Answer:
a) The sequence is geometric,b) [tex]t(n) = 2^{4-n}[/tex]----------------------------------------
Given sequence from zeros to 6th terms:
16, 8, 4, 2, 1, 1/2, 1/4It is a GP with the common ratio of 1/2 as each term is half the previous one.
Use the nth term equation, considering the first term is t(1) = 8, and common ratio r = 1/2:
[tex]t(n) = t(1)*r^{n-1}[/tex][tex]t(n) = 8*(1/2)^{n-1}=8*2^{1-n}=2^3*2^{1-n}=2^{4-n}[/tex]Answer:
a) Geometric sequence
[tex]\textsf{b)} \quad t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]
Step-by-step explanation:
An Arithmetic Sequence has a constant difference between each consecutive term.
A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.
Part (a)From inspection of the given table, t(n) halves each time n increases by 1.
Therefore, the sequence is geometric with a common ratio of 1/2.
Part (b)[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]
Given:
a = 8r = 1/2aₙ = t(n)Substitute the initial value (when n = 1) and the common ratio into the formula to create an equation for the nth term:
[tex]\implies t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]
Given that ∅ = 12.2° , calculate the area of the triange below
give your answer to 2 d.p.
Answer:
A = (1/4)√(4 + 11 + 14)√(-4 + 11 + 14)√(4 - 11 + 14)√(4 + 11 - 14)
A = (1/4)√29√21√7
= about 16.32 mm²
Answer:
16.27 mm² (see comment)
Step-by-step explanation:
You want the area of a triangle with side lengths 11 mm and 14 mm, and the angle between them 12.2°.
AreaThe area is given by the formula ...
A = 1/2ab·sin(C)
A = 1/2(11 mm)(14 mm)·sin(12.2°) ≈ 16.27 mm²
The area of the triangle is about 16.27 square millimeters.
__
Additional comment
If you use Heron's formula for the area from the three side lengths, you find it is about 16.32 mm². That's the trouble with over-specified geometrical figures. The result you get depends on which of the given values you use. (To get the area accurate to 4 sf, the angle needs to be specified to 4 sf: 12.24°.)
s = (4 +11 +14)/2 = 14.5
A = √(s(s -a)(s -b)(s -c))
A = √(14.5(14.5 -4)(14.5 -11)(14.5 -14)) = √(14.5·10.5·3.5·0.5) = √266.4375
A ≈ 16.32
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A head teacher shared 27 notebooks among 9 students.one of them students found that each of his books combined 155 pages.how many pages were in the books he received
The number of pages that each book contained is given as follows:
465 pages.
How to obtain the number of pages?The number of pages that each book contained is obtained applying the proportions in the context of the problem.
A head teacher shared 27 notebooks among 9 students, hence the number of the notebooks per student is given as follows:
27/9 = 3 notebooks per student.
One of them students found that each of his books combined 155 pages, hence the number of pages on the book is given as follows:
155 x 3 = 465 pages.
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What is the end behavior of this radical function? f(x) = -2½ + 7
Answer:
Step-by-step explanation:
The function you provided, f(x) = -2.5 + 7, represents a linear function rather than a radical function. A linear function has a constant slope and a constant y-intercept.
The end behavior of a linear function is determined by its slope. In this case, the slope of the function is 0 since there is no term involving x. When the slope is 0, it means the function is a horizontal line.
The function f(x) = -2.5 + 7 represents a horizontal line at y = 4.5. As x approaches positive infinity (∞) or negative infinity (-∞), the value of y remains constant at 4.5. Therefore, the end behavior of this linear function is that y approaches 4.5 as x approaches both positive and negative infinity.
In conclusion, the end behavior of the function f(x) = -2.5 + 7 is that y approaches 4.5 as x approaches positive and negative infinity.
Can anyone help me answer this question?
