The dimension of pen is 42 feet by 36 feet
Solution:
Let "x" be the width of pen
Let "y" be the length
Farmer wants each side of the pen to be at least 20 feet long
[tex]x\geq 20[/tex]
The farmer has 120 feet of fence to enclose an area of 1512 square feet
The amount of fencing is equal to the perimeter of fence which is 2 times the width plus only one length since the other side (length) is along the barn
2x + y = 120
Therefore,
y = 120 - 2x
The area of barn is given as 1512 square feet
The area of rectangle is given as:
[tex]Area = length \times width[/tex]
[tex]1512 = x \times y\\\\1512 = x \times (120-2x)\\\\1512 = 120x - 2x^2\\\\2x^2-120x + 1512 = 0[/tex]
Divide the entire equation by 2
[tex]x^2-60x + 756 = 0[/tex]
Factor the left side of equation
[tex]x^2-60x+756 = 0\\\\(x-18)(x-42) = 0[/tex]
Therefore, we get two values of "x"
x = 18 or x = 42
Since, [tex]x\geq 20[/tex]
Therefore, x = 42 is the solution
Thus, width = x = 42 feet
Length = y = 120 - 2x
y = 120 - 2(42)
y = 120 - 84
y = 36
Thus the dimension of pen is 42 feet by 36 feet
The area of a shape is the amount of space it occupies.
The dimension of the pen is 42 by 36 feet.
The perimeter is given as:
[tex]\mathbf{P = 120}[/tex]
Because one of the sides does not need fencing, the perimeter would be:
[tex]\mathbf{P = 2x + y}[/tex]
Make y the subject
[tex]\mathbf{y = P - 2x}[/tex]
Substitute 160 for P
[tex]\mathbf{y = 120 - 2x}[/tex]
The area of a pen is:
[tex]\mathbf{A = xy}[/tex]
Substitute [tex]\mathbf{y = 120 - 2x}[/tex]
[tex]\mathbf{A = x(120 -2x)}[/tex]
Substitute 1512 for Area
[tex]\mathbf{x(120 -2x) = 1512}[/tex]
Open brackets
[tex]\mathbf{120x -2x^2 = 1512}[/tex]
Rewrite as:
[tex]\mathbf{2x^2 -120x + 1512 = 0}[/tex]
Divide through by 2
[tex]\mathbf{x^2 -60x + 756 = 0}[/tex]
Expand
[tex]\mathbf{x^2 -18x - 42x + 756 = 0}[/tex]
Factorize
[tex]\mathbf{(x -18)(x - 42) = 0}[/tex]
Solve for x
[tex]\mathbf{x =18 \ or\ x = 42}[/tex]
The dimension must be at least 20.
So, we have:
[tex]\mathbf{x = 42}[/tex]
Recall that:
[tex]\mathbf{y = 120 - 2x}[/tex]
This gives:
[tex]\mathbf{y = 120 - 2 \times 42}[/tex]
[tex]\mathbf{y = 36}[/tex]
Hence, the dimension of the pen is 42 by 36 feet.
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When randomly selecting an adult, A denotes the event of selecting someone with blue eyes. What do Upper P(Upper A)P(A) and Upper P(Upper A overbar)PA represent?
Answer:
When randomly selecting an adult, A denotes the event of selecting someone with blue eyes. What do P(A) and P([math] \overline A [/math]) represent?
Step-by-step explanation:
Solve for x.
x−11−−−−−√=5
A 14
B 16
C 25
D 36
The value of x in the expression will be 36.
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is;
⇒ √x - 11 = 5
Now,
Solve the expression for x as;
⇒ √x - 11 = 5
Squaring on both side;
⇒ (√x - 11)² = 5²
⇒ x - 11 = 25
Add 11 both side;
⇒ x - 11 + 11 = 25 + 11
⇒ x = 36
Thus, The value of x in the expression will be 36.
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The correct answer is option D. 36.
We need to solve the equation:
√(x - 11) = 5
To solve this, follow these steps:
Square both sides to eliminate the square root.
√(x - 11)2 = 52
x - 11 = 25
Isolate x by adding 11 to both sides:
x - 11 + 11 = 25 + 11
x = 36
So, the correct answer is 36 (option D).
Robert is Running in a race that is 3 miles long. A Distance marker is placed evenly every 1/2 mile from the beginning of the race until the end. What is the total number of distance markers on the trail?
