalent is 1(24)0(23)0(22)0(21)1(20), which is 100012.
The length of a rectangular lawn is measured to twice its width. the perimeter of the lawn is given as 30 m. find the length and width of the lawn.
Answers
Prove that if one solution for a quadratic equation of the form x 2 + bx + c = 0 is rational (where b and c are rational), then the other solution is also rational. (use the fact that if the solutions of the equation are r and s, then x 2 + bx + c = (x − r)(x − s).)
Answers
Say r is rational. Suppose for a second, that s is not. Then, r+s is irrational. But this contradicts the fact that b is rational.
So, if one root is rational, then the other root is also rational
Final answer:
If one root of a quadratic equation with rational coefficients is rational, the other root must be rational too because the sum and product of the roots are related to the coefficients, which are also rational.
Explanation:
To prove that if one solution for a quadratic equation of the form x^2 + bx + c = 0 is rational, then the other solution is also rational, we can use the quadratic formula and properties of rational numbers. If the quadratic equation has rational coefficients and one rational solution, then the sum and product of the roots must also be rational. This is because a quadratic equation with roots r and s can be factored as (x - r)(x - s) = 0, which expands to x^2 - (r + s)x + rs = 0. Matching coefficients, we see that - (r + s) = b and rs = c. Since b and c are rational, r + s and rs must be rational as well.
Given that we have one rational root, let's say r, the sum of the roots r + s is rational, so s must also be rational because the difference of two rational numbers is rational. Hence, if one root of a quadratic equation with rational coefficients is rational, the other root must be rational as well.
Solve x3 = 64 over 27.
±8 over 3
8 over 3
±4 over 3
Answers
Cube root the answer
³√x³=³√64/27
x=4/3.
Check:
x=4/3
x³=(4/3)³
x³=64/27. As a result, the correct answer is 4/3 or 4 over 3. If you wonder why don't I choose the answer as ±4 over 3 because of this
±4/3
Choose positive first
(4/3)²=64/27 Correct. However, then choose negative signs
(-4/3)³=-64/27 Incorrect. Hope it help!
The solution of the given equation is ±8/3.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true.
The given equation is x³= 64/27
x=±∛(64/27)
x=±∛(8³/3³)
x=±8/3
Therefore, the solution of the given equation is ±8/3.
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If n is a positive integer and the product all the integeres from 1 to n inclusive is a multiple of 990 what is the least possible value of n
Answers
Since 990=2*3²*5*11 ==>n=11
The following two-way table shows the number of students of a school who have a video game and/or have a laptop:
Have Video Game Do Not Have Video Game Total
Have Laptop
15
50
65
Do Not Have Laptop
50
15
65
Total
65
65
130
Based on the table, how many students have both a laptop and a video game?
15
50
65
130
Answers
have laptop 15 50 65
dont have 50 15 65
total 65 65 130
have both a laptop and a game : 15
Based on the table given, the number of students who have both a laptop and a video game is 15 students.
Which students have both laptops and video games?In order to solve this question, look at the part of the table where the row on students who have laptops intersects with the column on those who have video games.
That part of the table has 15 in the cell. This means that 15 students have both video games and laptops.
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Which of the following equations represents a line that is parallel to the line below?
A.
y=1/3x-2
B.
y =-1/3x-2
C.
y=3x-2
y=-3x-2
Answers
so slope = (2-1) / (0-3) = 1/-3 = -1/3
parallel has same slope = -1/3
b = 2
equation
y = -1/3x + 2
answer
equation is y = -1/3x +2. I don't see any of those above matched
What is the probability that when a fair coin is flipped 25 times, there will be exactly five heads
Answers
A sum of money amounting to $4.25 consists of dimes and quarters if there are 26 coins in all how many are quarters
Answers
d= dimes
q = quarters
d+q=26 coins
rewrite as d=26-q
0.25q +0.10d=4.25
0.25q+0.10(26-q)=4.25
0.25q+2.6-0.10q=4.25
0.15q=1.65
q=11
d=26-11=15
11*0.25 = 2.75, 15*0.10=1.50, 2.75+1.50=4.25
there are 11 quarters
The law of cosines is a^2+b^2-2abcos(C). Find the value of 2abcos(C).
