Answer:Therefore, f(-x) = -x^3 - 3x^2 - 2x + 2.
Step-by-step explanation:o find f(-x), we replace every instance of x in the function f(x) with -x:
f(-x) = (-x)^3 - 3(-x)^2 + 2(-x) + 2
Simplifying, we get:
f(-x) = -x^3 - 3x^2 - 2x + 2
14. The distance between (x, 2) and (0, 6) is 5 units. Use the Distance Formula to determine
the value of x. Show all your work.
Answer:
The value of x is 3.
Step-by-step explanation:
The Distance Formula is:
d = √[(x2 - x1)² + (y2 - y1)²]
Using the given points, we have:
d = √[(0 - x)² + (6 - 2)²]
Simplifying inside the square root gives:
d = √[x² + 16]
We know that the distance between (x, 2) and (0, 6) is 5 units, so we can set up the equation:
5 = √[x² + 16]
Squaring both sides gives:
25 = x² + 16
Subtracting 16 from both sides gives:
9 = x²
Taking the square root of both sides gives:
x = ±3
Since we are looking for the value of x, we choose the positive solution:
x = 3
Therefore, the value of x is 3.
Factor by grouping. Then supply the term that is missing below. 4mn+3m+8n+6=(m+2)(?+3)
Answer:
4mn + 3m + 8n + 6 = (m + 2)(4n + 3)
The missing term is 4n.
Water is leaking out of an inverted conical tank at a rate of 11500.0 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 9.0 m and the the diameter at the top is 4.5 m. If the water level is rising at a rate of 21.0 cm/min when the height of the water is 2.0 m, find the rate at which water is being pumped into the tank in cubic centimeters per minute.
water is being pumped into the tank at a rate of 23.7 cm³/min.we can take the derivative of the volume formula with respect to time and solve it
what is derivative ?
In mathematics, a derivative is a measure of how much a function changes as its input value changes. More specifically, the derivative of a function is defined as the rate of change of the function at a particular point.
In the given question,
We can solve this problem using related rates, where we relate the rate of change of one variable to the rate of change of another variable.
Let's denote the height of the water in the tank by h, and the radius of the water surface by r. Then, we can use the formula for the volume of a cone to relate these variables:
V = (1/3)πr²h
We are given that the tank has height 9.0 m and diameter (and thus radius) 4.5 m at the top. We can use this information to find the relationship between h and r:
r = (1/2) × diameter = 2.25 m
h = 9.0 m - 2.0 m = 7.0 m
Now, we can take the derivative of the volume formula with respect to time:
dV/dt = (1/3)π(2r(dr/dt) + r²(dh/dt))
We want to find the rate at which water is being pumped into the tank, which is the rate of change of the volume with respect to time when the water level is rising at a rate of 21.0 cm/min. We are also given that water is leaking out of the tank at a rate of 11500.0 cm^3/min, so we can set these rates of change equal to each other:
dV/dt = (21.0 cm/min) × (1 m/100 cm)³ = 0.021 m³/min
11500.0 cm³/min = 11.5 × (1 m/100 cm)³ m³/min
Substituting these values and the values we found for r and h into the derivative formula, we get:
0.021 m³/min = (1/3)π(2(2.25 m)(dr/dt) + (2.25 m)²(11500.0 cm³/min)/(100 cm/m)³)
Simplifying and solving for dr/dt, we get:
dr/dt = [(0.021 m³/min) - (1/3)π(2(2.25 m)(11500.0 cm³/min)/(100 cm/m)³)] ÷ [(1/3)π(2(2.25 m))]
= 0.0237 m/min = 23.7 cm/min
Therefore, water is being pumped into the tank at a rate of 23.7 cm³/min.
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The standard deviation of test scores on a certain achievement test is 11.4. A random sample of 50 scores on this test had a mean of 75.3. Based on this
sample, find a 95% confidence interval for the true mean of all scores. Then give its lower limit and upper limit.
Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)
Lower limit:
Upper limit:
X
Ś
The lower limit is 72.1 and the upper limit is 78.5.
The sample mean is 75.3, and the standard error is 1.613.
