Answer:
1.8 cans of paint are needed
Step-by-step explanation:
A large statue has a volume of 135 m³ and an area of 81 m², while a small statue has a volume of 5 m³. You want to know the number of cans of paint required to paint the small statue, if each can covers 5 m².
Scale factorThe ratio of linear dimensions on the small statue to those on the large statue is the cube root of their volume ratio:
sf = ∛(5/135) = ∛(1/27) = 1/3
AreaThe area of the small statue is the larger statue's area multiplied by the square of the scale factor:
smaller area = larger area × (sf)²
smaller area = (81 m²)(1/3)² = 9 m²
PaintEach can of paint covers 5 m², so the number of cans required is ...
(9 m²)/(5 m²/can) = 1.8 cans
You will need 1.8 cans of paint to paint the small statue.
__
Additional comment
If you must pay for whole cans of paint, you must buy 2 cans of paint.
a person runs at a speed of 4.5 kilometres per hour. She runs 10 laps on the outside track and 10 laps on the inside track, the inside track has a diameter of 44m. The outside track is 3m thicker than the inside track, how much more minutes did it take her to run the outside lap?
Answer:
Below
Step-by-step explanation:
Assuming a circular track
Outside track is 3 m 'thicker ' so the diameter is (44 + 3 + 3) = 50 m
Outside track circumference = pi * d = pi * 50 = 50 pi m
Inside track circumference = 44 pi
Outside 10 laps = .500 pi KM time = .5 pi km / 4.5 km/hr = .3490658 hr
Inside 10 laps = .444 pi KM time = .444 pi km / 4.5 km/hr = .309974
Difference = .039095 hr =2.35 minutes
For isosceles trapezoid PQRS, give the coordinates of R without using any new variables.
For isosceles trapezoid PQRS the coordinates for point R are;; (a - b, c).
What are the coordinates of the trapezoid?We are told that the following figure is an isosceles trapezoid. This tells us that the two non - parallel sides of the given trapezoid are equal.
A parallel side of the trapezoid lies on the x-axis, which tells us that the opposite side is a horizontal line parallel to the x-axis and that the y-coordinate remains the same for all point lie on that side.
Since the opposite side has vertex (b, c), therefore the y-coordinate for point R is c.
The non-vertical sides are equal. Since the horizontal distance between (0,0) and (b, c) is b, then it means that the horizontal distance between point R and (a, 0) is b. So, the x-coordinate of R is (a - b).
Therefore we conclude that the coordinates for point R are (a - b, c).
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Find the area of a circle if the radius is 5 feet.
78.5 ft2
31.4 ft2
72 ft2
15 ft2
The area of the circle is obtained as 78.50 square feet. The correct option is (A).
What is a circle?A circle is a geometric shape, all of which points are equidistant from a fixed point called as the Centre.
The Centre of the circle does not lie on it.
Given that,
The radius of the circle is 5 feet.
The expression for the area of the circle is given as πr².
Substitute the value of r and take π = 3.14 to obtain,
Area = πr²
= 3.14 × 5²
= 3.14 × 25
= 78.50
Hence, the area of the circle with given radius is obtained as 78.50 square feet.
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It is believed that the best angle to fly a kite is 45°. If you fly a kite at this angle and le out 225 feet of string, approximately how high above the ground will the kite be?
159 feet
170 feet
223 feet
200 feet
The approximate height of the kite from the ground will be 159 feet
The best angle to fly the kite = 45 degrees
Th length of the string = 225 feet
To find the height above the ground to the kite, we have to use the trigonometric function
sin θ = Opposite side / Hypotenuse
The angle θ = 45 degrees
The length of the hypotenuse = 225 feet
Substitute the values in the equation of sin θ
sin 45 = Opposite side / 225
Opposite side = 225 × sin 45
= 159 feet
Therefore, the height of the kite from the ground is 159 feet
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There were 25 basketball players from three cities in the area who played in the all star game. If Georgetown had 12 all star players and Marketville had 11 all star players, how many all star players came from Claremont?
Using mathematical operations, we know that the all-star players of Claremont are 2.
What are mathematical operations?Calculating a value using operands and a math operator is referred to as performing a mathematical "operation."
The math operator's symbol has predetermined rules that must be applied to the supplied operands or numbers.
Mathematical operations include addition, subtraction, multiplication, division, and finding the root.
So, we know that 25 players in 3 cities.
1 city has 12 players.
Another city has 11 players.
So, the number of players that the 3rd city will have:
12 + 11 + x = 25
23 + x = 25
x = 25 - 23
x = 2
Therefore, using mathematical operations, we know that the all-star players of Claremont are 2.