Answer: the total number of distance markers on the trail is 6
Step-by-step explanation:
The total distance that Robert covers in the race is 3 miles.
If a distance marker is placed evenly every 1/2 mile from the beginning of the race until the end, then the total number of distance markers on the trail would be
3/0.5 = 6
Joey spent $.75 on a bus ticket. Then he spent 1/2 of his money on the soccer ball. After that, he spent 1/4 of his money on lunch. With the last of his money, Joey spent $14.95 for soccer tickets. In the end, Joey had 5 cents left. How much money did he start out with?
Joey started his day with $63.
Step-by-step explanation:
Let,
x be the amount Joey started with.
Amount left = 5 cent = $0.05
Amount spent on bus ticket = $0.75
Amount spent on soccer ball = [tex]\frac{1}{2}x[/tex]
Amount spent on lunch = [tex]\frac{1}{4}x[/tex]
Amount spent on soccer tickets = $14.95
According to given statement;
Total amount - bus ticket - soccer ball - lunch - soccer tickets = Amount left
[tex]x-0.75-\frac{1}{2}x-\frac{1}{4}x-14.95=0.05\\\\x-\frac{1}{2}x-\frac{1}{4}x=0.05+0.75+14.95\\\\\frac{4x-2x-x}{4}=15.75\\\\\frac{x}{4}=15.75[/tex]
Multiplying both sides by 4
[tex]4*\frac{x}{4}=15.75*4\\x=63[/tex]
Joey started his day with $63.
Keywords: variable, addition
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Jack decides to grow and sell bean plants. Let P represent the number of plants he will grow and sell. After considering his expenses, the expression -3p(p-10)-6p(p-10) represents his profit.
a. Rewrite and simplify the profit expression by factoring out the greatest common factor.
b. Rewrite the expression in simplest form with no parentheses.
Answer:
b. [tex]\displaystyle 90 - 9p^2[/tex]
a. [tex]\displaystyle -9[-10 + p^2][/tex]
Step-by-step explanation:
b. [tex]\displaystyle 90 - 9p^2 = -3p^2 + 30 - 6p^2 + 60 = -3p(p - 10) - 6p(p - 10)[/tex]
a. [tex]\displaystyle -9[-10 + p^2] = 90 - 9p^2[/tex]
I am joyous to assist you anytime.
Final answer:
The profit expression -3p(p-10)-6p(p-10) simplifies first by factoring out the greatest common factor to -9p(p-10), and then further simplifies to -9p² + 90p by distributing the -9p across the parentheses.
Explanation:
The student is tasked with simplifying the profit expression given by -3p(p-10)-6p(p-10), first by factoring out the greatest common factor, and then by writing the expression in the simplest form without parentheses.
To begin, we notice that both terms in the expression share a common factor of p(p-10), so we can factor this out:
-3p(p-10) - 6p(p-10) = -9p(p-10).
Upon factoring out the common factor, our expression simplifies to -9p(p-10). Next, we simplify this expression by removing the parentheses and multiplying through by the common factor:
-9p(p - 10) = -9p² + 90p.
Therefore, the simplified profit expression with no parentheses is -9p² + 90p.
need help asap
Solve the equation by completing the square
-2x^2+x=0
a. {0,-0.5}
b. {1,0}
c. {0.5,0}
d. {0}
Answer:
Option C) [tex]{\{0.5,0}\}[/tex] is correct
The solution to the given equation is [tex]{\{0.5,0}\}[/tex]
Step-by-step explanation:
Given quadratic equation is [tex]-2x^2+x=0[/tex]
To solve the equation by completing the square :
[tex]-2x^2+x=0[/tex]
[tex]-2(x^2-\frac{x}{2})=0[/tex]
[tex]x^2-\frac{x}{2}=0[/tex]
Rewritting the above equation as below :
[tex]x^2-2(x)\frac{1}{4}+\frac{1}{4^2}-\frac{1}{4^2}=0[/tex]
[tex](x-\frac{1}{4})^2-\frac{1}{4^2}=0[/tex]
[tex](x-\frac{1}{4})^2=\frac{1}{4^2}[/tex]
Taking square root on both sides we get
[tex]\sqrt{(x-\frac{1}{4})^2}=\sqrt{\frac{1}{4^2}}[/tex]
[tex](x-\frac{1}{4})=\pm\frac{1}{4}[/tex]
[tex]x=\pm\frac{1}{4}+\frac{1}{4}[/tex]
[tex]x=+\frac{1}{4}+\frac{1}{4}[/tex] and [tex]x=-\frac{1}{4}+\frac{1}{4}[/tex]
[tex]x=\frac{2}{4}[/tex] and [tex]x=0[/tex]
Therefore [tex]x=\frac{1}{2}[/tex] and x=0
[tex]x=0.5[/tex] and x=0
Therefore the solution to the given equation is [tex]{\{0.5,0}\}[/tex]
Option C) [tex]{\{0.5,0}\}[/tex] is correct
Simplify the following expressions by combining like terms.
