The surface area of the sphere is given by the equation
[tex]A=4\pi * r^{2}[/tex],
where A is the surface area and r is the radius.
We want to find the volume of the sphere, which is given by the equation
[tex]V = \frac{4}{3} * \pi * r^{3}[/tex],
where V is the volume and r is the radius.
Looking at these equations, we see that they both involve the sphere's radius. If we know what r is, we can calculate the volume.
We know that the sphere's surface area is [tex]36 \pi[/tex]. Plugging that in for A in the surface area equation, we get
[tex]36 \pi=4\pi * r^{2}[/tex], then divide by [tex]\pi[/tex]
[tex]36 = 4 * r^{2}[/tex], then divide by 4
[tex]r^{2} = 9[/tex], then take the square root of both sides
[tex]r = 3[/tex]
So the radius of the sphere is 3. Plugging this into the volume equation,
[tex]V = \frac{4}{3} * \pi * 3^{3}[/tex], simplify terms
[tex]V = \frac{4}{3} * \pi * 27[/tex], multiply [tex]\frac{4}{3}[/tex] by 27
[tex]V = 36 * \pi[/tex]
So the volume of the sphere is [tex]36\pi[/tex].
Answer:
36π ft^3
Step-by-step explanation:
The surface area (S) of a sphere can be defined as:
S = 4×π×r^2 = 36×π
Solve for r to get the radius of the sphere:
r = ([tex]\sqrt{36/4}[/tex] = 3
The voluem (V) of a sphere can be defined as:
V= (4/3)×π×r^3
The volume of the sphere is:
V = (4/3)×π×(3^3) = 36π ft^3
The volume of the sphere can be calculated from the surface area given.
Find the value of X such that the data set has the given mean. 99, 123, 105, 114, 107, X; mean 108
Answer:
The value of x is 100
Step-by-step explanation:
* Lets revise how to find the missing number of the given Mean
- Add up the numbers you know.
- Set up your equation by adding the sum of the numbers plus “x”
- Divide the sum by the number of numbers given.
- Equate them by the value of the mean
- Solving for "x"
- Check the Answer
* Now lets solve the problem
- The numbers are 99 , 123 , 105 , 114 , 107
- The sum of the numbers = 99 + 123 + 105 + 114 + 107 = 548
- The equation is ⇒ (548 + x)/6 = 108 ⇒ × 6 both sides
∴ 548 + x = 648 ⇒ subtract 548 from both sides
∴ x = 100
* Now lets check the answer
- The mean = (99 + 123 + 105 + 114 + 107 + 100)/6 = 648/6 = 108
∴ The answer x = 100 is correct
* The value of x is 100
Answer:
The value of X = 100
Step-by-step explanation:
Points to remember
Mean of a data set is given by,
Mean = (sum of data)/number of data
To find the value of X
It is given a data set,99, 123, 105, 114, 107, X
Sum of data set = 99 + 123 + 105 + 114 + 107 + X = 548 + X
Mean = 108
Mean = (sum of data)/number of data
108 = (548 + X)/6
548 + X = 108 * 6 = 648
X = 648 - 548 = 100
Therefore the value of X = 100
Which has the greater area: a 6 ‐centimeter by 4 1 2 ‐centimeter rectangle or a square with a side that measures 5 centimeters? How much more area does that figure have? The has the greater area. Its area is square centimeters greater.
Answer:
The rectangle has the greater area
Is area is [tex]2\ cm^{2}[/tex] greater
Step-by-step explanation:
we know that
The area of rectangle is equal to
[tex]A=(6)*(4\frac{1}{2})=(6)*(\frac{9}{2})=27\ cm^{2}[/tex]
The area of the square is equal to
[tex]A=5^{2}=25\ cm^{2}[/tex]
therefore
The rectangle has the greater area
Find the difference
[tex]27\ cm^{2}-25\ cm^{2}=2\ cm^{2}[/tex]
Is area is [tex]2\ cm^{2}[/tex] greater
There was a sample of 650 milligrams of a radioactive substance to start a study. Since then, the sample has decayed by 7.8% each year.
Let t be the number of years since the start of the study. Let y be the mass of the sample in milligrams. Write an exponential function showing the relationship between t
and y
.