Define y as an explicit function of x; x + y + y^2 = x^2
We have two explicit functions of x for y:
[tex]y = -1/2 + \sqrt{(x^2 - x + 1/4)} \\or\\y = -1/2 - \sqrt{(x^2 - x + 1/4)}[/tex]
To define y as an explicit function of x, we need to solve for y in terms of x in the given equation:
[tex]x + y + y^2 = x^2[/tex]
First, let's simplify the equation by moving all the terms to one side:
[tex]y^2 + y + (x - x^2) = 0[/tex]
Now, we can use the quadratic formula to solve for y:
[tex]y = (-b + \sqrt{(b^2 - 4ac)} ) / 2a[/tex]
where a = 1, b = 1, and [tex]c = x - x^2.[/tex]Substituting these values, we get:
[tex]y = (-1 + \sqrt{(1 - 4(x - x^2)} )) / 2[/tex]
Simplifying further:
[tex]y = (-1 + \sqrt{(1 - 4x + 4x^2)} ) / 2\\y = (-1 + \sqrt{(4x^2 - 4x + 1)} ) / 2\\y = (-1 + 2\sqrt{(x^2 - x + 1/4)} ) / 2\\y = -1/2 + \sqrt{(x^2 - x + 1/4)}[/tex]
Therefore, we have two explicit functions of x for y:
[tex]y = -1/2 + \sqrt{(x^2 - x + 1/4)} \\or\\y = -1/2 - \sqrt{(x^2 - x + 1/4)}[/tex]
Either of these expressions represents y as an explicit function of x.
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Which model represents the expression 87 - 42?
The model that represents the expression 87 - 42 is (d)
Identifying the model that represents the expression 87 - 42?From the question, we have the following parameters that can be used in our computation:
87 - 42
Using their place values, we have
87 = 8 tens 7 units
42 = 4 tens 2 units
This means that
87 - 42 = 8 tens 7 units - 4 tens 2 units
Subtract the tens
87 - 42 = 4 tens 7 units - 2 units
Subtract the units
87 - 42 = 4 tens 5 units
The model that represents the expression 87 - 42 is 4 tens 5 units
This is represented by model (d) bottom right
Hence, the model that represents the expression 87 - 42 is (d)
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You Draw two marbles (without replacement) from a bag containing 4 green 2 yellow and 6 red marbles.what is the probability that both marbles are yellow? round to the nearest thousand
The probability of drawing two yellow marbles is 0.015 to the nearest thousandth.
What is the probability?The probability of drawing two yellow marbles after drawing a yellow marble on the first draw and without replacement drawing another yellow marble is determined as follows:
Total number of marbles in the bag = 4 + 2 + 6
Total number of marbles in the bag = 12 marbles in the bag.
P(Yellow on first draw) = 2/12 or 1/6
P(Yellow on second draw | Yellow on first draw) = 1/11
The probability of both marbles being yellow will be:
P(Both marbles yellow) = (1/6) * (1/11)
P(Both marbles yellow) = 1/66 or 0.015
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6x^2=-3x+1 to the nearest hundredth
The solutions to the quadratic equation 6x² = -3x + 1 to the nearest hundredth are -0.73 and 0.23.
What are the solutions to the quadratic equation?Given the quadratic equation in the question:
6x² = -3x + 1
To solve the quadratic equation 6x² = -3x + 1, we can rearrange it into standard form, where one side is set to zero:
6x² + 3x - 1 = 0
Now we can solve the equation using the quadratic formula, which states that for an equation in the form ax² + bx + c = 0, the solutions for x are given by:
[tex]x = \frac{-b \±\sqrt{b^2-4ac} }{2a}[/tex]
Here; a = 6, b = 3, and c = -1.
Let's substitute these values into the quadratic formula:
[tex]x = \frac{-b \±\sqrt{b^2-4ac} }{2a}\\\\ x= \frac{-3 \±\sqrt{3^2-4\ *\ 6\ *\ -1} }{2*6}\\\\x = \frac{-3 \±\sqrt{9+24} }{12}\\\\x = \frac{-3 \±\sqrt{33} }{12}\\\\x = -0.73, \ x=0.23[/tex]
Therefore, the values of x are -0.73 and 0.23.
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Find the 9th term of the geometric sequence 4 , − 16 , 64 , . . . 4,−16,64,...
The 9th term of the geometric sequence 4, -16, 64, ... is 262144.
To find the 9th term of the geometric sequence 4, -16, 64, ... , we need to determine the common ratio (r) of the sequence.