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Sydney is designing a new board game, and is trying to figure out all the possible
outcomes. How many different possible outcomes are there if she spins a spinner
with four equal-sized sections labeled Red, Green, Blue, Orange and flips a coin?
Please help
Answer:
8
Step-by-step explanation:
For each different outcome of the spinner, there are 2 different outcomes of the coin flip.
4 × 2 = 8
Answer: 8
In Exercises 1 and 2, complete the proof.
1. Given PQ=RS
Prove PR=QS
2. Given ∠1 is a complement of ∠2.
∠2≅∠3
Prove ∠1 is a complement of ∠ 3
In Exercises 3-6, name the property that the statement illustrates.
3. If PQ≅ ST and ST≅UV, then PQ≅UV
1. For the given line segment if PQ = RS then it is proved that PR = QS .
2. From the given angles if ∠1 is complement to ∠2 (∠1 + ∠2 = 90° ) then angle 1 is complement to angle 3.
As given in the question,
1. Given : PQ = RS
To prove : PR = QS
Proof:
PQ = RS ( given )
Add both the side QR we get,
PQ + QR = QR + RS
From the diagram, it is given P - Q - R - S are collinear points
PR = QS ( PQ + QR = PR and QR + RS = QS)
Hence, PQ = RS implies that PR = QS.
2. Given: ∠1 is a complement of ∠2
∠2 ≅ ∠3
To prove : ∠1 is a complement of ∠3
proof: ∠1 is a complement of ∠2
⇒∠1 + ∠2 = 90°
⇒∠1 + ∠3 = 90° ( given ∠2 = ∠3 )
⇒∠1 is a complement of ∠3.
Hence, it is proved that ∠1 is a complement of ∠2 implies that ∠1 is a complement of ∠3.
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olve the equation 2x + 3y = 5 for x.
A. x equals negative 3 times y plus 5 over 2
B. x equals negative 3 over 2 times x plus 5
C. x equals the quantity negative 3 times y plus 5 end quantity over 2
D. x equals the quantity 3 times y plus 5 end quantity over 2
By solving a linear equation, it can be calculated that
x = [tex]\frac{-3y + 5}{2}[/tex]
Option A is correct
What is a linear equation?
At first, it is important to know about algebraic expression.
Algebraic expression consist of variables and numbers connected with addition, subtraction, multiplication and division.
Now we need to know about equation
Equation shows the equality between two algebraic expressions by connecting the two algebraic expressions by an equal to sign.
A one degree equation is known as linear equation.
Linear equation is a very important tool in mathematics
The given linear equation is
2x + 3y = 5
Now,
2x + 3y = 5
2x = -3y + 5
x = [tex]\frac{-3y + 5}{2}[/tex]
Option A is correct
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14. What is the ordered pair for point 7
A-045)
B.(4.50)
C. 5,0
D. 0,5
Pls helpp i really need help pleaseee
Answer:
P/mh
Step-by-step explanation:
P = mgh to solve for g Divide both sides by mh
[tex]\frac{P}{mh}[/tex] = [tex]\frac{mgh}{mh}[/tex] On the right side of the equation the m and h cancel out and you are left with
[tex]\frac{P}{mh }[/tex] = g
Use the drawing tool(5) to form the correct answer on the provided number line.
Draw a line segment with a length of 1.5 and a midpoint of -0.5
A line segment with a length of 1.5 and a midpoint of -0.5 is given below:
In the given question, we have to use the drawing tool to form the correct answer on the provided number line.
We have to draw a line segment with a length of 1.5 and a midpoint of -0.5.
As we know that;
Line section on a number line is referred to as a number line segment. A line segment varies from the a lines because a line continues infinitely in both directions, while a line segment is limited by end points. The shortest distance between any two points is a section of a line.
So a line segment with a length of 1.5 and a midpoint of -0.5 is given below:
A line segment a length of 1.5 and a midpoint of -0.5 is notify in the color of red in the graph below.
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Suppose that you have $4000 to invest. Which investment yields the greater return over a 10 year period: 7.83% compounded daily or 7.9% compounded quarterly?
Find the total amount of the investment after 10 years if $4000 is invested at 7.83% compounded daily.
The investment yields a greater return over a 10-year period if he invested 7.83% compounded daily.
What is compound interest?A loan or deposit's interest is computed using the starting principle and the interest payments from the ago decade as compound interest.
We know that the compound interest is given as
A = P(1 + r)ⁿ
Where A is the amount, P is the initial amount, r is the rate of interest, and n is the number of years.
Suppose that you have $4000 to invest.