Answer:
y = 650·0.922^t
Step-by-step explanation:
At the end of each year, 92.2% of the amount at the beginning of the year remains. That is, the beginning amount is multiplied by 0.922. The exponent t in 0.922^t tells how many times (years) that multiplication has taken place. At the end of t years, the amount remaining in milligrams (y) is ...
y = 650·0.922^t
Find two numbers that have the maximum possible product and a sum of 7
Call these numbers [tex]x,y[/tex]. Then [tex]x+y=7[/tex] or [tex]y=7-x[/tex].
We want to maximize their product,
[tex]f(x,y)=xy\implies f(x,7-x)=F(x)=7x-x^2[/tex]
We could consider the derivative, but I think that's overkill. Instead, let's complete the square:
[tex]7x-x^2=-\left(x^2-7x+\dfrac{49}4\right)+\dfrac{49}4=\dfrac{49}4-\left(x-\dfrac72\right)^2[/tex]
whose graph is a parabola opening downward with vertex at [tex]\left(\dfrac72,\dfrac{49}4\right)[/tex], so that the maximum product is [tex]\dfrac{49}4[/tex].
Now if [tex]x=\dfrac72[/tex], it follows that [tex]y=7-\dfrac72=\dfrac72[/tex].
The two numbers that have the maximum possible product and a sum of 7are 3.5 and 3.5
System of equationsLet the two number be x and y
If the sum of the numbers is 7, then;
x + y = 7 ........................ 1
If their product is at maximum, then;
xy = P ............................. 2
From equation 1, y = 7 - x
Substitute into equation 2 to have:
x(7-x) = P
P = 7x - x²
If the function is at maximum, then;
dP/dx = 7 - 2x
0 = 7 - 2x
x = 7/2
x = 3.5
Recall that x + y = 7
x = 7 - y
x = 3.5
Hence the two numbers are 3.5 and 3.5
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Scientist released 5 foxes into a new habitat in year 0. Each year, there were four times as many foxes as the year before. How many foxes were there after x years? Write a function to represent this scenario
Answer:
The function that this scenario represents is:
[tex]P(x) = 5(4) ^ x[/tex]
Step-by-step explanation:
The initial number of foxes was 5. The following year they had
year 1: [tex]5 * (4) = 20[/tex] foxes
year 2: [tex]5 * 4 * (4) = 80[/tex] foxes
year 3: [tex]5 * 4 * 4 * (4) = 320[/tex] foxes
year x: [tex]5 * 4 ^ x[/tex] foxes
Then the equation that models the situation is an equation of exponential growth. Where P(x) is the population of foxes in year x.
So:
[tex]P(x) = 5(4) ^ x[/tex]
You have 4 different trophies to arrange on the top shelf of a bookcase. How many ways are there to arrange the trophies?
Answer:
4!=4 x 3 x 2 x 1=24
So the answer is 24.
Answer:
24 ways.
Step-by-step explanation:
If i have n different trophies to arrange on top shelf of a book case then the number of ways in which we can arrange the trophies will be
= n!
When n = 4
Then number of ways in which we can arrange the books
= 4!
= 4 × 3 × 2 × 1
= 24
Therefore, answer is 24 ways.
Find an equation of the tangent line to the curve at the given point. y = x , (4, 2) Step 1 To find the equation of a line, we need the slope of the line and a point on the line. Since we are requested to find the equation of the tangent line at the point (4, 2), we know that (4, 2) is a point on the line. So we just need to find its slope. The slope of a tangent line to f(x) at x = a can be found using the formula m tan = lim x→a f(x)-f(a)/ x-a
Answer:the answer is C
Step-by-step explanation:
Combine the following expressions.
[tex]\sqrt{3y^2} + 4\sqrt{12y^2} - y\sqrt{75}[/tex]
ANSWER
[tex]12y \sqrt{3} [/tex]
EXPLANATION
The given expression is
[tex]\sqrt{3y^2} + 4\sqrt{12y^2} - y\sqrt{75}[/tex]
We identity and remove the perfect squares to obtain
[tex]y\sqrt{3} + 16y\sqrt{3} - 5y\sqrt{3}[/tex]
We now observe that, the three terms are all similar.
