a) 9-12/2
b) 27-13/²2
a) Option a) 9 - 1/2 is equal to 17/2.
b) Option b) 27 - 2/3 is equal to 79/3.
a) The expression 9 - 1/2 can be simplified by finding a common denominator for the terms. The common denominator for 9 and 1/2 is 2.
Multiplying 9 by 2/2, we get:
9 * (2/2) = 18/2
So, the expression 9 - 1/2 can be simplified to:
18/2 - 1/2 = 17/2
Therefore, option a) 9 - 1/2 is equal to 17/2.
b) The expression 27 - 2/3 can be simplified in a similar manner by finding a common denominator for the terms. The common denominator for 27 and 2/3 is 3.
Multiplying 27 by 3/3, we get:
27 * (3/3) = 81/3
So, the expression 27 - 2/3 can be simplified to:
81/3 - 2/3 = 79/3
Therefore, option b) 27 - 2/3 is equal to 79/3.
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A math student has a plan to solve the following system by the elimination method. To eliminate the x-terms, he wants to multiply the top equation by 7. What should he multiply the second equation by so that when he adds the equations, the x-terms are eliminated? -3x-7y=-56 and -7x+10y=1
Answer:
-3
Step-by-step explanation:
You want the multiplier of the second equation that would result in eliminating the x-terms when the first equation is multiplied by 7 and added to the multiplied second equation.
-3x -7y = -56-7x +10y = 1MultiplierThe desired multiplier will have the effect of making the coefficient of x be zero when the multiplications and addition are carried out. If k is that multiplier, the resulting x-term will be ...
7(-3x) +k(-7x) = 0
-21x -7kx = 0 . . . . . . simplify
3 +k = 0 . . . . . . . . . divide by -7x
k = -3 . . . . . . . . . . subtract 3
The multiplier of the second equation should be -3.
__
Additional comments
Carrying out the suggested multiplication and addition, we have ...
7(-3x -7y) -3(-7x +10y) = 7(-56) -3(1)
-49y -30y = -395
y = -395/-79 = 5
The solution is (x, y) = (7, 5).
In general, the multipliers will be the reverse of the coefficients of the variable, with one of them negated. Here the coefficients of x are {-3, -7}. When these are reversed, you have {-7, -3}. When the first is negated, the multipliers of the two equations are {7, -3}. That is, the second equation should be multiplied by -3, as we found above.
Note that if you subtract the multiplied equations instead of adding, you can use the reversed coefficients without negating one of them. The choice of where the minus sign appears (multiplication or subtraction) will depend on your comfort level with minus signs.
The number of minus signs in this system can be reduced by multiplying the first equation by -1 to get 3x +7y = 56.
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The following is a list of shoe sizes for a group of 13 people.
4.5, 9.5, 8, 6.5, 10, 7, 8.5, 6, 7.5, 9, 6, 7, 11
Which of the following box plots best represents the numerical data?
A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
A box plot using a number line from 3 to 11.25 with tick marks every one-fourth unit. The box extends from 6.25 to 8.75 on the number line. A line in the box is at 7.25. The lines outside the box end at 4.5 and 10. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
A box plot using a number line from 3 to 13 with tick marks every one-half unit. The box extends from 6.5 to 9 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 12. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
A box plot using a number line from 3 to 12.5 with tick marks every one-fourth unit. The box extends from 6.25 to 8.75 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 10.5. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
The box plot that best represents the numerical data is: A. A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
How to complete the five number summary of a data set?In order to determine the five-number summary for the survey, we would arrange the data set in an ascending order:
4.5,6,6,6.5,7,7,7.5,8,8.5,9,9.5,10,11
Based on the information provided about the list of shoe sizes for a group of 13 people, we would use a graphical method (box plot) to determine the five-number summary for the given data set as follows:
Minimum (Min) = 4.5.
First quartile (Q₁) = 6.25.
Median (Med) = 7.5.
Third quartile (Q₃) = 9.25.
Maximum (Max) = 11.