To do this, we can divide any term by its preceding term:
-16 / 4 = -4
64 / -16 = -4
We see that the common ratio (r) is -4.
To find the 9th term, we can use the formula for the nth term of a geometric sequence:
Tn = a * r^(n-1)
Where Tn is the nth term, a is the first term, r is the common ratio, and n is the term number.
In this case, the first term a is 4, the common ratio r is -4, and we want to find the 9th term.
T9 = 4 * (-4)^(9-1)
T9 = 4 * (-4)^8
T9 = 4 * 65536
T9 = 262144
Therefore, the 9th term of the geometric sequence 4, -16, 64, ... is 262144.
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Do it please i will reward brainlest
Answer:
Step-by-step explanation:
Mrs. Garcia invests a total of $6331 in two savings accounts. One account yields 8.5% simple interest and the other 8% simple interest. Find the amount placed in each account if she receives a total of $517.68 in interest after one year.
Mrs. Garcia invested $2240 in the 8.5% account and $4091 in the 8% account.
Let x be the amount invested in the 8.5% account, and y be the amount invested in the 8% account. Since the total investment is $6331, we have x + y = 6331.
The total interest received is $517.68, which can be expressed as 0.085x + 0.08y = 517.68, where 0.085 and 0.08 are the decimal equivalents of the interest rates.
We can now solve this system of equations to find x and y. One possible method is to use substitution, where we solve for one variable in terms of the other from one of the equations, and substitute it into the other equation. From x + y = 6331, we have y = 6331 - x. Substituting this into the second equation, we get:
0.085x + 0.08(6331 - x) = 517.68
Simplifying and solving for x, we get:
0.005x + 506.48 = 517.68
0.005x = 11.2
x = 2240
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Find the area of a triangle with the base of 3x²y2 and a height of 4x4y³. Use the formula: A=bh
The area of the triangle is 6x³y⁵
What is area of a triangle?The space enclosed by the boundary of a plane figure is called its area.
A triangle is a polygon with three sides having three vertices.
There are different types of triangle, scalene triangle, equailteral triangle, isosceles triangle, right triangle e.t.c
The area of a triangle is expressed as ;
A = 1/2 bh
where b is the base and h is the height of the of the triangle.
Base = 3x²y²
height = 4x4y³
A = 1/2 × 3x²y² × 4x4y³
A = 1/2 × 12x³y⁵
A = 6x³y⁵
Therefore the area of the triangle is 6x³y⁵
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The area of the Triangle is [tex]48x^{3} y^{4}[/tex]
What is Triangle?Triangle is a two-dimensional three-sided polygon, which has three vertices, three sides and three angles. It is a shape formed when three straight lines meet.
How to determine this
Area of triangle = 1/2 base * height as given
Where area of triangle = ?
Base = [tex]3x^{2} y2[/tex]
i.e 3 * 2 [tex]x^{2} y[/tex]
Base, b = [tex]6x^{2} y[/tex]
Height = [tex]4x4y^{3}[/tex]
i.e [tex]4x[/tex] * [tex]4y^{3}[/tex]
Height,b = [tex]16xy^{3}[/tex]
Area of triangle = 1/2 * [tex]6x^{2} y[/tex] * [tex]16xy^{3}[/tex]
Area = 1/2 * 96* [tex]x^{2+1}[/tex] * [tex]y^{1+3}[/tex]
Area = 1/2 * 96 * [tex]x^{3}[/tex] * [tex]y^{4}[/tex]
Area = 48 * [tex]x^{3} y^{4}[/tex]
Area of the triangle = [tex]48x^{3} y^{4}[/tex]
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A softball player hits a pitched ball when it is 4 feet above the ground. The initial velocity is 75 feet per second. Use the formula h=-16t^2+vt+s. How long will it take for the ball to hit the ground?
If the initial velocity is 75 feet per second, it will take approximately 5.125 seconds for the ball to hit the ground.
The given formula h= -16t²+vt+s represents the height (h) of an object thrown vertically in the air at time (t), with initial velocity (v) and initial height (s). In this case, we are given that the initial height of the softball is 4 feet and the initial velocity is 75 feet per second.
We want to find out how long it will take for the ball to hit the ground, which means we want to find the time (t) when the height (h) is 0.