We combine the similar terms to get:
[tex]y\sqrt{3} + 16y\sqrt{3} - 5y\sqrt{3} = 12y \sqrt{3} [/tex]
Answer:
4y square root of 3
Step-by-step explanation:
What are the solution(s) to the quadratic equation 50 – x2 = 0?
x = ±2
x = ±6
x = ±5
no real solution
Answer:
No real solution
Step-by-step explanation:
There is no perfect squares in 50
ANSWER
[tex]x = \pm5\sqrt{2} [/tex]
EXPLANATION
The given equation is:
[tex]50 - {x}^{2} = 0[/tex]
Group the constant term to get:
[tex]- {x}^{2} = 0 - 50[/tex]
[tex] - {x}^{2}=- 50[/tex]
[tex]{x}^{2}=50[/tex]
Take square root to get:
[tex]x = \pm \sqrt{50} [/tex]
We simplify further to remove the perfect square.
[tex]x = \pm \sqrt{25 \times 2} [/tex]
[tex]x = \pm \sqrt{25} \times \sqrt{2} [/tex]
[tex]x = \pm5\sqrt{2} [/tex]
*WORD PROBLEM* A jar contains dimes and quarters. Twenty percent of the coins in the jar are dimes. A number generator simulates randomly selecting 10 coins from the jar. The number generator is used 12 times and the number of dimes in each trial is shown in the dot plot.
Which description is correct about the number generator is fair or not?
-The number generator is not fair. The dot plot shows a distribution that is skewed right.
-The number generator is fair. The dot plot shows a distribution that is skewed right.
-The number generator is fair. It shows that 20% of the coins selected are dimes most of the time.
-The number generator is not fair. In one of the experiments, no dimes are chosen.
Step-by-step explanation:
20% of the coins are dimes, and 10 coins are selected, so the expected number of dimes is 0.20 * 10 = 2.
So the number generator is fair. It shows that 20% of the coins selected are dimes most of the time.
The correct description is given by the number generator is fair. It shows that 20% of the coins selected are dimes most of the time.
The correct option is (c).
What is distribution?The distribution is a mathematical function that describes the relationship of observations of different heights. A distribution is simply a collection of data, or scores, on a variable
As 20% of the coins are dimes, and 10 coins are selected,
So, the expected number of dimes is,
=[tex]20[/tex] % of 10
= [tex]\frac{20}{100} * 10[/tex]
= 0.20 * 10
= 2.
Thus, the number expected number of dimes is 2
So, the number generator is fair. It shows that 20% of the coins selected are dimes most of the time.
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One number is 10 times as large as another, and their difference is 81. Find the numbers.
If x represents the smaller number, then the larger number is
10
10x
X - 10
Answer:
90
Step-by-step explanation:
10x - x = 81
9x = 81
x = 9
larger number = 10 x 9 = 90
For this case we have that "x" is the variable that represents the smallest number to find. Let and the variable that represents the largest number, then:[tex]y = 10x\\y-x = 81[/tex]
Substituting the first equation in the second:
[tex]10x-x = 81\\9x = 81\\x = \frac {81} {9}\\x = 9[/tex]
So, the biggest number is:
[tex]y = 10 * 9 \\y = 90[/tex]
Answer:
10x
90
Find the area of the triangle to the nearest tenth.
11 mm
330
14 mm
The area of the triangle is approximately
Answer:
42.4
Step-by-step explanation:
1) use this formula of Area of the triangle:
[tex]A=\frac{1}{2} a*b*sin(a,b)[/tex]
2) using the formula described above:
A=0.5*14*11*0.55≈42.4 (mm²)
A circle has a radius of 10 inches. Find the approximate length of the arc intersected by a central angle of 2(pi)/3
[tex]\bf \textit{arc's length}\\\\ s=r\theta ~~ \begin{cases} r=&radius\\ \theta =&angle~in\\ &radians\\ \cline{1-2} r=&10\\ \theta =&\frac{2\pi }{3} \end{cases}\implies s=10\left( \cfrac{2\pi }{3} \right)\implies s=\cfrac{20\pi }{3}\implies s\approx 20.94[/tex]
Answer:
C
Step-by-step explanation:
The absolute value of any complex number a + bi is the ___________ from (a, b) to (0, 0) in the complex plane.
Answer:
distance
Step-by-step explanation:
Usually the absolute value of a complex number is called its magnitude. The squared magnitude is the algebraic quantity that's preferable to work with.