A. 37
B. -40
C. 40
D. 20
Answers
and surely you know how much that is.
3. Elizabeth opened a library with 19,000 books in the year 1998. The number of books increases at a rate of 6.49% each year. Use a graph to predict the number of books in 2020.
A) ≈ 71,160
B) ≈ 75,779
C) ≈ 80,697
D) ≈ 66,824
Answers
V (t)=V0 (1+r)^t
V (t) number of books in 2020 ?
V0 books in the year 1998 19000
R rate of growth 0.0649
T time 2,020−1,998=22 years
V (22)=19,000×(1+0.0649)^(22)
V (22)=75,778.81 Round your answer to get 75779
Answer: ≈ 75,779
Step-by-step explanation:
PLEASE HELP!!!!!!!
f(x)=x^2−3x+9
g(x)=3x^3+2x^2−4x−9
Find (f−g)(x)
Select one:
a. −3x^3−x^2+x+18
b. 3x^3+3x^2−7x
c. 3x^3+x^2−x−18
d. 3x^3−x^2−x
Answers
-----------------------------
Work Shown:
(f-g)(x) = f(x) - g(x)
(f-g)(x) = [ f(x) ] - [ g(x) ]
(f-g)(x) = [ x^2-3x+9 ] - [ 3x^3+2x^2-4x-9 ]
(f-g)(x) = x^2-3x+9 -3x^3-2x^2+4x+9
(f-g)(x) = -3x^3+(x^2-2x^2)+(-3x+4x)+(9+9)
(f-g)(x) = -3x^3-x^2+x+18
Bicycle city makes custom bicycles. They charge $160 plus $80 for each day that it takes to build the bicycle. If you have $480 to spend on your new bicycle, how many days can it take Bicycle City to build the bike?
Answers
80d≤320 divide both sides by 80
d≤4
So the maximum number of days it can take them to build the bicycle so that $480 will cover the cost.
43% of adults say cashews are their favorite kind of nut. you randomly select 12 adults and ask each to name his or her favorite nut. find the probability that the number who say cashews are their favorite nut is (a) exactly three, (b) at least four, and (c) at most two. if convenient, use technology to find the probabilities.
Answers
This question involves the concept of binomial probability in statistics. The scenarios involve calculating the probability of a certain number of successes (cashew preference) among a set number of trials (12 adults). It's recommended to use a calculator or software tool to perform the calculations.
Explanation:The subject of the query is known as binomial probability which is part of statistics in mathematics. Here, we need to find the probability of 'success' (in this case, the preference of cashews) exactly a set number of times when conducting a set number of trials (12 adults).
(a) Exactly 3: In this scenario, you want exactly 3 out of 12 adults to prefer cashews. Using binomial probability formula, it would be [tex]12C3\times (0.43)^3 \times (0.57)^9.[/tex](b) At least 4: Here, you want 4 or more adults to prefer cashews. You can either calculate separate probabilities for exactly 4, 5, 6, and so on up to 12, and then add them together. Or, you can use the complement rule: 1 - (P(0) + P(1) + P(2) + P(3)).(c) At most 2: In this case, you want 2 or fewer adults to prefer cashews. Similar to (b), you take the probability of 0, 1, and 2 'successes' and add them together.
It's also important to note, the results in these examples would be more accurate if you use a calculator or software like Excel to handle the computations.
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The probability of exactly 3 cashews in a random sample of 12 adults is approximately 19.4%.
The probability of at least 4 cashews is approximately 15.0%.
The probability of at most 2 cashews is approximately 72.0%.
These probabilities were calculated using the binomial probability formula and considering the complementary event for cases with "at least" or "at most" conditions. Technology like calculators or statistical software can be helpful in performing these calculations efficiently.