10z-11z+2-5=
Answer:
-1z-3
Step-by-step explanation:
10z-11z = -1z
+2-5= -3
= -z-3
Answer:
-z - 3
Step-by-step explanation:
10z - 11z + 2 - 5
Simplifying:
10z - 11z = -z
+2 - 5 = -3
Theresfore; 10z - 11z +2 - 5 = -z - 3
A _______ variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure. A ▼ relative key probability random variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.
Answer:
random variable
Step-by-step explanation:
Random variable also known as stochastic variables are variables that are used in statistics and probability, The value of a random variable must be determined by chance which remain uncertain and depends on the outcome of a random process or experiment.
Unlike algebra, when random variable is introduced into a mathematical model, it is represented by capital letter and it has a single numerical value.
Complete the following statement. The number of iPhones sold today at an Apple store is an example of ________ data.
a. interval,
b. classified as discrete ratio,
c. classified as continuous interval,
d. classified as continuous ratio,
e. classified as discrete
Answer: Option 'e' is correct.
Step-by-step explanation:
Since we know that
Discrete data are those data which is countable and it is expressed in exact numerical form.
Continuous data are those data which is lie within the interval and it varies so it does not expressed in exact numerical form.
Here, the number of iphones sold today at an Apple store is countable and it can be expressed in exact numbers.
So, it is considered as discrete.
Hence, Option 'e' is correct.
Final answer:
The number of iPhones sold today at an Apple store represents an example of quantitative discrete data.
Explanation:
The number of iPhones sold today at an Apple store is an example of classified as discrete data. This is because the number of iPhones can only take on specific numerical values and cannot be a fraction or a decimal. In other words, you cannot sell half an iPhone, so the data is quantitative discrete.
I got 13.7 is that correct
Answer:
The answer to your question is 10.5 in
Step-by-step explanation:
Data
First triangle
height = x
base = 7 in
Second triangle
height = 12 in
base = 8 in
Process
1.- Use proportions to solve this problem, relates the sides of a triangle with the same sides of the other triangle.
[tex]\frac{x}{7} = \frac{12}{8}[/tex]
2.- Multiply both sides by 7
[tex]7\frac{x}{7} = \frac{7(12)}{8}[/tex]
3.- Simplify
x = [tex]\frac{84}{8}[/tex]
4.- Result
x = 10.5 in
Determine algebraically whether the function is even, odd, or neither even nor odd.
f(x) = -9 x^3 + 8x
Odd
Even
Neither
Answer: The function will be ODD.
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QuezoMartiinez
The function f(x) = -9x³ + 8x is odd.
Option A is the correct answer.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
A function f(x) is even if f(-x) = f(x).
A function is odd if f(-x) = -f(x).
Using the above definition.
f(x) = -9x³ + 8x
Put x = -x
f(-x) = -9 x (-x³) + 8 x (-x)
f(-x) = 9x³ - 8x
f(-x) = - ( -9x³ + 8x)
f(-x) = -f(x)
This means,
The function is odd.
Thus,
f(x) is odd.
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I need help with 2b.
The topic is rational equations and restrictions are 2 and 0.
As my answer I got 0 = 8 but I’m unsure if that’s correct...
Answer:
The answer to your question
a) Values excluded = -2, 0
b) solutions = -2/3, 2
Step-by-step explanation:
Equation
[tex]\frac{4x}{2x + 4} = \frac{x+ 2}{2x}[/tex]
a) x is restricted when the denominators equal zero, so let's get them.