Let
[tex]z = a+bi[/tex]
[tex]|z|^2 = z^* z = zz^* = |a+bi|^2= (a+bi)(a-bi) = a^2 - i^2 b^2 = a^2+b^2[/tex]
[tex]|a+bi| = \sqrt{a^2+b^2}[/tex]
That's the distance from the origin to (a,b)
The absolute value of any complex number a + bi is the distance from (a, b) to (0, 0) in the complex plane.
What is a Complex number?This is a number which is in the form of a + bi in which a and b are expressed as real numbers.
|a + bi| = √a²+b² depicts the magnitude which therefore expresses the distance from the origin.
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20. The sum of two consecutive even integers is 158. Find the least of the two integers.
A. 78
B. 156
C. 80
D. –78
Answer:
78
Step-by-step explanation:
The tricky part of this is figuring out how to assign the unknowns. We are told that we are working with two consecutive even integers. Consecutive means "next to" or "in order" and sum means to add. If we use 2 and 4 as examples of our 2 consecutive even integers and assign x to 2, then in order to get from 2 to 4 we have to add 2. So the lesser of the 2 integers is x, and the next one in order will be x + 2. (2 and 4 are just used as examples; they mean nothing to the solving of this particular problem. You could pick any 2 even consecutive integers and find the same rule applies. All we are doing here with the example numbers is finding a rule for our integers.) Now we have the 2 expressions for the integers, we will add them together and set the sum equal to 158:
x + (x + 2) = 158
The parenthesis are unnecessary since we are adding, so when we combine like terms we get
2x + 2 = 158 and
2x = 156 and
x = 78
That means that the lesser of the 2 integers in 78, and the next one in order would be 80, and 78 + 80 = 158
Answer:
2A + (2A +2) = 158
4A = 156
A= 39
2A = 78 and (2A + 2) = 80
Answer is A
Step-by-step explanation:
Let ABC be a right triangle with mLC = 90°. Given tan LA =0.5, find tan LB.
Answer:
tan(∠B) = 2
Step-by-step explanation:
In a right triangle, the relationships of the tangents of the acute angles is ...
tan(∠B) = cot(∠A) = 1/tan(∠A)
For tan(∠A) = 0.5, this means ...
tan(∠B) = 1/0.5 = 2
Question Part Points Submissions Used Use a Double- or Half-Angle Formula to solve the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.) cos(2θ) + sin^2(θ) = 0
[tex]\bf \textit{Double Angle Identities} \\\\ cos(2\theta)= \begin{cases} cos^2(\theta)-sin^2(\theta)\\ \boxed{1-2sin^2(\theta)}\\ 2cos^2(\theta)-1 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ cos(2\theta )+sin^2(\theta )=0\implies \boxed{1-2sin^2(\theta)}+sin^2(\theta )=0 \\\\\\ 1-sin^2(\theta )=0\implies 1=sin^2(\theta )\implies \pm\sqrt{1}=sin(\theta ) \\\\\\ \pm 1=sin(\theta )\implies sin^{-1}(\pm 1) = \theta \implies \theta = \begin{cases} \frac{\pi }{2}\\\\ \frac{3\pi }{2} \end{cases}[/tex]
To solve the equation cos(2θ) + sin^2(θ) = 0 using a double- or half-angle formula, rewrite the equation and combine like terms to simplify.
Explanation:To solve the equation cos(2θ) + sin^2(θ) = 0 using a double- or half-angle formula, we can rewrite the equation as:
2cos^2(θ) - 1 + sin^2(θ) = 0
Using the identity sin^2(θ) = 1 - cos^2(θ), we can substitute and simplify:
2cos^2(θ) - 1 + (1 - cos^2(θ)) = 0
Combining like terms, we have:
-cos^2(θ) + cos^2(θ) = 0
This simplifies to:
0 = 0
Since 0 = 0 is a true statement, the equation is satisfied for all values of θ within the interval [0, 2π).
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Barry buys 6 loaves of bread. Each loaf weighs 1 1 4 pounds. What is the total weight of the bread in pounds? in ounces? A. 6 1 4 pounds; 100 ounces B. 7 1 2 pounds; 112 ounces C. 7 1 2 pounds; 120 ounces D. 9 pounds; 144 ounces
Answer:
C. 7 1/2 pounds; 120 ounces
Step-by-step explanation:
I'll assume you wanted to write each loaf weighs 1 and 1/4 (1.25) pounds and not 114 pounds... since that makes sense considering the choices for answer.