Substituting the given values into the formula, we get:
0 = -16t² + 75t + 4
This is a quadratic equation in standard form, which we can solve using the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
Where a=-16, b=75, and c=4. Substituting these values into the formula, we get:
t = (-75 ± √(75² - 4(-16)(4))) / 2(-16)
t = (-75 ± √(5625 + 256)) / (-32)
t = (-75 ± √(5881)) / (-32)
We can simplify the expression under the square root as follows:
√(5881) = √(49121) = 711 = 77
So we have:
t = (-75 ± 77) / (-32)
Simplifying further, we get two possible solutions:
t = 0.5 seconds or t = 5.125 seconds
Since the softball player hits the ball when it is 4 feet above the ground, we can disregard the solution t=0.5 seconds (which corresponds to when the ball is at its maximum height) and conclude that it will take approximately 5.125 seconds for the ball to hit the ground.
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A plane takes off from an airport andtravels 13 miles on its path.
if the plane is 12 milesfrom its takeoff poin horizontally, what is its height?
The height of the plane is 5 miles.
To solve this problem, we can visualize it as a right triangle. The horizontal distance traveled by the plane forms the base of the triangle, which is 12 miles. The total distance traveled by the plane forms the hypotenuse of the triangle, which is 13 miles. We need to find the height, which corresponds to the vertical side of the triangle.
Using the Pythagorean theorem, we can calculate the height as follows:
height^2 + 12^2 = 13^2
height^2 + 144 = 169
height^2 = 169 - 144
height^2 = 25
Taking the square root of both sides, we get:
height = √25
height = 5
Therefore, the height of the plane is 5 miles.
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(10)
In 2008, the average new car price was approximately $27,700. In 2010,
the average new car price had increased to $29,200. Assuming a linear
relationship, what will be the approximate new car price in 2014?
A $33,700
B. $32,200
C. $30,700
D. $29,950
The approximate price of the new car in 2014 is:
B. $32,200
How to find the approximate new car price in 2014?The general form of a linear equation is given by:
y = mx + c
where y is the future price of the car, x is the number of years, m is the rate of change of price and c is the initial price of the car
c = $27,700
m = ($29,200 - $27,700)/(2010 - 2008)
m = 1500/2
m = $750 per year
In 2014, x = 2014 - 2008 = 6 years
Substituting into y = mx + c:
y = 750(6) + 27,700
y = 4500 + 27700
y = $32,200
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Explain why you cannot use the product of powers property to simplify (3z + y)^3. Be specific.
Any badd answer will be reported
The product of powers of exponents cannot be used to simplify the binomial expansion
Given data ,
Let the binomial expansion be represented as A
A = ( 3z + y )³
According to the property of products of powers, exponents can be multiplied when a power is increased to a higher power.
The product of powers characteristic cannot be applied to the equation (3z + y)³. This is due to the fact that (3z + y)³ is a binomial raised to the power of 3, not just a power of a single word.
These terms cannot be simplified further using the product of powers property because they involve different variables or variable combinations.
In this case, it is more appropriate to expand the expression using the binomial expansion
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Find the median and mean of the data set below: 9,23,38,45,14
Answer:
mean, 25.8 median 23
Step-by-step explanation:
Arrange the data in an ascending order and the median is the middle value. If the number of values is an even number, the median will be the average of the two middle numbers.
The mean of a set of numbers is the sum divided by the number of terms.
Plsss help plsss help
Answer:
It's B
Step-by-step explanation:
Look at the sides with the number compare them with each other and you will find they are the most similar ones with each other.
if 4.2 pounds of strawberry sells for an income of $12 what is 24,990 pounds income
Answer: $71,400
Step-by-step explanation:
We will set up a proportion with income in the numerator and pounds in the denominator.
[tex]\displaystyle \frac{\$12}{4.2\;lbs} =\frac{\$x}{24,990\;lbs}[/tex]
Then we will cross-multiply.
4.2 * x = 12 * 24,990
4.2x = 299,880
Lastly, we will divide both sides of the equation by 4.2.
x = $71,400
The income from 24,990 pounds of strawberries would be approximately $71,690.14.