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pls help i have finals tomorrow and i want to know how to answer this question
The values in the expression is as follows:
a = 2
b = 0
c = -1
How to solve an expression?The expression can be solve using the exponential law. Therefore,
g = 2³ × 3 × 7²
h = 2 × 3 × 7³
Therefore, let's solve the following:
g/h = 2ᵃ × 3ᵇ × 7ⁿ
Therefore,
g = 2³ × 3 × 7² = 2 × 2 × 2 × 3 × 7 × 7
h = 2 × 3 × 7 × 7 × 7
Hence,
g / h = 2 × 2 × 2 × 3 × 7 × 7 / 2 × 3 × 7 × 7 × 7
Hence,
g / h = 2 × 2 / 7
g . h = 2² × 3° × 7⁻¹
Hence,
a = 2
b = 0
c = -1
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Determine the limit in the following equation.
The limit of the expression lim (x² - √x⁴ + 3x²) as x approaches any value is indeterminate (∞ - ∞), except when x approaches zero, where the limit is 0.
How did we get the value?To find the limit of the expression lim (x² - √x⁴ + 3x²) as x approaches a certain value, we can simplify the expression and evaluate the limit.
First, let's simplify the expression:
lim (x² - √x⁴ + 3x²)
= lim (4x² - x² - √x⁴)
= lim (3x² - √x⁴)
Now, let's consider the behavior of the expression as x approaches a value.
As x approaches any finite value, the term 3x² will approach a finite value.
For the term √x⁴, as x approaches a finite value, the square root of x⁴ will approach the absolute value of x².
Therefore, the limit becomes:
lim (3x² - √x⁴) = lim (3x² - |x²|)
Next, let's consider the different cases as x approaches positive infinity, negative infinity, and zero.
1. As x approaches positive infinity, the term 3x² will tend to positive infinity, and |x²| will also tend to positive infinity. Thus, the expression becomes:
lim (3x² - |x²|) = lim (∞ - ∞)
In this case, the limit is indeterminate (∞ - ∞).
2. As x approaches negative infinity, the term 3x² will tend to positive infinity, and |x²| will also tend to positive infinity. Thus, the expression becomes:
lim (3x² - |x²|) = lim (∞ - ∞)
Again, in this case, the limit is indeterminate (∞ - ∞).
3. As x approaches zero, the term 3x² will tend to zero, and |x²| will also tend to zero. Thus, the expression becomes:
lim (3x² - |x²|) = lim (0 - 0) = 0
Therefore, the limit of the expression lim (x² - √x⁴ + 3x²) as x approaches any value is indeterminate (∞ - ∞), except when x approaches zero, where the limit is 0.
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5 [11 marks] Question 4 - Exponents 1. A factory makes 1 000 000 000 yo-yos. Write the number of yo-yos in exponential form. (1) 2. Mrs. Penn has a vegetable garden. There are 8 rows and 8 columns in each bed. Each row and column is 1cm wide. Work out the number of small squares, each 1cm by 1cm in the grid and write it in exponential form. (1) [2 marks]
1.) The exponential form for the number of yo-yos, 1,000,000,000, is [tex]10^9[/tex]. This represents 10 raised to the power of the number of zeros in the original number.
2.) For Mrs. Penn's vegetable garden, with 8 rows and 8 columns of 1cm squares, the total number of small squares is 64. This can be expressed as [tex]2^6[/tex] , where 2 is the base number and 6 represents the exponent obtained from repeated multiplication.
1.) To write the number of yo-yos in exponential form, we can use the base 10 since we have a decimal system. The given number of yo-yos is 1,000,000,000. We can express it as 10 raised to the power of a certain exponent that represents the number of zeros in the original number.
In this case, the number has 9 zeros, so we can write it as [tex]10^9[/tex]. The exponential form for the number of yo-yos is [tex]10^9.[/tex]
2.) In Mrs. Penn's vegetable garden, there are 8 rows and 8 columns in each bed. Each row and column is 1cm wide. We need to calculate the number of small squares, each measuring 1cm by 1cm, in the grid.
Since there are 8 rows and 8 columns, we can multiply these two numbers together to find the total number of small squares. 8 multiplied by 8 equals 64.
The exponential form represents repeated multiplication of the base number. In this case, the base number is 64 since we have 64 small squares. To write it in exponential form, we need to determine the exponent that represents the number of times 64 is multiplied by itself.