So, to get the total weight of the 6 loafs, we multiply by 6 the weight of one...
TP = 6 * 1.25 = 7.5 or 7 1/2 pounds
To get that in ounces, we just have to remember there are 16 ounces per pound... so
TO = 7.5 lbs * 16 ounces/lbs = 120 ounces
A triangular plot of land is shown.
What is the longest dimension of the plot?
Enter your answer in the box.
Round only your final answer to the nearest foot.
___
Check the picture below.
make sure your calculator is in Degree mode.
–x + 5y = –2; {(7, 1.6), (5, 0.6), (6, 3.6), (4, –1.4)}
a.
{(5, 0.6)}
c.
{(6, 3.6)}
b.
{(4, –1.4)}
d.
{(7, 1.6)}
Answer:
a. {(5, 0.6)}
Step-by-step explanation:
A graph is a useful tool. It can help you find that the set of points that satisfies both functions is ...
{(5, 0.6)}
_____
Comment on the graph
Strictly speaking, the points connecting the dots of the second function are not part of the function. They are there simply to provide a visual aid in locating the points. As we see, the line goes through point (5, 0.6) exactly.
The graph of f(x) = x^2 is shown.
Use the parabola tool to graph g(x).
9(x)= (1 + x)^2 – 2
graph the parabola by first plotting its vertex and then plotting a second point on the parabola
Answer:
see the graph below
Step-by-step explanation:
The function ...
g(x) = f(x+1) -2
tells you that the graph of f(x) is shifted 1 unit to the left and 2 units down. That means the vertex will be (-1, -2) and another point will be (0, -1).
Point (1, 1) is on the graph of the f(x) parabola. You can pick any other point you like and shift it left 1 and down 2 to find a point on g(x).
The graph of the function g(x) = (1 + x)² - 2 is added as an attachment
Sketching the graph of the functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = x²
Also, we have
g(x) = (1 + x)² - 2
The above function is a quadratic function that has been transformed from f(x) = x² as follows
Shifted to the right by 1 unitShifted down by 2 unitsNext, we plot the graph using a graphing tool by taking note of the above transformations
The graph of the function is added as an attachment
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Help me pleaseeeeeee?
Answer:
-7x+3y+3
Step-by-step explanation:
(4-5+4)=3
(-2x-5x)=-7x
(-4y+7y)=3y
For this case we must simplify the following expression:
[tex]4-5-2x-4y + 4-5x + 7y[/tex]
We must combine similar terms, taking into account that equal signs are added and the same sign is placed, while different signs are subtracted and the sign of the greater is placed:
[tex]4-5 + 4-2x-5x-4y + 7y =\\3-7x + 3y[/tex]
Answer:
[tex]3-7x + 3y[/tex]
A circle with radius r is inscribed into a right triangle. Find the perimeter of the triangle if the length of the hypotenuse is 24 cm and r=4 cm;
Answer: 56
Step-by-step explanation:
4=(a+b-24)/2
32=a+b
32+24=56
Answer:
The perimeter of the triangle is 56 cm.
Step-by-step explanation:
Given:
The length of the hypotenuse is 24 cm
Radius of the circle r=4 cm
To Find:
The perimeter of the triangle=?
Solution:
Let us see the below diagram representing situation.
Here, first we have drawn the radius of circle perpendicular to the sides of triangle,
We have Taken PB = 4 because, it is a side of square formed there, and we take OC as x.
Then, QC also becomes x as they are tangents from single point, so they are equal.
Now, AQ becomes 24 – x as AC is 24 according to given information.
So, perimeter = AB + BC + CA
Perimeter = (24 – x + 4) + (4 + x) + (24 – x + x)
Perimeter = 28 – x + 4 + x + 24
Perimeter = 28 + 28 = 56
Hence, the perimeter of the triangle is 56 cm.