To find the income from 24,990 pounds of strawberries, we need to use a proportion:
4.2 pounds of strawberries sells for $12, so 1 pound of strawberries sells for $12/4.2 = $2.86 (rounded to two decimal places).
Therefore, 24,990 pounds of strawberries would sell for:
$2.86 x 24,990 = $71,690.14 (rounded to two decimal places).
5
Enter the correct answer in the box.
Solve the quadratic equation by completing the square.
2x² + 12x = 66
Fill in the values of a and b to complete the solutions.
x=a-√b
x=a+√b
The values of a and b are a = -3 and b = 42. The solutions for x can be written as:
x = -3 - √42
x = -3 + √42
To solve the quadratic equation 2x² + 12x = 66 by completing the square, we need to follow these steps:
Step 1: Move the constant term to the right side:
2x² + 12x - 66 = 0
Step 2: Divide the equation by the leading coefficient (2):
x² + 6x - 33 = 0
Step 3: To complete the square, we take half of the coefficient of x, square it, and add it to both sides of the equation:
x² + 6x + (6/2)² = 33 + (6/2)²
x² + 6x + 9 = 33 + 9
x² + 6x + 9 = 42
Step 4: Rewrite the left side as a perfect square:
(x + 3)² = 42
Step 5: Take the square root of both sides:
√(x + 3)² = ±√42
x + 3 = ±√42
Step 6: Solve for x:
x = -3 ± √42.
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When comparing the f(x) = –x2 + 2x and g(x) = log(2x + 1), on which interval are both functions positive
(–∞, 0)
(0, 2)
(2, ∞)
(–∞, ∞)
Answer:
They are both positive on (2, ∞)
Step-by-step explanation:
g(x) > 0
log(2x + 1) > 0
2x + 1 > 0
x > –1/2
(-1/2, ∞)
Which statement is the converse of the following conditional? If a polygon has three sides, then it is a triangle. A. If a polygon does not have three sides, then it is not a triangle. B. If a polygon is not a triangle, then it does not have three sides. C. If a polygon is a triangle, then it does not have three sides. D. If a polygon is a triangle, then it has three sides.
Answer:
Step-by-step explanation:
The converse of the conditional statement "If a polygon has three sides, then it is a triangle" is:
D. If a polygon is a triangle, then it has three sides.
In the original conditional statement, the "if" part is "a polygon has three sides," and the "then" part is "it is a triangle." The converse switches the positions of the "if" and "then" parts, resulting in the statement "If a polygon is a triangle, then it has three sides."
hey i have a question about this assignment and want to check if my answers are right
The value of x from the given triangle with angle 55° and side 27 units is 15.5 units.
A) By using Pythagoras theorem,
x²=16²+6²
x²=292
x=17.1 units
B) By using Pythagoras theorem,
23²=x²+9²
x²=529-81
x²=448
x=21.2 units
C) By using Pythagoras theorem,
10²=x²+8.5²
x²=100-72.25
x²=27.75
x=5.3 units
D) By using Pythagoras theorem,
x²=12²+15²
x²=369
x=19.2 units
E) Here, cos60°=6√3/x
1/2=6√3/x
x=12√3 units
F) Here, cos45°=√10/x
1/√2=√10/x
x=10 units
G) Here, sin60°=30/x
2/√3=30/x
x=15√3 units
H) Here, cos30°=x/16
2/√3=x/16
x=32/√3 units
I) Here, sinx=25/26
x=74°
J) Here, sin45°=x/16√2
1/√2=x/16√2
x=16 units
K) tan34°=x/28
0.6745=x/28
x=18.886
L) tanx°=17/18.5
tanx°=0.9189
x=42.5°
M) Here, cosx=9/12
x=41.4°
N) Here, sin16°=4/x
0.2756=4/x
x=4/0.2756 units
x=14.5
O) Here, cos55°=x/27
0.5735=x/27
x=15.4845
Therefore, the value of x from the given triangle with angle 55° and side 27 units is 15.5 units.
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A rectangle has an area of 114cm squared and a perimeter of 50cm. What are the dimensions
If rectangle has an area of 114cm squared and a perimeter of 50 cm, the dimensions of the rectangle are approximately 5 cm by 22.8 cm.