What is standard deviation?The standard deviation is a measurement of how differently distributed a set of data values are from their mean or average. Finding the square root of the variance, which is the sum of the squared deviations between each data point and the mean, is how it is determined.
To find the confidence interval, we can use the formula:
Confidence interval = sample mean ± (critical value) x (standard error)
where the standard error is the standard deviation of the sample mean and the critical value is based on the desired confidence level and the sample size.
Since we want a 95% confidence interval and the sample size is 50, we can find the critical value from a t-distribution with 49 degrees of freedom (n-1). Using a t-distribution instead of a normal distribution is appropriate here because the population standard deviation is not known.
From a t-distribution table or calculator, the critical value for a 95% confidence interval with 49 degrees of freedom is approximately 2.009.
The standard error can be found using the formula:
standard error = standard deviation / sqrt(sample size)
Substituting the given values, we get:
standard error = 11.4 / sqrt(50) = 1.613
Now we can plug in the values into the formula for the confidence interval:
Confidence interval = 75.3 ± 2.009 x 1.613 = (72.14, 78.46)
Therefore, the lower limit is 72.1 and the upper limit is 78.5. The sample mean is 75.3, denoted by X-bar and the standard error is 1.613, denoted by Ś.
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Write the standard form of the following polynomial.
(2x-3)²-3(x-2)(x+4)-7
X² ? X+ ?
Please help!!!
Natalie gathered a random sample of flower bouquets at the florist. She calculated data on different variables. For one data that she collected, she constructed a bar graph
Which of the following variables did she use?
O Type of flowers in each bouquet
O Amount of water needed for each bouquet
O Number of flowers in each bouquet
O Price for each bouquet
The variables that Natalie used in the data collected was C. Number of flowers in each bouquet.
How to find the variable ?The bar graph that Natalie crafted utilized the variable "number of flowers in each bouquet." This particular type of graph reveals the frequency or count associated with every number of flowers contained within a single bouquet.
As an illustration, let us suppose there were 20 bouquets holding 5 flowers, 30 bouquets possessing 10 flowers, and 15 bouquets that had 15 flowers each; consequently, the bar chart would exhibit exactly three bars symbolizing the count for all three categories as outlined above.
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In IXL I need help im bad :(
Answer:
Wilma will pay $23.44 total.
Step-by-step explanation:
Cost of sweater = (+) $24.49
Discount = (-) $5
Shipping cost = (+) $3.95
Solve:
24.49 - 5 = 19.49
19.49 + 3.95 = 23.44
Wilma will pay $23.44 total.
what is the value for 43.7 x 0.25
Answer:
43.7 x 0.25 is 10.925
Step-by-step explanation:
To multiply decimals, we can use the following steps:
1. Multiply the numbers as if they were whole numbers, ignoring the decimal points.
2. Count the total number of decimal places in the factors being multiplied. This will be the number of decimal places in the product.
3. Place the decimal point in the product so that it has the same number of decimal places as the total counted in step 2.
Using these steps, we can find the product of 43.7 and 0.25 as follows:
1. 437 x 25 = 10925
2. There are two decimal places in 43.7 and two decimal places in 0.25, so there are a total of four decimal places in the factors being multiplied.
3. Place the decimal point in the product so that it has four decimal places: 10.925
Therefore, the value of 43.7 x 0.25 is 10.925.
Assertion: 7V5, V2+21 are the irrational number Reason: every integer is a rational number
Assertion: Yes, √2 is an irrational number.
Reason: The decimal expansion of √2 is 1.41421356237 which is a non-recurring and non-terminating number.
How to solveBoth (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A).
Assertion: Yes, √2 is an irrational number.
Reason: The decimal expansion of √2 is 1.41421356237 which is a non-recurring and non-terminating number.
All real numbers that are not rational numbers are referred to as irrational numbers in mathematics. In other words, it is impossible to describe an irrational number as the ratio of two integers.