At a mortgage company, 60% of calls are answered by an attendant. The remaining 40% of callers leave their phone numbers. Of these 40%, 75% receive a return phone call the same day. The remaining 25% receive a return call the next day. Of those who initially spoke to an attendant, 80% will apply for a mortgage. Of those who received a return call the same day, 60% will apply. Of those who received a return call the next day, 40% will apply. Calculate the probability that a person initially spoke to an attendant, given that he or she applied for a mortgage.
Answer:
24/35, about 69%
Step-by-step explanation:
The data given can be put into a 2-way table (attached). It shows that 0.48 of all calls were answered and resulted in a mortgage application. Altogether, 0.70 of all calls resulted in a mortgage application. Thus the conditional probability of interest is ...
p(spoke to attendant | applied for a mortgage) = p(spoke & applied)/p(applied)
= 0.48/0.70 = 24/35 ≈ 69%
The probability can be calculated by dividing the probability of speaking to an attendant and applying for a mortgage by the probability of applying for a mortgage.
Explanation:To calculate the probability that a person initially spoke to an attendant, given that he or she applied for a mortgage, we need to use conditional probability. Let's break it down step by step:
Find the probability that a person initially spoke to an attendant, which is given as 60%.Find the probability that a person spoke to an attendant and applied for a mortgage, which is given as 80% of those who initially spoke to an attendant.Find the probability that a person applied for a mortgage, which can be calculated by multiplying the probability of speaking to an attendant and applying for a mortgage by the probability of speaking to an attendant.Use these probabilities to calculate the conditional probability by dividing the probability of speaking to an attendant and applying for a mortgage by the probability of applying for a mortgage.Let's do the calculations:
Probability of speaking to an attendant = 60% = 0.6
Probability of attending and applying for a mortgage = 80% of 60% = 0.8 * 0.6 = 0.48
Probability of applying for a mortgage = (Probability of attending and applying for a mortgage) + (Probability of receiving a same-day return call and applying for a mortgage) + (Probability of receiving a next-day return call and applying for a mortgage)
= 0.48 + (0.75 * 0.4 * 0.6) + (0.25 * 0.4 * 0.4) = 0.48 + 0.18 + 0.04 = 0.7
Conditional probability = (Probability of attending and applying for a mortgage) / (Probability of applying for a mortgage) = 0.48 / 0.7 = 0.6857
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Consider the following equation. 3x4 − 8x3 + 6 = 0, [2, 3] (a) Explain how we know that the given equation must have a root in the given interval. Let f(x) = 3x4 − 8x3 + 6. The polynomial f is continuous on [2, 3], f(2) = < 0, and f(3) = > 0, so by the Intermediate Value Theorem, there is a number c in (2, 3) such that f(c) = . In other words, the equation 3x4 − 8x3 + 6 = 0 has a root in [2, 3]. (b) Use Newton's method to approximate the root correct to six decimal places.
Answer:
a) see your problem statement for the explanation
b) 2.54539334183
Step-by-step explanation:
(b) Many graphing calculators have a derivative function that lets you define the Newton's Method iterator as a function. That iterator is ...
x' = x - f(x)/f'(x)
where x' is the next "guess" and f'(x) is the derivative of f(x). In the attached, we use g(x) instead of x' for the iterated value.
Here, our f(x) is ...
f(x) = 3x^4 -8x^3 +6
An expression for f'(x) is
f'(x) = 12x^3 -24x^2
but we don't need to know that when we use the calculator's derivative function.
When we start with x=2.545 from the point displayed on the graph, the iteration function g(x) in the attached immediately shows the next decimal digits to be 393. Thus, after 1 iteration starting with 4 significant digits, we have a result good to the desired 6 significant digits: 2.545393. (The interactive nature of this calculator means we can copy additional digits from the iterated value to g(x) until the iterated value changes no more. We have shown that the iterator output is equal to the iterator input, but we get the same output for only 7 significant digits of input.)
___
Alternate iterator function
If we were calculating the iterated value by hand, we might want to write the iterator as a rational function in Horner form.
g(x) = x - (3x^4 -8x^3 +6)/(12x^3 -24x^2) = (9x^4 -16x^3 -6)/(12x^3 -24x^2)
g(x) = ((9x -16)x^3 -6)/((12x -24)x^2) . . . . iterator suitable for hand calculation
The equation must have a root in the interval [2, 3] based on the Intermediate Value Theorem. Newton's method can be used to approximate the root to six decimal places.