Let's assume the length of the rectangle is "l" and the width is "w". We can start by using the formula for the area of a rectangle, which is A = lw. From the given information, we know that the area is 114cm².
So, we have:
lw = 114
Next, we can use the formula for the perimeter of a rectangle, which is P = 2l + 2w. From the given information, we know that the perimeter is 50cm.
So, we have:
2l + 2w = 50
We now have two equations with two variables, which we can solve using substitution or elimination. Let's use substitution by solving the first equation for l:
l = 114/w
We can then substitute this expression for l in the second equation:
2(114/w) + 2w = 50
Multiplying both sides by w to eliminate the fraction, we get:
228 + 2w² = 50w
Rearranging and simplifying, we get a quadratic equation:
2w² - 50w + 228 = 0
We can solve for w using the quadratic formula:
w = [50 ± √(50² - 4(2)(228))]/(2(2)) ≈ 11.4 or 5
Since the length and width must be positive, we can discard the solution w = 11.4. Therefore, the width of the rectangle is approximately 5 cm. We can then use the equation lw = 114 to solve for the length:
l(5) = 114
l ≈ 22.8
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A volunteer walks 1 mile to a dog
shelter. She walks 4 dogs for 1/2mile
each. Then she walks 1 mile
home. She does this each day for
3 days, How many miles does she
walk in all?
The total cost of a tie and a pair of pants was $87.18. If the price of the tie was $2.12 less than the pair of pants, what was the price of the tie?
This is a linear equation problem with two variables. One way to solve it is by using the substitution method. Let x be the price of the tie and y be the price of the pair of pants. Then we have:
$$
\begin{aligned}
x + y &= 87.18 \\
x &= y - 2.12
\end{aligned}
$$
Substituting x into the first equation, we get:
$$
\begin{aligned}
(y - 2.12) + y &= 87.18 \\
2y - 2.12 &= 87.18 \\
2y &= 89.30 \\
y &= 44.65
\end{aligned}
$$
Therefore, the price of the pair of pants is $44.65. To find the price of the tie, we plug y into the second equation:
$$
\begin{aligned}
x &= y - 2.12 \\
x &= 44.65 - 2.12 \\
x &= 42.53
\end{aligned}
$$
Therefore, the price of the tie is $42.53.
Answer:
42.53
Step-by-step explanation:
write equation
tie + pants = 87.18
since ties cost 2.12 less than pants,
tie = pants -2.12
now substitute back into first equation
pants - 2.12 + pants = 87.18
solve for pants by adding 2.12 to the other side
divide by 2 since there are 2 pants on one side
find pants and then substitute value into the tie equation to find the price of the tie
Suppose the prices of a certain model of new homes are normally distributed with a mean of 150,000. Use the 68-95-99.7 rule to find the percentage of buyers who paid between $149,000 and $151,000 if the standard deviation is $1000
The percentage of buyers is approximately 68.26% of buyers of new houses paid between [tex]$149,000[/tex] and [tex]$151,000[/tex] .
We are given that the prices of the new homes are normally distributed with a mean of [tex]$150,000[/tex] and a standard deviation of $1000.
Using the 68-95-99.7 rule, we know that: approximately 68% of the data falls within one standard deviation of the mean approximately 95% of the data falls within two standard deviations of the mean, approximately 99.7% of the data falls within three standard deviations of the mean.
In order to determine the proportion of customers who spent between $149,000 and , we must first determine the z-scores for these values:
z1 = (149,000 - 150,000) / 1000 = -1 z2 = (151,000 - 150,000) / 1000 = 1
Now, we can determine the proportion of data that falls between z1 and z2 using the z-table or a calculator. The region to the left of z1 is 0.1587, and the area to the left of z2 is 0.8413, according to the z-table. Thus, the region bounded by z1 and z2 is:
0.8413 - 0.1587 = 0.6826
We can get the percentage of consumers who spent between by multiplying this by 100% is [tex]$149,000[/tex] and [tex]$151,000[/tex]:
0.6826 x 100% = 68.26%
Therefore, the standard deviation of customers who paid between is [tex]$149,000[/tex] and [tex]$151,000[/tex] for this model of new homes.