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The Complete Question
Assertion (A): V2 is an irrational number. Reason (R) : Decimal expansion of an irrational number is non-recurring and non terminating???? a) Both (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A) b) Both (A) and ( R) are true but Reason (R) is not a correct explanation of (A) c) Assertion (A) is true and Reason (R) is false d) Assertion (A) is false and Reason (R) is true
A rainwater collection system uses a cylindrical storage tank with a diameter of 50 cm and a height of 80 cm what is the total volume of water in cubic centimeters that can be collected
Answer:
157,000
Step-by-step explanation:
π r² h
The volume of the following square pyramid is
48^3. What is the length of 'l'? Round your answer to the nearest hundredth.
The length of 'l' is approximately equal to 331.13 units
Volume refers to the amount of space occupied by a three-dimensional object. In other words, it is the measure of how much space an object takes up.
The formula to calculate the volume of a square pyramid is:
Volume = (1/3) x Base Area x Height
where the base area is the area of the square base, and the height is the perpendicular distance from the apex to the base.
Let's substitute the given values in the formula and solve for the height:
48³ = (1/3) x l² x h
Multiplying both sides by 3, we get:
3 x 48³ = l² x h
Dividing both sides by l², we get:
h = (3 x 48³) / l²
Now, we can substitute the value of h in the original formula to get:
48³ = (1/3) x l² x [(3 x 48³) / l²]
Simplifying this equation, we get:
48³ = 48³
This equation is true, which means our solution is correct. Therefore, the length of 'l' is equal to the square root of (3 x 48³), which is approximately equal to 331.13 units (rounded to the nearest hundredth).
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The scatter plot shows the amount of money, y, Sally was paid in each of her jobs versus the number of hours x she worked in each job.
A scatter plot with number of hours worked on the X axis and amount of money paid on the Y axis. The data points are: two, 40, and, three, 70, and, four, 110, and, five, 120, and, six, 150, and, seven, 140, and, eight, 180, and, nine, 190.
Part A
Which equation best models a line of fit for the data?
A. y = 24x + 10
B. y = 21x + 11
C. y = −18x + 15
D. y = 20x
Part B
The slope of the line of fit means that for every additional hour that Sally works, she earns an additional
dollars, on average.
Using the given data, the equation that best models a line of fit for the data is y = 20x. The slope of the line of fit indicates that for each additional hour that Sally works, she earns an additional $20, on average. So, the correct option is D).
For finding the equation of the line of best fit and the slope
Firstly, Find the mean of x and y.
Mean of x = (2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 8 = 5.5
Mean of y = (40 + 70 + 110 + 120 + 150 + 140 + 180 + 190) / 8 = 127.5
Calculate the deviations from the mean for x and y.
Deviation from mean of x = x - mean of x
= [2 - 5.5, 3 - 5.5, 4 - 5.5, 5 - 5.5, 6 - 5.5, 7 - 5.5, 8 - 5.5, 9 - 5.5]
= [-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5]
Deviation from mean of y = y - mean of y
= [40 - 127.5, 70 - 127.5, 110 - 127.5, 120 - 127.5, 150 - 127.5, 140 - 127.5, 180 - 127.5, 190 - 127.5]
= [-87.5, -57.5, -17.5, -7.5, 22.5, 12.5, 52.5, 62.5]
Calculate the product of the deviations for each data point.
Product of deviations = deviation from mean of x * deviation from mean of y
[(2-5.5) * (40-127.5)] + [(3-5.5) * (70-127.5)] + [(4-5.5) * (110-127.5)] + [(5-5.5) * (120-127.5)] + [(6-5.5) * (150-127.5)] + [(7-5.5) * (140-127.5)] + [(8-5.5) * (180-127.5)] + [(9-5.5) * (190-127.5)]
= [-135.625, -51.875, 32.625, 28.125, 17.8125, 26.25, 137.8125, 229.0625]
Calculate the sum of the deviations of x squared.
Sum of deviations of x squared = (deviation from mean of x)²
= (-3.5)² + (-2.5)² + (-1.5)² + (-0.5)² + 0.5² + 1.5² + 2.5² + 3.5²
= 70
Calculate the slope of the line of best fit.
slope = sum of products of deviations / sum of deviations of x squared
= (−135.625 + (−51.875) + 32.625 + 28.125 + 17.8125 + 26.25 + 137.8125 + 229.0625) / 70
= 20
Calculate the y-intercept of the line of best fit.
y-intercept = mean of y - slope * mean of x
= 127.5 - 20 * 5.5
= 10
The equation of the line of best fit in slope-intercept form.
y = slope * x + y-intercept
= 20x + 10
Therefore, the equation that best models a line of fit for the data is D) y = 20x, and the slope is 20. So, the correct answer is D).