Explanation:To show that the given equation must have a root in the interval [2, 3], we use the Intermediate Value Theorem. The function f(x) = 3x4 − 8x3 + 6 is continuous on [2, 3], and f(2) < 0 while f(3) > 0. Therefore, by the Intermediate Value Theorem, there must be a number c in the interval (2, 3) such that f(c) = 0. This implies that the equation 3x4 − 8x3 + 6 = 0 has a root in the interval [2, 3].
To approximate the root of the equation using Newton's method, we start with an initial guess, let's say x0 = 2.5. We iterate using the formula xi+1 = xi - f(xi)/f'(xi) until we obtain the desired level of accuracy. By repeating this process, we can approximate the root of the equation correct to six decimal places.
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An empty container has a mass of 600 grams. When 8 cans of equal mass are put into the container, the filled container has a mass of 7 kilograms. What is the mass of each can in grams? A. 750 grams B. 800 grams C. 900 grams D. 950 grams
Answer: Option B
[tex]x = 800\ gr[/tex]
Step-by-step explanation:
We know that the mass of the empty container is 600 gr
And when they put 8 cans of equal mass then the final weight of the container is 7 kilograms or 7000 grams
If we call x the weight of the cans then this situation can be represented by the following linear equation.
[tex]8x + 600 = 7000[/tex]
Now we solve the equation for the variable x.
[tex]8x = 7000 -600\\\\8x = 6400\\\\x = \frac{6400}{8}\\\\[/tex]
[tex]x = 800\ gr[/tex]
Samantha wanted to fill her new fish tank with goldfish and guppies. She bought 26 total fish. Each goldfish cost $3, and each guppy cost $4. She spent $86 total. How many guppies did she buy? 4 7 8 16
Answer:
8 guppies
Step-by-step explanation:
Let:
x = number of goldfishes
y = number of guppies
We can come up with two equations:
x + y = 26
3x +4y = 86
We use the first equation to come up with a solution for one of the unknowns:
x = 26 - y
We can use this to substitute the x on the second equation:
[tex]3x + 4y = 86\\\\3(26-y) + 4y=86\\\\78 - 3y + 4y = 86\\\\78 + y = 86\\\\y = 86 - 78\\\\y = 8[/tex]
So she bought 8 guppies.
The total guppies Samantha buys are 8 and the total goldfish she buys are 18 and this can be determined by forming the linear equation.
Given :
Samantha wanted to fill her new fish tank with goldfish and guppies. She bought 26 total fish. Each goldfish cost $3, and each guppy cost $4. She spent $86 in total.Let the total number of goldfish be 'x' and the total number of guppies be 'y'. Then the linear equation that represents the total number of fish is given by:
x + y = 26
x = 26 - y ---- (1)
The linear equation that represents the total spent $86 is given by:
3x + 4y = 86 --- (2)
Now, substitute the value of 'x' in equation (2).
3(26 - y) + 4y = 86
Simplify the above equation.
78 - 3y + 4y = 86
y = 86 - 78
y = 8
Now, substitute the value of 'y' in equation (1).
x = 26 - 8
x = 18
So, Samantha buys 8 guppies.
For more information, refer to the link given below:
https://brainly.com/question/21835898
Two symptoms are associated with a certain disease. There is a 95% probability that at least one of the symptoms occurs; in addition, the first symptom occurs with 50% probability, the second symptom occurs with 45% probability. Based on these probability results, answer the following two questions 1) Are the two events "first symptom occurs" and "second symptom occurs" mutually exclusive (i.e. disjoint)? 2) Are the two events "first symptom occurs" and "second symptom occurs" independent? For each question, clearly state YES or NO and provide a brief written explanation that includes the appropriate numerical support.
Answer:
YES
Step-by-step explanation:
which of the following is an irrational number?
A. √1
B. √49
C. √9
D. √80
Answer:
D
Step-by-step explanation:
Only perfect squares can be rational numbers. the square root of 80 is the only one that is not a perfect square.
D. Square Root of 80
State the range of the following functions: f(x)=x^2+4
Answer:
[4, ∞)
Step-by-step explanation:
The vertex of the upward-opening quadratic is (0, 4), so the function will take on values of 4 or more. The range is 4 to infinity, inclusive of 4.