Determining the Probability of Nut Preference in a Random Sample
In this scenario, we're analyzing the probability of specific outcomes when randomly selecting 12 adults and inquiring about their favorite nut, knowing that 43% prefer cashews. Here's a breakdown of the requested probabilities:
(a) Exactly Three Cashew Preferences:
Imagine having 12 slots to fill with "cashew" or "other." We need 3 "cashew" slots and 9 "other" slots. Using the binomial probability formula, the probability of this specific arrangement is:
P(3 cashews in 12 trials) = (12 choose 3) * (0.43)^3 * (0.57)^9 ≈ 0.194
(b) At Least Four Cashew Preferences:
This includes scenarios with 4, 5, 6, 7, 8, 9, 10, 11, or all 12 adults preferring cashews. We can calculate the probability for each case and sum them up, but a simpler approach is to find the probability of the opposite event (fewer than 4 cashews) and subtract it from 1:
P(at least 4 cashews) = 1 - P(fewer than 4 cashews)
P(at least 4 cashews) ≈ 1 - (0.194 + 0.237 + 0.184 + 0.115 + 0.063 + 0.034 + 0.015 + 0.006 + 0.002) ≈ 0.150
(c) At Most Two Cashew Preferences:
This includes scenarios with 0, 1, or 2 adults preferring cashews. Similar to (b), we can calculate the probability for each case and sum them up:
P(at most 2 cashews) = P(0 cashews) + P(1 cashew) + P(2 cashews)
P(at most 2 cashews) ≈ 0.237 + 0.299 + 0.184 ≈ 0.720
Is #7 correct? Please explain.
Answers
What are the phase shift and period for the function y = 3cos[4(θ + 60°)] − 2?
Phase shift = right 60°, period = 90°
Phase shift = right 60°, period = −90°
Phase shift = left 60°, period = 90°
Phase shift = left 60°, period = −90°
Answers
How do I simplify this problem?
Answers
(15x^2h+15xh^2+5h^3)/h= 15x^2+15xh+h^2
Adam has $450. he spends $210 on food. later he divides all the money into four parts out of which three parts were distributed and one part he keeps for himself. then he found $50 on the road. write the final expression and find the money he has left?
Answers
First, subtract $210 from it (spent by Adam on food)
$450 - $210 = $240
Then, divide the remaining amount of money by 4
$240 / 4 = $60
Since, it was only one part that he kept to himself, the amount of money left with him, x, is
x = $60
Then, the total amount of money he has including that which he only picked from the road is,
T = $60 + $50 = $110
In the end, Adam had $110.
PLEASE HELP ON THESE ILL GIVE 20 POINTS AND A BRAINLIEST IF YOUR CORRECT!!
Variation is a term that is used to describe __________.
A.
how repetitive a data set is
B.
how large or small a data set is
C.
how spread out or scattered a data set is
D.
how different a data set is from other data sets
Answers
Find the arc length of a central angle of pi/4 in a circle whose radius is 8 inches
Answers
C=Θ/360πd
where:
Θ=45
d=8*2=16 in
hence;
C=45/360*π*16
=6.3 inches
Last year, there were n pies baked for the bake sale. This year, there were 156 pies baked. Using n, write an expression for the total number of pies baked in the two years
Answers
Number of pies this year: 156
n + 156 = total number of pies baked in 2 years.
A trapezoid has two right angles and bases that measure 16m and 8m. The right triangle formed by an altitude has a hypotenuse of 4 square root 5m. Sketch the trapezoid. What are its perimeter and area?
Answers
length of the rectangle = 8 m
base of triangle = 16 - 8 = 8 m
hypotenuse = 4√5 m
width of the rectangle = √(4√5 - 8) = 4 m
Perimeter = 8 + 16 + 4√5 + 4 = 28 + 4√5 m
Area = area of triangle + area of rectangle
Area = 8(4) / 2 + 8(4) = 48 m^2
Final answer:
The trapezoid forms a right-angled triangle with one additional rectangle. Its area is found to be 48m^2, and its approximate perimeter is 36.944m, by adding the lengths of all its sides together.
Explanation:
To find the perimeter and area of a trapezoid with two right angles and bases of 16m and 8m, we must first visualize the trapezoid. This trapezoid appears like a right-angled triangle with an additional rectangle attached to its hypotenuse.