Since 64 is 2 raised to the power of [tex]6 (2^6)[/tex], we can express it as [tex](2^6)^1[/tex]. Simplifying this, we get [tex]2^(6*1)[/tex] which equals [tex]2^6.[/tex]
Therefore, the exponential form for the number of small squares in the grid is [tex]2^6.[/tex]
In summary, the number of yo-yos in exponential form is [tex]10^9[/tex], and the number of small squares in the grid is [tex]2^6.[/tex]
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GEOMETRY 80POINTS
ty
Answer:
37.98
Step-by-step explanation:
Divide. (4x^3− 12x + 11) ÷ (2x − 2)
1. The initial odometer reading of a cab is 369 km. It travelled for 2 hours and the final odometer reading showed 469 km. Find the approximate average speed of the cab.
The approximate average speed of the cab is 50 km/h.
To find the approximate average speed of the cab, we can use the formula:
Average Speed = Total Distance / Total Time
Given that the initial odometer reading is 369 km and the final reading is 469 km, the total distance covered by the cab is:
Total Distance = Final Odometer Reading - Initial Odometer Reading
Total Distance = 469 km - 369 km
Total Distance = 100 Km.
The cab traveled for 2 hours, so the total time is:
Total Time = 2 hours
Now, we can substitute the values into the average speed formula:
Average Speed = Total Distance / Total Time
Average Speed = 100 km / 2 hours
Average Speed = 50 km/h
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O Velocity stays the same...
Distance increases slower than time.
00
QUESTION 4
Why did you not compute a slope for the accelerated motion graph?
We didn't have enough data.
It graphed as a straight line.
The curve wasn't smooth enough.
A curve has many different slopes.
QUESTION 5
Which of the following is a vector?
For accelerated motion, changes in distance compare with equal changes in velocity as follows:
B. Distance increases faster than time.
A reason why a person will not compute a slope for the accelerated motion graph is that it graphed as a straight line.
Details about accelerated motionAccelerated motion is a form of motion in which the motion is not equal and the object moving does not complete the same distances in the same intervals of time.
The graph formed in the case of an accelerated motion is not a straight line. So, a reason why a person would not compute a slope for the accelerated motion is that it graphed as a straight line.
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Determine the perimeter of a soccer field with a length of 97 metres and a width of 69 metres
Answer: Therefore, the perimeter of the soccer field is 332 meters.
Step-by-step explanation:
To determine the perimeter of a soccer field with a length of 97 meters and a width of 69 meters, we can use the formula for the perimeter of a rectangle, which is given by:
Perimeter = 2 * (length + width)
Plugging in the values, we have:
Perimeter = 2 * (97 + 69)
Perimeter = 2 * 166
Perimeter = 332 meters
The commuting time (the average number of hours spent commuting each week) for students at a particular university is normally distributed with a mean of 63 mins and standard deviation of 9.61.
The probability that a randomly student at the university has a commuting time between 55 and 70 mins is about:
The probability that a randomly student at the university has a commuting time between 55 and 70 mins is about 0.56. Given information:The commuting time (the average number of hours spent commuting each week) for students at a particular university is normally distributed with a mean of 63 mins and standard deviation of 9.61.
Find: We are to determine the probability that a randomly student at the university has a commuting time between 55 and 70 mins.
Here,μ = 63 min σ = 9.61 min. We have to find the probability of a random student has commuting time between 55 and 70 min. That is P(55 ≤ X ≤ 70).First, we need to convert the given range to Standard Normal Distribution form.i.e., z-score for X = 55 and X = 70.Z-score formula:z = (X - μ) / σFor X = 55z = (55 - 63) / 9.61z = -0.83For X = 70z = (70 - 63) / 9.61z = 0.73. We need to find the probability of a random student has a z-score between -0.83 and 0.73.P(-0.83 < z < 0.73)
Using standard normal table or calculator, we can find the probability P(-0.83 < z < 0.73) = P(z < 0.73) - P(z < -0.83)= 0.7665 - 0.2033≈ 0.56
Thus, the probability that a randomly student at the university has a commuting time between 55 and 70 mins is about 0.56.
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Select the correct answer.
The number of hours that 20 people spent watching television per day, in relation to age, is graphed. This quadratic equation represents the model
for the set of data.
y = 0.004z²0.314z + 7.5
Based on the model, approximately how much time does an 18-year-old spend watching television each day?