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In AABC, AB=32, AC = 23, and BC= 20. What is mZA?
The measure of m∠A in the triangle is:
m∠A = 38.44°
How to find the measure of m∠A?The cosine rule is used for solving triangles which are not right-angled in which two sides and the included angle are given. The following are cosine rule formula for angles:
cos(A) = (b² + c² − a²)/2bc
cos(B) = (c² + a² − b²)/ 2ac
cos(C) = (a² + b² − c²)/2ab
Where a, b and c are the length of the sides, and A, B, and C are the measures of the angles
We have:
a = BC = 20
b = AC = 23
c = AB = 32
Substituting into:
cos(A) = (b² + c² − a²)/2bc
cos(A) = (23² + 32² − 20²)/(2*23*32)
cos(A) = 0.7833
A = cos⁻¹(0.7833)
A = 38.44°
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a certain radioactive isotope has leaked into a small stream. one hundred days after the leak 8% of the original amount of substance remained. Determine the half life of this radioactive isotope
Answer:
The half-life of a radioactive substance is the time it takes for half of the initial amount of the substance to decay. We can use the fact that 8% of the original amount remains after 100 days to determine the half-life of the isotope.
Let's assume that the initial amount of the substance is 1 unit (it could be any amount, but we're assuming 1 unit for simplicity). After one half-life, half of the original amount remains, or 0.5 units. After two half-lives, half of the remaining amount remains, or 0.25 units. After three half-lives, half of the remaining amount remains, or 0.125 units. We can see that the amount of substance remaining after each half-life is half of the previous amount.
We can use this information to set up the following equation:
0.08 = (1/2)^n
where n is the number of half-lives that have elapsed. We want to solve for n.
Taking the logarithm of both sides, we get:
log(0.08) = n*log(1/2)
Solving for n, we get:
n = log(0.08) / log(1/2) = 3.42
So the number of half-lives that have elapsed is approximately 3.42. Since we know that 100 days is the time for three half-lives (from the previous calculation), we can find the half-life by dividing 100 days by 3.42:
Half-life = 100 days / 3.42 = 29.2 days (rounded to one decimal place)
Therefore, the half-life of the radioactive isotope that leaked into the stream is approximately 29.2 days.
Alfred has a coupon for 35 cents off a 32-ounce bottle of detergent that sells for $1.39. Another brand offers a 20-ounce bottle for 79 cents. If he uses the coupon, which will be the better buy?
The detergent with the coupon is the better buy since it has a lower price per ounce.
To determine which option is the better buy, we need to compare the prices per ounce for each detergent brand.
First, let's calculate the price per ounce for the 32-ounce bottle of detergent after applying the coupon:
Price per ounce = (Price - Coupon) / Ounces
Price per ounce = ($1.39 - $0.35) / 32
Price per ounce = $1.04 / 32
Price per ounce ≈ $0.0325
Next, let's calculate the price per ounce for the 20-ounce bottle of the other brand:
Price per ounce = Price / Ounces
Price per ounce = $0.79 / 20
Price per ounce = $0.0395
Comparing the two price per ounce values, we can see that the price per ounce for the detergent with the coupon is approximately $0.0325, while the price per ounce for the other brand is $0.0395.
Therefore, the detergent with the coupon is the better buy since it has a lower price per ounce.
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Determine the value of real parameters p in such a way that the equation 3x2−24x+p=0 has one root equal to triple of the second root
has one root equal to triple of the second root.
The value of the parameter p that satisfies the given conditions is 36.
Let the roots of the quadratic equation [tex]3x^2 - 24x + p = 0[/tex] be denoted by α and β, where α is the root that is triple the value of β.
Then we have:
α = 3β
The sum and product of the roots of the quadratic equation are given by:
α + β = 8 (from the coefficient of x in the linear term)
αβ = p/3 (from the constant term)
Substituting α = 3β in the first equation gives:
3β + β = 8
4β = 8
β = 2
Therefore, α = 6.
So the roots of the quadratic equation are α = 6 and β = 2.
The product of the roots is:
αβ = 6 × 2 = 12
From the equation αβ = p/3, we have:
p/3 = 12
p = 36
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