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If the dealership you are buying the car from is offering 6.3% interest on a 5 year loan, what will your monthly payment be? How am I able to solve this when the selling price is $56,937.
Answer:
Step-by-step explanation:
To calculate the monthly payment on a loan, you can use the formula for the monthly payment on an annuity: P = (r * PV) / (1 - (1 + r)^(-n)), where P is the monthly payment, r is the monthly interest rate, PV is the present value of the loan (i.e., the amount borrowed), and n is the total number of monthly payments.
In this case, the amount borrowed is $56,937 and the interest rate is 6.3% per year. To convert this to a monthly interest rate, we divide by 12: r = 0.063 / 12 = 0.00525. The loan term is 5 years, so the total number of monthly payments is n = 5 * 12 = 60.
Substituting these values into our formula for the monthly payment, we get: P = (0.00525 * 56937) / (1 - (1 + 0.00525)^(-60)) ≈ $1102.53. So, the monthly payment on this loan would be approximately $1102.53.
Raman is watching the movie 4 times and he fislokw s the value wirh
Answer:learn how to spell :)
Step-by-step explanation:
Find the equation of the line that passes through the point (3,−5) and is perpendicular to the line y=1/5x-2
Solve x²-3x+ 5 = 0.
A.
3+√-29
+VE
2
and
3-
2
-29
B. 3+√29 and 3-√29
O c. 3+√-11 and 3-√-11
2
2
D. 3+√11 and 3-√11
Using the quadratic formula to solve the equation x²-3x+ 5 = 0, the resultant answer is (D) 3+√11/2 and 3-√11/2.
What is the quadratic formula?The quadratic formula in elementary algebra is a formula that yields the answer to a quadratic problem.
In addition to the quadratic formula, other methods of solving quadratic equations include factoring, completing the square, graphing, and others.
A second-order equation of the form ax² + bx + c = 0 denotes a quadratic equation, where a, b, and c are real number coefficients and a 0.
So, we have the equation:
x²-3x+ 5 = 0
Now, solve it using the quadratic formula as follows:
x²-3x+ 5 = 0
a = 1
b = -3
c = 5
x = -(-3)±√(-3)² -4*1*5/2
Solve this further:
x = 3±√11/2
Therefore, using the quadratic formula to solve the equation x²-3x+ 5 = 0, the resultant answer is (D) 3+√11/2 and 3-√11/2.
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Correct question:
Solve x²-3x+ 5 = 0.
A.3+√-29 +VE/2 and 3- 2-29
B. 3+√29 and 3-√29
C. 3+√-11 and 3-√-11/2
D. 3+√11/2 and 3-√11/2
Math is hard!!
Point A is shown on the number line below.
What is the location of point A?
PLSSS HELP ITS DUE TOMMROW I WILL GIVE U MORE POINTS WHEN I GET MORE PLSSS
Answer:
[tex]A_{\text{red}} = 1766.25 \text{ in}^2[/tex]
Step-by-step explanation:
We can see that there are four bands total on the target, two of which are red bands. With this information, we can solve for the radius of the circle within each band, since we know that each band is the same width:
[tex]r_1 = \dfrac{1}{4} \cdot 30 = 7.5\\[/tex]
[tex]r_2 = \dfrac{2}{4} \cdot 30 = 15\\[/tex]
[tex]r_3 = \dfrac{3}{4} \cdot 30 = 22.5\\[/tex]
[tex]r_4 = 30[/tex]
We can now find the area of each red band by subtracting:
the area of the circle within the white band just inward of the red band
-- from --
the area of the circle within the red band.