O A.
OB.
O C.
O D.
3 hours
2 hours
7.5 hours
0.5 hour
Based on the quadratic function, an 18 year old would spend 3 hours watching television.
Using the quadratic function given :
y = 0.004z²-0.314z + 7.5The age is represented as the variable , 'z'
substitute z = 18 into the equation
y = (0.004*18²) - 0.314(18) + 7.5
y = 3.144
y = 3 hours approximately
Hence, an 18 year old spend approximately 18 hours watching television.
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The product of 3, and a number increased by -7, is -36
┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈
✦ The number is - 5
┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈
[tex]\begin{gathered} \; \sf{\color{pink}{Let \; the \; other \; number \; be \; (x)::}} \\ \end{gathered}[/tex]
Atq,,
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x + ( - 7) \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x - 7 \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x - 21 = - 36} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 36 + 21} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 15} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{x = \dfrac{\cancel{ - 15}}{\cancel{ \: 3}}} \qquad \bigg \lgroup \sf{Cancelling \: by \: 3} \bigg \rgroup \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered} \; \color{pink} :\dashrightarrow \underline{\color{pink}\boxed{\colorbox{black}{x = - 5}}} \: \pmb{\bigstar} \\ \\ \end{gathered}[/tex]
The answer is:
z = -5Work/explanation:
The product means we multiply two numbers.
Here, we multiply 3 and a number increased by -7; let that number be z.
So we have
[tex]\sf{3(z+(-7)}[/tex]
simplify:
[tex]\sf{3(z-7)}[/tex]
This equals -36
[tex]\sf{3(z-7)=-36}[/tex]
[tex]\hspace{300}\above2[/tex]
[tex]\frak{solving~for~z}[/tex]
Distribute
[tex]\sf{3z-21=-36}[/tex]
Add 21 on each side
[tex]\sf{3z=-36+21}[/tex]
[tex]\sf{3z=-15}[/tex]
Divide each side by 3
[tex]\boxed{\boxed{\sf{z=-5}}}[/tex]
GEOMETRY 50POINTS
FIND x
Combining the results of a given triangle, we can conclude that the value of 'x' must be greater than -22 and also less than 52. So, the possible range for 'x' is -22 < x < 52.
To find the value of 'x' in a triangle with side lengths 'x', 37, and 15, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
In this case, we have:
x + 37 > 15 (Sum of x and 37 is greater than 15)
x + 15 > 37 (Sum of x and 15 is greater than 37)
37 + 15 > x (Sum of 37 and 15 is greater than x)
From the first inequality, we can subtract 37 from both sides:
x > 15 - 37
x > -22
From the second inequality, we can subtract 15 from both sides:
x > 37 - 15
x > 22
From the third inequality, we can subtract 15 from both sides:
52 > x
Combining the results, we can conclude that the value of 'x' must be greater than -22 and also less than 52. So, the possible range for 'x' is -22 < x < 52.
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A bank deposit paying simple interest grew from an initial amount of $1300 to $1365 in 3 months. Find the interest rate.
%/year
Answer:
the interest rate. : 5%
[(1365-1300)/1300]*100 = 5%
Step-by-step explanation:
Determine which postulate or theorem can be used to prove that
ALMN=ANOL.
M
A. SAS
B. ASA
C. AAS
D. SSS
The answer is A. SAS.
If a parallelogram is given and we want to prove that ALMN is congruent to ANOL, we can use the SAS (Side-Angle-Side) postulate.
The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
In the case of the parallelogram, we know that AL is congruent to NO (opposite sides of a parallelogram are congruent) and LM is congruent to OL (opposite sides of a parallelogram are congruent). Additionally, angle LAM is congruent to angle LAO (opposite angles of a parallelogram are congruent).
By using the SAS postulate, we have two sides and the included angle that are congruent in both triangles. Therefore, we can conclude that triangle ALMN is congruent to triangle ANOL.
So, the answer is A. SAS.