2x + 4 = 0 2x = 0
2x = - 4 x = 0 / 2
x = -4 / 2 x = 0
x = -2
x must be different to -2, 0
b) [tex]\frac{4x}{2x + 4} = \frac{x+ 2}{2x}[/tex]
[tex]\frac{2x}{x + 2} = \frac{x + 2}{2x}[/tex]
4x² = (x + 2)²
4x² = x² + 4x + 4
4x² - x² - 4x - 4 = 0
3x² - 4x - 4 = 0
3x² - 6x + 2x - 4 = 0
3x(x - 2) + 2(x - 2) = 0
(x - 2)(3x + 2) = 0
x₁ - 2 = 0 3x₂ + 2 = 0
x₁ = 2 x₂ = -2/3
A parachute rate during a free fall reaches 70 meters per second. What is the rate in feet per second? At this rate how many feet will the parachute fall during 20 seconds of free fall
Answer:
229.659 ft/s4,593.18 ftStep-by-step explanation:
1 foot is 0.3048 meters, so ...
70 m/s = (70 m/s)×(1 ft)/(0.3048 m) ≈ 229.659 ft/s
At that speed, the distance covered in 20 seconds is ...
(20 s)(229.659 ft/s) = 4,593.18 ft
The parachutist will fall 4593 feet during 20 seconds of free fall at 229.7 ft per second.
_____
Final answers are rounded. Intermediate values are not.
How many different ways are there to select 24 donuts if there are 7 types of donuts available (and donuts are only distinguished by their type).?
Answer:
593,775 ways
Step-by-step explanation:
24 donuts have to be selected from 7 different varieties of donuts
n = 7
r= 24
Repetition is allowed
C(n+r-1, r) = C(7 + 24 - 1 , 24)
= C(30,24)
Recall that C(n,r) = n! /(n-r)! r!
C(30,24) = 30!/(30 - 24)! 24!
= 30!/(6!24!)
= 593,775 ways
Answer: 346,789 ways
Step-by-step explanation:
Using the formular for permutation to calculate :
P= n!/r!(n-r)!
We need to select 24 apples from 7 types of apples
n=24 ,r=7
Permutation =24!/7!(24-7)!
Permutation =6.2×10^23/1.793×10^18
P= 346 ,789 ways
A manufacturing company with 450 employees begins a new project line and must increase their number of employees by 18% how many total employees does the company have now
Answer: the company now have 531 employees.
Step-by-step explanation:
The initial number of employees that the company had was 450.
The company began a new project line and must increase their number of employees by 18% . it means that the number of new employees that the company employed would be
18/100 × 450 = 0.18 × 450 = 81
Therefore, the total number of employees that the company have now would be
450 + 81 = 531
Can someone please help me with my homework page 17, 19, and 20. Thank you!
Answer:
17.B
19. C
20.C
Step-by-step explanation:
17.
[tex]\frac{m^2n^3}{p^3} \div \frac{mp}{n^2}\\=\frac{m^2n^3}{p^3} \times \frac{n^2}{mp}\\=\frac{mn^5}{p^4}[/tex]
19.
[tex]w^{2} +w-20=w^2+5w-4w-20=w(w+5)-4(w+5)=(w+5)(w-4)\\(w-3)(w-4)=w(w-4)-3(w-4)=w^2-4w-3w+12=w^2-7w+12\\reqd.~fraction ~is~ \frac{w^2-7w+12}{w^2+w-20}[/tex]
20.
[tex]\frac{w^2+5w+6}{w^2-w-12} =\frac{w^2+3w+2w+6}{w^2-4w+3w-12} =\frac{w(w+3)+2(w+3)}{w(w-4)+3(w-4)} =\frac{(w+3)(w+2)}{(w-4)(w+3)} =\frac{w+2}{w-4}[/tex]
A 28ft ladder leans against a wall so that the base of the ladder is 5ft away from the base of the wall. How far up the wall does the ladder reach?
Round to the nearest tenth.
Answer:
the top of the ladder is 27.5ft above the ground.
Step-by-step explanation:
The ladder makes an angle, θ with the ground thus forming a right angle triangle with the wall of the house.
The length of the ladder represents the hypotenuse of the right angle triangle.
The ground distance from the base of the wall to the foot of the ladder represents the adjacent side of the right angle triangle.