Finding the area of the outer red band:
[tex]A_{\text{outer red}} = A(\text{circle within outer red}) - A(\text{circle within outer white})[/tex]
[tex]A_{\text{outer red}} = A(\odot \text{ with radius }r_4) - A(\odot \text{ with radius } r_3)[/tex]
↓ substituting the radius values into the circle area formula ([tex]\pi r^2[/tex])
[tex]A_{\text{outer red}} = (\pi \cdot 30^2) - (\pi \cdot 22.5^2)[/tex]
↓ using 3.14 for [tex]\pi[/tex]
[tex]A_{\text{outer red}} = (3.14 \cdot 30^2) - (3.14 \cdot 22.5^2)[/tex]
↓ evaluating the right side
[tex]A_{\text{outer red}} = 2826 - 1589.625[/tex]
[tex]A_{\text{outer red}} = 1236.375 \text{ in}^2[/tex]
Finding the area of the inner red band:
[tex]A_{\text{inner red}} = A(\text{circle within inner red}) - A(\text{circle within inner white})[/tex]
[tex]A_{\text{inner red}} = A(\odot \text{ with radius }r_2) - A(\odot \text{ with radius } r_1)[/tex]
↓ substituting the radius values into the circle area formula ([tex]\pi r^2[/tex])
[tex]A_{\text{inner red}} = (\pi \cdot 15^2) - (\pi \cdot 7.5^2)[/tex]
↓ using 3.14 for π
[tex]A_{\text{inner red}} = (3.14 \cdot 15^2) - (3.14 \cdot 7.5^2)[/tex]
↓ evaluating the right side
[tex]A_{\text{inner red}} = 706.5 - 176.625[/tex]
[tex]A_{\text{inner red}} = 529.875 \text{ in}^2[/tex]
Finally, we can find the area of all of the red on the target by adding the area of the outer and inner red bands.
[tex]A_{\text{red}} = A_{\text{outer red}} + A_{\text{inner red}}[/tex]
[tex]A_{\text{red}} = 1236.375 \text{ in}^2 + 529.875 \text{ in}^2[/tex]
[tex]\boxed{A_{\text{red}} = 1766.25 \text{ in}^2}[/tex]
HELP ITS AN EMERGENCY
Given the expression: 4x2 + 18x − 10
Part A: What is the greatest common factor? Explain how to find it. (3 points)
Part B: Factor the expression completely. Show all necessary steps. (5 points)
Part C: Check your factoring from Part B by multiplying. Show all necessary steps. (2 points)
Answer:
Part A ----> 2
Part B ----> 2 (2x - 1) (x + 5)
Part C -----> See explanation below
Step-by-step explanation:
Given expression is
4x² + 18x - 10
Part A
The greatest common factor (GCF) of the terms 4x², 18x and 10 is the largest number that divides into all three numbers without a remainder
To find this
Find the prime factors of each of the terms
4x² = 2 · 2 · x · x
18x = 2 · 3 · 3 · x
10 = 2 · 5
The prime factor common to all 3 is 2
Therefore the GCF of the expression is 2
Part B
The factored expression can be obtained by factoring out the GCF
4x² + 18x - 10 = 2(2x² + 9x - 5)
We can further factor the term in parentheses:
2x² + 9x - 5
To do this,
Break the expression into groups:
= (2x² - x) + (10x - 5)
Factor x from 2x² - x: x(2x - 1)
Factor 5 from 10x - 5: 5(2x - 1)
Therefore
(2x² - x) + (10x - 5) = x(2x - 1) + 5(2x - 1)
Factor out the common term 2x -1 to get
(2x - 1)(x + 5)
Therefore
4x² + 18x + 10 = 2 (2x - 1) (x + 5)
Part C
Checking if factorization is correct
Multiply (2x - 1)(x + 5) using the FOIL method
= 2x ·x + 2x · 5 + (-1) · x + (-1) · 5
= 2x² + 10x -1x -5
= 2x² + 9x - 5
Multiply the whole expression by 2
2 · 2x² + ² · 9x - 2 · 5
= 4x² + 18x + 10
which is the original expression
So the factorization is correct
A bug crawls 5 1/2 feet in 28.6 seconds. At that pace, how many seconds does it take the bug to crawl one foot?
Answer:
5.2 seconds
Step-by-step explanation:
To get one foot, we need to divide by 5.5
Set up a proportion:
[tex]\frac{5.5}{5.5}=\frac{28.6}{5.5}[/tex]
Solve:
[tex]1ft.=5.2secs.[/tex]
Beth has three as much money as Dennis. Dennis has $20 more than Will. If all of them had a total of $1260, how much money does Beth have?