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8. Given the figure at right, which of the following is a
true statement?
a. sin(0) = ²/
b. tan(N) =
C. cos(0) =
d. cos(N) =
12
6√5
6
6√5
6√5
6√5
Answer:
Step-by-step explanation:
a. sin(0) = 0
The sine of 0 degrees is 0.
b. tan(N) = 6√5
We don't have enough information to determine the value of tan(N) without knowing the specific value of N.
c. cos(0) = 1
The cosine of 0 degrees is 1.
d. cos(N) = 6√5/6
Again, we can't determine the specific value of cos(N) without knowing the value of N.
A small college has 204 student athletes. The number of students who play soccer is 52. The number of students who play volleyball is 31. The probability that a student plays in both volleyball and soccer is 5/204.What is the probability that a randomly selected student athlete in this school: Plays both soccer and volleyball? Plays volleyball?
To calculate the probabilities, we can use the following information:
Total number of student athletes = 204
Number of students who play soccer = 52
Number of students who play volleyball = 31
Probability of a student playing both soccer and volleyball = 5/204
1. Probability that a student plays both soccer and volleyball:
Let's denote the probability of playing both soccer and volleyball as P(Soccer and Volleyball). From the given information, we know that the number of students who play both soccer and volleyball is 5.
P(Soccer and Volleyball) = Number of students who play both soccer and volleyball / Total number of student athletes
P(Soccer and Volleyball) = 5 / 204
2. Probability that a student plays volleyball:
We want to find the probability of a student playing volleyball, denoted as P(Volleyball).
P(Volleyball) = Number of students who play volleyball / Total number of student athletes
P(Volleyball) = 31 / 204
Therefore, the probability that a randomly selected student athlete in this school plays both soccer and volleyball is 5/204, and the probability that they play volleyball is 31/204.
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This year, the number of raffle tickets sold for a school's extracurricular activities fundraiser is 848. It is estimated that the number of raffle tickets sold will increase by 5% each year. Find the total number of raffle tickets sold at the end of 9 years.
Select the correct answer below:
9,158
9,351
9,818
10,666
This year, the number of raffle tickets sold for a school's extracurricular activities fundraiser is 848. It is estimated that the number of raffle tickets sold will increase by 5% each year.
The total number of raffle tickets sold at the end of 9 years is approximately 9,818.
To find the total number of raffle tickets sold at the end of 9 years, we need to calculate the number of tickets sold each year and sum them up.
Starting with the initial number of tickets sold, which is 848, we will increase this number by 5% each year for a total of 9 years.
Year 1: 848 + (5% of 848) = 848 + 42.4 = 890.4
Year 2: 890.4 + (5% of 890.4) = 890.4 + 44.52 = 934.92
Year 3: 934.92 + (5% of 934.92) = 934.92 + 46.746 = 981.666
Year 9: Ticket sales at the end of 9 years = Number of tickets sold in Year 8 + (5% of Year 8 sales)
Year 9: Total = 1,399.585 + 69.97925 = 1,469.56425 ≈ 1,469.56
The total number of raffle tickets sold at the end of 9 years is approximately 1,469.56.
The correct option is 9,818.
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Charmaine is buying a new car. Her bank offers her a loan of $20,000 with a 6.25% annual interest rate compounded quarterly, or every 3 months. Which of the following equations could model the bank’s offer? Select all that apply.
Answer:
[tex]{A = 20000(1 + \frac{0.0625}{4})^{4t}}[/tex]
Step-by-step explanation:
The question asks us to find an expression for compound interest for the given scenario.
To do this, we have to use the following formula for compound interest:
[tex]\boxed{A = P(1 + \frac{r}{n})^{nt}}[/tex]
where:
• A ⇒ final amount
• P ⇒ principal amount = $20,000
• r ⇒ interest rate (decimal) = [tex]\frac{6.25}{100}[/tex] = 0.0625
• n ⇒ number of times interest is compounded per year = 4
• t ⇒ time in years
Therefore, if we substitute the data above into the formula, we can find the required expression:
[tex]{A = 20000(1 + \frac{0.0625}{4})^{4t}}[/tex]
answer following question
Answer:
Option (C), 8 am
Step-by-step explanation:
Newton's Law of Cooling is a mathematical model that describes the cooling process of an object. It states that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature.
The equation representing Newton's Law of Cooling is:
[tex]\dfrac{dT}{dt} = -k (T_0 - T_A)[/tex]
Where...