Therefore, to determine how high up is the top of the ladder on the wall, x, we would apply Pythagoras theorem which is expressed as
Hypotenuse² = opposite side² + adjacent side²
28² = x² + 5²
784 = x² + 25
x² = 784 - 25 = 759
x = √759 = 27.5ft
Using the Pythagorean theorem, the ladder reaches approximately 27.5 feet up the wall when the base is 5 feet from the wall.
Explanation:The question involves finding the height up a wall that a 28ft ladder reaches when its base is 5ft from the wall. To solve this, we will use the Pythagorean theorem which relates the sides of a right-angled triangle. The ladder, wall, and ground form a right-angled triangle with the ladder as the hypotenuse and the wall and ground as the other two sides.
Let h be the height the ladder reaches up the wall. Using the Pythagorean theorem, we have:
28² = h² + 5²
Solving for h:
28² - 5² = h²
784 - 25 = h²
759 = h²
h ≈ √759
h ≈ 27.5 (to the nearest tenth)
Therefore, the ladder reaches approximately 27.5 feet up the wall.
A gardener is mowing a 20 by 40 yard rectangular pasture using a diagonal pattern. He mows from one corner of the pasture to the corner diagonally opposite. What is the length of this pass with the mower? Give your answer in simplified form. Recall from the Pythagorean Theorem that, for a right triangle, the square of the length of the diagonal is equal to the sum of the squares of the lengths of the sides.
A. 10 times square root 20
B. 20 times square root 2
C. 400 times square root 5
D. 20 times square root 5
20 * (sqrt) 5 is the correct answer or “D”
recently took this trust me 100% i’m just here to help anybody :)
Answer:
d
Step-by-step explanation:
using Pythagorean theorem you would do a^2 + b^2 = c^2 so 20^2 + 40^2 = 2000 do the square root of 2000 and you get 20* square-root of 5
The probability that an event will not happen is Upper P (Upper E prime) = 0.94. Find the probability that the event will happen.
Answer: P(Upper E' prime) = 0.06
Step-by-step explanation:
Not happen:P(Upper E prime) = 0.94
Happen: P(Upper E' prime) = ?
P(Upper E' prime) + P(Upper E' prime) = 1
Therefore
P(Upper E' prime) + 0.94= 1
P(Upper E' prime) =1- 0.94
P(Upper E' prime) = 0.06
identify the image of triangle XYZ for a composition of a 30° rotation and a 330° rotation both about point z
Answer:
As
30° + 330° = 360°
So, the figure has rotated one complete revolution as
Step-by-step explanation:
Considering the image of triangle XYZ for a composition of a
30° rotationand a
330° rotation both about point zAs
30° + 330° = 360°
It means, the figure has rotated one complete revolution. And now the figure is very much back to starting position.
Keywords: image, triangle, rotation, revolution
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Answer: The first image is the triangle XYZ and the second is the composition of 30° and 330° (360°) as a rotation about point z
Im in desperate need of help!
Answer:
umm, my regards, how should I help lol
Assume that the mean of a normal distribution of IQ scores is 102 and the standard deviation is 15. You have been told that your test score is 1 standard deviation above the mean. What is your IQ test score?
Answer: your IQ test score is 117
Step-by-step explanation:
Assume that the mean of a normal distribution of IQ scores is 102 and the standard deviation is 15.
If you have been told that your test score is 1 standard deviation above the mean, it means that your score would be
102 + 15 = 117
According to the empirical rule, 68% of data falls within the first standard deviation from the mean. This means that 117 falls within 68% of the scores.
Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth. x2 – 6 = x
A. x = 2, 3
B. x = –2, 3
C. x = 2, –3
D. x = –2, –3.
Answer:
3, -2
Step-by-step explanation:
Find the value of x.