Answer:
Beth has $752
Step-by-step explanation:
First we lay out what we know:
Beth: 3x+20
Will: x
Dennis: x+20
Next we write the equation:
3x+20+20+x+x=1260
Then we add any numbers that can be added to each other:
5x+40=1260
Then we solve,
We first minus 40 from both sides of the equation,
5x+40=1260
-40 -40
---------------------
5x=. 1220
Last we divide 1220 by 5,
5x=1220
÷5. ÷ 5
----------------
x=244
Now that we know that x is 244 we fill in the blanks,
Beth: 732+20
Will: 244
Dennis: 244+20
Then we just solve the equations,
Beth: 732+20=752
Will: 244
Dennis: 244+20=264
That's it,
if you want to check your work you can add them all up and if they equal 1260 it's correct.
752+244+264=1260
Your welcome, have a good day/night.
Write a polynomial function in standard form with real coefficients whose zeros include 1,4i, and -4i
Answer:
f(x) = x^3 - x^2 + 16x - 16
Step-by-step explanation:
If a polynomial has the zeros 1, 4i, and -4i, then it must have the factors (x - 1), (x - 4i), and (x + 4i). This is because a factor of (x - a) produces a root of x = a.
To find the polynomial, we can multiply these factors together:
(x - 1)(x - 4i)(x + 4i)
= (x - 1)(x^2 - (4i)^2)
= (x - 1)(x^2 + 16)
= x^3 + 16x - x^2 - 16
So the polynomial function in standard form with real coefficients whose zeros include 1, 4i, and -4i is:
f(x) = x^3 - x^2 + 16x - 16
‼️‼️‼️‼️WILL MARK BRAINLIEST‼️‼️‼️‼️
Answer:
S = 2π(14^2) + 2π(14)(154)
= 2π(196) + 2π(2,156)
= 4,704π = 14,778.1 ft^2
Using 3.14 for π:
S = 4,704(3.14) = 14,770.6 ft^2
Casey is going to wear a gray sportscoat and is trying to decide what tie he should wear to work. In his closet, he has 29 ties, 15 of which he feels go well with the sport coat. If Casey selects one tie at random, determine the probability and the odds of the tie going well or not going well with the sport coat.
The probability of Casey selecting a tie that goes well with the sport coat is 0.517.
The probability of Casey selecting a tie that does not go well with the sport coat is 0.483
What are the probability and the odds of the tie going well or not going well with the sport coat?The probability of Casey selecting a tie that goes well with the sport coat is:
P(goes well) = 15/29
P(goes well) ≈ 0.517
The probability of Casey selecting a tie that does not go well with the sport coat is:
P(does not go well) = 1 - P(goes well)
P(does not go well) = 1 - 15/29 = 14/29
P(does not go well) ≈ 0.483
The odds in favor of the tie going well with the sport coat are:
P(goes well) : P(does not go well) = 15/29 : 14/29
P(goes well) : P(does not go well) = 15:14
The odds against the tie going well with the sport coat are:
P(does not go well) : P(goes well) = 14/29 : 15/29
P(does not go well) : P(goes well) = 14:15
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Find the side lengths of each triangle.
Answer:
9) 36 36 and 36 (all three sides equal)
10) 3.1 3.1 and 3.3 (two sides equal)
Step-by-step explanation:
#9
This is an equilateral triangle so all three sides are equal
Expressions for two of the sides are 6y and 4y + 12
Set these expressions equal to each other and solve for y:
6y = 4y + 12
6y - 4y = 12
2y = 12
y =6
So the side with expression 6y becomes 6 · 6 = 36
Since it is an equilateral triangle, each of the three sides is 36
#10
This is an isosceles triangle with side 2x + 1.7 = x + 2.4
Solve for x:
2x + 1.7 = x + 2.4
2x - x = 2.4 - 1.7
x = 0.7
So the side with x + 2.4 = 0.7 + 2.4 = 3.1 same as the side with 2x + 1.7
The third side with 4x + 0.5 = 4(0.7) + 0.5 = 2.8 + 0.5 = 3.3
So the sides of this triangle are 3.1, 3.1 and 3.3
PLEASE HELP SOLVE 30 PTS
Answer:
The polynomial has a unique zero in the interval [1,2] by using intermediate value theorem
The ratio of two numbers is 7 to 3 and the sum of their squares is 232. Find the numbers.