"dT/dt" is the rate of change of temperature with respect to time."k" is the cooling constant."T_0" is the temperature of the object."T_a" is the surrounding temperature.After solving the differential equation we get the following function:
[tex]T(t)=T_A+(T_0-T_A)e^{-kt}[/tex]
[tex]\hrulefill[/tex]
Given:
[tex]T_0=98.6 \ \textdegree F \ \text{(This is the average human body temperature)}\\\\T_f=T(t)=80\ \textdegree F \\\\T_A=40 \ \textdegree F \\\\k=0.1947[/tex]
Find:
[tex]T(??)= \ 80 \ \textdegree F[/tex]
Substituting the values into the formula:
[tex]T(t)=T_A+(T_0-T_A)e^{-kt}\\\\\\\Longrightarrow 80=40+(98.6-40)e^{-0.1947t}\\\\\\\Longrightarrow 80=40+58.6e^{-0.1947t}\\\\\\\Longrightarrow 40=58.6e^{-0.1947t}\\\\\\\Longrightarrow 0.682594=e^{-0.1947t}\\\\\\\Longrightarrow \ln(0.682594)=-0.1947t\\\\\\\Longrightarrow t=\dfrac{\ln(0.682594)}{-0.1947} \\\\\\\therefore \boxed{t \approx 2 \ \text{hours}}[/tex]
Thus, we can conclude the time of death was at 8 am.
6 a) Complete the table of values for y=x 0.5 1 2 3 X y 6 3 4 5 1.2 6
Answer:
Step-by-step explanation:
x=0.5, y=12.
x=3, y=2.
x=4, y=1.5.
x=6, y=1.
At what point(s) A through E is the rate of change of f(x) equal to zero?
The points where the rate of change of f(x) equal to zero are A, C and E
How to determine the point where the rates is 0From the question, we have the following parameters that can be used in our computation:
The graph
The point where the rates is 0 are the points where movement is at a constant
using the above as a guide, we have the following:
The points are A, C and E
Hence, the point where the rates is 0 are A, C and E
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The product of two irrational numbers is
rational. (Sometimes,Never,always)?
The product of two irrational numbers can be either rational or irrational, depending on the specific irrational numbers being multiplied. It is not always rational, nor is it never rational.
The product of two irrational numbers can be either rational or irrational, depending on the specific irrational numbers being multiplied. It is not always rational, nor is it never rational.
Consider the square root of 2 (√2) and the square root of 3 (√3), both of which are irrational numbers. When you multiply √2 and √3, you get √6, which is also an irrational number. In this case, the product of two irrational numbers is irrational.
However, there are cases where the product of two irrational numbers can be rational. For example, consider √2 and its reciprocal (1/√2), both of which are irrational. When you multiply these two numbers, you get 1, which is a rational number. So, in this case, the product of two irrational numbers is rational.
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Trent has an 8-foot tall tent in the shape of square based pyramid with a base length of 14 feet. If one bottle of waterproof spray covers 76 square feet, how many bottles will he need to waterproof his tent.
Trent will need approximately 2.86 bottles of waterproof spray to cover his tent.
To calculate the number of bottles of waterproof spray Trent will need to cover his tent, we first need to find the surface area of the tent.
The surface area of a square-based pyramid is given by the formula:
Surface Area = Base Area + (0.5 x Perimeter of Base x Slant Height)
The base of the pyramid is a square with a side length of 14 feet, so the base area is:
Base Area = (Side Length)^2 = 14^2 = 196 square feet
To find the slant height of the pyramid, we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by one side of the base, the height of the pyramid, and the slant height. The height of the pyramid is given as 8 feet, and half the length of the base is 7 feet.
Using the Pythagorean theorem:
[tex]Slant Height^2 = (Half Base Length)^2 + Height^2[/tex]
[tex]Slant Height^2 = 7^2 + 8^2Slant Height^2 = 49 + 64Slant Height^2 = 113Slant Height ≈ √113 ≈ 10.63 feet[/tex]
Now we can calculate the surface area of the tent:
Surface Area = 196 + (0.5 x 4 x 10.63)
Surface Area = 196 + (2 x 10.63)
Surface Area = 196 + 21.26
Surface Area ≈ 217.26 square feet
Since each bottle of waterproof spray covers 76 square feet, we can divide the total surface area of the tent by the coverage of each bottle to find the number of bottles needed:
Number of Bottles = Surface Area / Coverage per Bottle
Number of Bottles = 217.26 / 76
Number of Bottles ≈ 2.86
Therefore, Trent will need approximately 2.86 bottles of waterproof spray to cover his tent. Since we can't have a fraction of a bottle, he will need to round up to the nearest whole number. Therefore, Trent will need 3 bottles of waterproof spray to fully waterproof his tent.