We are given the hypotenuse of the altitude's right triangle is 4√5m, thanks to Pythagoras' theorem, we can find the two legs (which are the altitude h and the difference in bases). Let's call the altitude h and the difference in bases 'd'. Now, we know that the length of the longer leg of the right triangle is 16m - 8m = 8m.
Using the Pythagorean theorem where hypotenuse2 = altitude2 + difference in bases2, we have (4√5)2 = h2 + 82. Solving for 'h', we have h = √(80 - 64) = √16 = 4m. The area of a trapezoid is given by the formula A = (1/2) × (sum of the bases) × (height), which in this case is A = (1/2) × (16m + 8m) × 4m = 48m2.
For the perimeter, it can be calculated by adding the lengths of all sides. So, perimeter = 16m + 8m + 4m + 4√5m = 28m + 4√5m. To find the approximate value of 4√5m, we can calculate 4 × 2.236 (since √5 = 2.236), which gives us approximately 8.944m. Adding this to 28m gives us a perimeter of approximately 36.944m.
cot^2x-csc^2x=-1 for all values of x true or falsse
Answers
sin²x + cos²x =1
1 + tan²x = sec² x
1 + cot² x = csc² x
These are all derived from circle geometry on the cartesian plane. Now, the useful trigonometric property to be used is the third one. Rearranging this, we come up with
cot²x - csc²x = -1
This coincided with the given equation. Therefore, this is true. This is because it is already established from the pythagorean theorems.
The original equation cot^2x - csc^2x = -1 is true for all values of x.
How to determine if cot^2x-csc^2x=-1 for all values of xThe equation cot^2x - csc^2x = -1 is true for certain values of x, but not for all values of x.
To see why, let's break down the equation using trigonometric identities:
cot^2x - csc^2x = -1
Using the reciprocal identities, we can rewrite cot^2x and csc^2x in terms of sine and cosine:
(cos^2x / sin^2x) - (1 / sin^2x) = -1
Now, let's simplify:
(cos^2x - 1) / sin^2x = -1
Using the Pythagorean identity cos^2x + sin^2x = 1, we can substitute cos^2x with (1 - sin^2x):
(1 - sin^2x - 1) / sin^2x = -1
-sin^2x / sin^2x = -1
Now, we can cancel out the sin^2x terms:
-1 = -1
This equation holds true for all values of x. Therefore, the original equation cot^2x - csc^2x = -1 is true for all values of x.
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A cube is packed with decorative pebbles. If the cube has a side length of 6 inches, and each pebble weighs on average 0.5 lb per cubic inch, what is the total weight of the pebbles in the cube?
Answers
V = l³ , where l = 6 in
V = 6³ = 216 in³
Each pebble weights on average is 0.5 lb per cubic inch.
Weight = 216 in³ · 0.5 lb/in = 108 lb
Answer:
The total weight of the pebbles in the cube is 108 lb.
The tip of an 11-inch wiper blade wipes a path that is 31 inches long. What is the angle of rotation of the blade in radians to the nearest tenth?
Answers
31 = C*11
C = 31/11 = 2.8 radians to nearest tenth
2.8 radians is the answer.
Find the value of n in the equation 6.2n – 3.7n = 85 + 45. A. 16 B. 52 C. 13.13 D. 325
Answers
2.5n=130
N=130/2.5
N=52
The answer is B. Hope this helps!
Find two positive numbers such that their product is 192 and their sum is a minimum pre-calc
Answers
xy = 192
Let's rearrange this to make the equation explicit: y = 192/x
Then, the other equation is for the sum. Let S be the sum of the two positive numbers.
S = x + y
Now, substitute the other equation into this one:
S = x + 192/x
Here's where we apply calculus. We can determine the minimum S by getting the first derivative of S with respect to y and equating to zero. Thus,
dS/dx = 1 - 192x⁻² = 0
1 = 192x⁻²
x² = 192
x = +/- √192
But since we are finding the positive number, x = + √192
Then, we use this to the first equation:
y = 192/√192 = √192
Therefore, the two positive numbers are equal which is √192 or 13.86.