logx 8 = 0.5
А. 4
В. 16
C.32
D. 64
Answer:
The answer to your question is letter D. x = 64
Step-by-step explanation:
Equation
log x 8 = 0.5
Express the log as a exponential equation
x[tex]x^{0.5}[/tex] = 8
Express the power as a fraction
[tex]x^{1/2} = 8[/tex]
[tex]\sqrt{x^{2}}[/tex]
Look for a number which squared root is 8
[tex]\sqrt{64} = 8[/tex]
Then
x = 64
Answer:
Please mark as brainliest :)
Step-by-step explanation:
If you invested $100 at an interest rate of 5% per year compounded quarterly, how much will the investment be worth at the end of two years? *
50 points
Answer: the investment would be
$110.45
Step-by-step explanation:
Initial amount deposited into the account is $100 This means that the principal is
P = 100
It was compounded quarterly. This means that it was compounded 4 times in a year. So
n = 4
The rate at which the principal was compounded is 5%. So
r = 5/100 = 0.05
It was compounded for 2 years. So
t = 2
The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of t years. Therefore
A = 100 (1+0.05/4)^4×2
A = 100 (1.0125)^8
A = $110.45
Consider triangle GHJ. Triangle G H J is shown. Angle G H J is a right angle. The length of the hypotenuse is 10 and the length of another side is 5. What is the length of line segment HJ? 5 units 5 StartRoot 3 EndRoot units 10 units 10 StartRoot 3 EndRoot units
Length of segment HJ is 5√3 units
Step-by-step explanation:
Here, apply the Pythagorean relationship where sum of squares of the sides equals the square of the hypotenuse.
Given that the hypotenuse, c= 10 units, and one of the side, a=5 units, then the other side can be calculated as;
a²+b²=c²
5²+b²=10²
b²=10²-5²
b²=100-25
b²=75
b=√75 = √3*25 = √3 *√25 = √3 *5 = 5√3
Length of segment HJ is 5√3 units
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Answer:
B
Step-by-step explanation:
The Hall family and the Nguyen family each used their sprinklers last summer. The water output rate for the Hall family's sprinkler was 35L per hour. The water output rate for the Nguyen family's sprinkler was 15L per hour. The families used their sprinklers for a combined total of 50 hours, resulting in a total water output of 1050L. How long was each sprinkler used?
Answer:
Nguyen family uses sprinkler for 35 hours while the Hall family uses sprinkler for 15 hours
Step-by-step explanation:
Let h and n be the number of hours that the Hall and Nguyen family individually use their sprinkler for, respectively. Since they use for a combined total of 50 hours, h + n = 50, or h = 50 - n
Also their total water output is 1050 L, we can have the following equation
35h + 15n = 1050
Substitute h = 50 - n and we have
35(50 - n) + 15n = 1050
1750 - 35n + 15n = 1050
35n - 15n = 1750 - 1050
20n = 700
n = 35 hours
h = 50 - n = 50 - 35 = 15 hours
The Rivera family went out to eat at a restaurant to celebrate a birthday. The bill for the meal, including tax, came to $126.50. They left a tip in the amount of 18% of the bill. How much did the Rivera family pay for the meal, including tip?
Answer:
$149.27
Step-by-step explanation:
18% of $126.50 is $22.77 which then added to 126.50=$149.27
Sam bought 2 granola bars and Haley Bought 5 granola bars each granola bar is the same price right in expression for the total price of the granola bars then simplify the expression to create an equivalent expression
Answer:
Step-by-step explanation:
figure out the answer by figuring out how much each bar is
Answer:
2b + 5b = b(2+5) = 7b
Step-by-step explanation:
Have a good day :)
Sam 2 granola bars
ADD
Haley 5 granola bars
Equal 7 granola bars
a. Show that the following statement forms are all logically equivalent. p → q ∨ r, p ∧ ∼q → r, and p ∧ ∼r → q b. Use the logical equivalences established in part (a) to rewrite the following sentence in two different ways. (Assume that n represents a fixed integer.) If n is prime, then n is odd or n is 2.
Final answer:
The statement forms p → q ∨ r, p ∧ ¬q → r, and p ∧ ¬r → q are shown to be logically equivalent using principles of logical equivalence and contraposition. The original statement 'If n is prime, then n is odd or n is 2' can thus be rephrased in different ways while retaining the same meaning.
Explanation:
Logical Equivalences
To show that the statement forms p → q ∨ r, p ∧ ¬q → r, and p ∧ ¬r → q are all logically equivalent, we need to understand that logical equivalence means that the statements have the same truth value in every possible scenario. The implication p → q is equivalent to p∨ q, as a truth table would confirm. Likewise, by using the principle of contraposition, which states that p → q is equivalent to ¬q → ¬p, we can transform the given implications accordingly.
Considering the sentence "If n is prime, then n is odd or n is 2" and denoting p as "n is prime", q as "n is odd", and r as "n is 2", the original statement can be represented as p → q ∨ r. Using logical equivalences established earlier, two different ways to rewrite this sentence could be:
The logical equivalences show that all these forms express the same condition from different perspectives but with the same meaning.