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What is the five-number summary for the data set? 73, 62, 90, 28, 45, 90
Answer:
it's easy
Step-by-step explanation:
first take a deep breath and then search it
What is the difference between relational understanding and Instructional understanding in mathematics?
a. Find the slope of x^3+y^3-65xy=0 at the points (4,16) and (16,4).
b. At what point other than the origin does the curve have a horizontal tangent line?
c. Find the coordinates of the point other than the origin where the curve has a vertical tangent line.
a. The slope of the curve at the point (4,16) is approximately 1.165, and at the point (16,4) is approximately -0.496.
b. The curve has a horizontal tangent line at the points(0,0) and (3,27).
c. The curve has a vertical tangent lineat the points (0,0) and (65/2, (65/2)³).
How is this so?a. To find the slope of the curve given by the equation x³ + y³ - 65xy = 0 at the points (4,16) and (16,4),we can differentiate the equation implicitly with respect to x and solve for dy/dx.
Differentiating the equation with respect to x, we have -
3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0
To find the slope at a specific point, substitute the x and y coordinates into the equation and solve for dy/dx.
For the point (4,16) -
3(4)² + 3(16)²(dy/dx) - 65(16) - 65(4)(dy/dx) = 0
48 + 768(dy/dx) - 1040 - 260(dy/dx) = 0
508(dy/dx) = 592
(dy/dx) = 592/508
(dy/dx) ≈ 1.165
For the point (16,4) -
3(16)² + 3(4)²(dy/dx) - 65(4) - 65(16)(dy/dx) = 0
768 + 48(dy/dx) - 260 - 1040(dy/dx) = 0
(-992)(dy/dx) = 492
(dy/dx) = 492/(-992)
(dy/dx) ≈ -0.496
Thus, the slope of the curve at the point (4,16) isapproximately 1.165, and at the point (16,4) is approximately -0.496.
b. To find the point where the curve has a horizontal tangent line, we need to find the x-coordinate(s)where dy/dx equals zero.
This means the slope is zero and the tangent line is horizontal.
From the derivative we obtained earlier -
3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0
Setting dy/dx equal to zero -
3x² - 65y = 0
Substituting y = x³/65 into the equation -
3x² - 65(x³/65) = 0
3x² - x³ = 0
Factoring out an x² -
x²(3 - x) = 0
This equation has two solutions - x = 0 and x = 3.
hence, the curve has a horizontal tangent line at the points(0,0) and (3,27).
c. To find the point where the curve has a vertical tangent line, we need to find the x-coordinate(s) where the derivative is undefinedor approaches infinity.
From the derivative -
3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0
To find the vertical tangent line, dy/dx should be undefined or infinite. This occurs when the denominator of dy/dx is zero.
Setting the denominator equal to zero: -
65x = 65y
x = y
Substituting this condition back into the original equation -
x³ + x³ - 65x² = 0
2x³ - 65x² = 0
x²(2x - 65) = 0
This equation has two solutions - x = 0 and x = 65/2.
Therefore, the curve has a vertical tangent line at the points (0,0)
and(65/2, (65/2)³).
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On a piece of paper, graph y<-3/4x+2. Then determine which answer choice
matches the graph you drew.
The graph of the linear inequality is given by the image presented at the end of the answer.
How to define a linear function?The slope-intercept equation for a linear function is presented as follows:
y = mx + b
In which:
m is the slope.b is the intercept.The function for this problem is given as follows:
y = -3x/4 + 2.
Hence the graph crosses the y-axis at y = 2, and when x increases by 4, y decays by 3.
The inequality is given as follows:
y < -3x/4 + 2.
Meaning that points below the dashed line are the solution to the inequality.
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