Let $a_1, a_2, a_3,\dots$ be an arithmetic sequence.
If $a_1 + a_3 + a_5 = -12$ and $a_1a_3a_5 = 80$, find all possible values of $a_{10}$.
(There are multiple)
The possible values of [tex]$a_{10}$[/tex] are [tex]$-\frac{263}{4}$[/tex]and [tex]$-\frac{13}{5}$[/tex].
Since [tex]$a_1, a_2, a_3,\dots$[/tex] is an arithmetic sequence, we can write[tex]$a_3 = a_1 + d$[/tex] and [tex]$a_5 = a_1 + 2d$[/tex] where [tex]$d$[/tex] is the common difference between consecutive terms. Then the given equations become[tex]$3a_1 + 4d = -12$ and $a_1(a_1 + d)(a_1 + 2d) = 80$.[/tex] Simplifying the second equation gives $a_[tex]1^3 + 3da_1^2 + 2d^2a_1 - 80 = 0$.[/tex]
We can solve for [tex]$d$[/tex] in the first equation: [tex]$d = \frac{-3a_1-12}{4} = -\frac{3}{4}a_1 - 3$[/tex]. Substituting this into the second equation yields a cubic equation in terms of[tex]$a_1$[/tex]:
[tex]a\frac{3}{1}-[/tex] [tex]\frac{9}{4} a\frac{2}{1} -[/tex] [tex]\frac{15}{4} a_{1}- 80=0[/tex]
Using synthetic division or another method, we can find that [tex]$a_1 = -5$[/tex] is a root of this equation. Dividing by [tex]$a_1 + 5$[/tex] yields the quadratic [tex]$a_1^2 - \frac{1}{4}a_1 - 16 = 0$[/tex], which has roots [tex]$a_1 = -4$[/tex] and [tex]$a_1 = 4/5$[/tex].Therefore, the possible values of the common difference [tex]$d$[/tex] are [tex]$-\frac{27}{4}$[/tex] and [tex]\frac{4}{5}$[/tex]
Using [tex]$a_1 = -5$[/tex] and [tex]$d = -\frac{27}{4}$[/tex], we find that [tex]$a_{10} = a_1 + 9d = -5 - \frac{243}{4} = -\frac{263}{4}$.[/tex]
Using [tex]$a_1 = -5$[/tex] and [tex]$d = \frac{4}{5}$[/tex], we find that [tex]$a_{10} = a_1 + 9d = -5 + \frac{36}{5} = -\frac{13}{5}$.[/tex]
Therefore, the possible values of [tex]$a_{10}$[/tex] are [tex]$-\frac{263}{4}$[/tex]and [tex]$-\frac{13}{5}$[/tex].
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A card is drawn from a deck of 52 cards. What is the probability that it is a 3 or a spade?
Answer:
P = 4/13 = 0.308
Step-by-step explanation:
3 cards 3
13 spade cards (includes the card 3 of spades)
[tex]P=(3+13)/52= 16/52 = 4/13=0.308[/tex]
Hope this helps.
Transactions on Furnell's credit card are shown for the month of June. The interest rate is 1.4% per month.
June 1 Balance $352.12
June 4 Sears $331.89
June 8 eBay $81.58
June 15 Outback $30
June 18 Payment $200
Find the average daily balance $
184.89
Find the interest for the month $
Find the balance for the following month $
The balance for the following month is $598.18.
How to solveThe average daily balance for June is $184.89.
To calculate the interest for the month, multiply the average daily balance by the interest rate: $184.89 * 1.4% = $2.59.
To find the balance for the following month, add the initial balance, transactions, and interest, then subtract the payment: $352.12 + $331.89 + $81.58 + $30 + $2.59 - $200 = $598.18.
The balance for the following month is $598.18.
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