The two numbers with a product equal to 192 such that their sum is minimized are:
A = √192 and B = √192
How to find the two numbers?
Let's define A and B as our two numbers, we know that their product must be equal to 192, then we have:
A*B = 192.
Now, the sum of these two numbers is:
A + B.
And we want to minimize this, to do it, we need to use the first equation to rewrite one variable in terms of the other. For example, if we isolate A, we get:
A = 192/B
Replacing this in the sum, we get:
192/B + B
To minimize this, we need to find the values of B that make 0 the differentiation of the above expression.
The differentiation is:
-192/B^2 + 1
Then we need to solve:
-192/B^2 + 1 = 0
192/B^2 = 1
192 = B^2
√192 = B
To get the value of A, we use:
A = 192/B = 192/√192 = √192
Then we can conclude that the two positive numbers such that their product is 192, and their sum is minimized, is:
A = √192 and B = √192.
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If the polynomial x5 − 105 can be split as the product of the polynomials
x − 10 and a, what is a?
Answers
The mathematical expression that can be used in order to properly express the given scenario is,
x⁵ - 10⁵ = (x - 10)(a)
The value of a, as stated above, can be calculated by the equation,
a = (x⁵ -10⁵) / (x - 10)
There are two ways to solve the problem: synthetic division and the long way. Each of which will yield the same answer. Simplifying the value of a will give us a final answer of,
a = x⁴ + 10x³ + 100x² + 1000x
the value of a is
[tex]\( a = \frac{9845}{x - 10} \)[/tex].
To find a, we can use polynomial long division or synthetic division to divide [tex]\( x^5 - 105 \)[/tex] by [tex]\( x - 10 \)[/tex]. The remainder should be zero if [tex]\( x - 10 \)[/tex] is a factor of [tex]\( x^5 - 105 \)[/tex].
Let's perform polynomial long division:
____________________________
x - 10 | x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 105
- (x^5 - 10x^4)
____________________________
10x^4 + 0x^3 + 0x^2 + 0x - 105
- (10x^4 - 100x^3)
____________________________
100x^3 + 0x^2 + 0x - 105
- (100x^3 - 1000x^2)
___________________________
1000x^2 + 0x - 105
- (1000x^2 - 10000x)
___________________________
995x - 105
- (995x - 9950)
___________________________
9845
```
Since the remainder is a constant term, it's clear that [tex]\( x - 10 \)[/tex] is a factor of [tex]\( x^5 - 105 \)[/tex]. So, [tex]\( a = \frac{9845}{x - 10} \)[/tex].
Therefore, [tex]\( a = \frac{9845}{x - 10} \)[/tex].
The midpoint of a segment is (4,3) and one endpoint is (10,8) Find the coordinates of the other endpoint.
Answers
Answer:
[tex](-2,-2)[/tex].
Step-by-step explanation:
Let us assume that coordinates of other endpoint are [tex](x_1,y_1)[/tex]
We have been given that the midpoint of a segment is (4,3) and one endpoint is (10,8). We are asked to find the coordinates of the other endpoint.
We will use midpoint formula to solve our given problem.
[tex]x\text{-coordinate of midpoint}=\frac{x_1+x_2}{2}[/tex]
[tex]y\text{-coordinate of midpoint}=\frac{y_1+y_2}{2}[/tex]
Upon using our given information, we will get:
[tex]4=\frac{x_1+10}{2}[/tex]
[tex]4\cdot 2=\frac{x_1+10}{2}\cdot 2[/tex]
[tex]8=x_1+10[/tex]
[tex]8-10=x_1+10-10[/tex]
[tex]x_1=-2[/tex]
Similarly, we will find y-coordinate.
[tex]3=\frac{y_1+8}{2}[/tex]
[tex]3\cdot 2=\frac{y_1+8}{2}\cdot 2[/tex]
[tex]6=y_1+8[/tex]
[tex]6-8=y_1+8-8[/tex]
[tex]y_1=-2[/tex]
Therefore, the coordinates of other endpoint would be [tex](-2,-2)[/tex].