The total amount of the investment after 10 years in each case will be given as,
At = r = 7.83% compounded daily, then we have
[tex]\rm A = \ \$4,000 \times \left (1 + \dfrac{0.0783}{365} \right)^{10\times 365}[/tex]
A = $4,000 × (1.000214521)³⁶⁵⁰
A = $4,000 × 2.1878
A = $8,751.37
At = r = 7.9% compounded quarterly, then we have
[tex]\rm A = \ \$4,000 \times \left (1 + \dfrac{0.079}{4} \right)^{10\times 4}[/tex]
A = $4,000 × (1.01975)⁴⁰
A = $4,000 × 2.186
A = $8,745.98
The investment yields a greater return over a 10-year period if he invested 7.83% compounded daily.
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in a recent year, 34.1% of all registered doctors were female. If there were 56,600 female registered doctors that year, what was the total number of registered doctors?
Round your answer to the nearest whole number.
324309
doctors were registered that year.
Explanation:
The questions says, "18.1% of all registered doctors are females"
(Consider, total number of registered doctors as
x
)
that's,
18.1
%
of
x
are female.
and
18.1
%
of
x
is
58
,
700
.
Mathematically,
18.1
100
⋅
x
=
58700
x
=
58700
⋅
100
18.1
x
=
324309
doctors were registered that year.
Please help ASAP!!
50 Points + Brainliest if correct!
(Also, I know that my answer is incorrect)
Answer:
5
Step-by-step explanation:
We have f(x) = 10x^3 and g(x) = 4x^(8/3) given. We need to find f/g, evaluating at x = 8. So how do we do this?
First, let's put our x value into the function f. f(8) = 10*8^3 = 5120.
Next, let's put our x value into the function g. g(8) = 4*8^(8/3) = 1024.
Now we just need to find f/g, which is just the result from our function f divided by our result from function g. This is f(8) / g(8). Substituting our solutions in, we have 5120/1024 = 5.
Answer:
5120/1024 = 5
Step-by-step explanation:
5120/1024 = 5
Helppp me reallll fast
Answer:
Reflection over the line y=x
Measure the angles of ABC and its three images, and record the measurements in the table. Round each answer to the nearest degree.
Type your response here:
ABC
Measure
A'B'C'
Measure
A'1B'1C'1
Measure
A'2B'2C'2
Measure
ABC
A'B'C'
A'1B'1C'1
A'2B'2C'2
ACB
A'C'B'
A'1C'1B'1
A'2C'2B'2
CAB
C'A'B'
C'1A'1B'1
C'2A'2B'2
The approximate measures of the angles of ΔABC and the images formed by translation, reflection, and rotation of ΔABC, which are rigid transformations are;
ΔABC [tex]{}[/tex] ΔA'B'C' ΔA''B''C'' ΔA'''B'''C'''
∠A = [tex]{}[/tex] 22° ∠A' = 22° ∠A'' = 22° ∠A''' = 22°
∠B = [tex]{}[/tex] 120° ∠B' = 120° ∠B = 120° ∠B''' = 120°
∠C [tex]{}[/tex] = 38° ∠C' = 38° ∠C = 38° ∠C''' = 38°
What is a rigid transformation?A rigid transformation, also known as congruence transformation, in geometry, is one in which the length of the pre-image is preserved, such that the image and the pre-image have the same side lengths, and are therefore congruent.
The coordinates of the vertices of triangle ΔABC, obtained from a similar question online are;
A(-6, -1), B(-3, -3), C(-1, -2)
The lengths of the sides of triangle ΔABC are;
AB = √((-3 - (-6))² + (-3 - (-1))²) = √(13)
BC = √((-1 - (-3))² + (-2 - (-3))²) = √5
AC = √((-1 - (-6))² + (-2 - (-1))²) = √(26)
The law of cosines can be used to fined the measure of an angle of the triangle as follows;
Let a represent the side BC, let b represent the side AC, and let c represent the side AB, the formula for the law of cosines is presented as follows;
a² = b² + c² - 2·b·c·cos(A)
Where;
∠A = The angle at the vertex A
Therefore;
(√5)² = (√(26))² + (√(13))² - 2 × √(26) × √(13) × cos(A)
5 = 26 + 13 - 2 × √(26) × √(13) × cos(A)
cos(A) = (5 - (26 + 13))÷ (-2 × √(26) × √(13))
∠A = arccos((5 - (26 + 13))÷ (-2 × √(26) × √(13))) ≈ 22°
Similarly, we get:
cos(B) = (26 - (5 + 13))÷ (-2 × √(5) × √(13))
∠B = arccos((26 - (5 + 13))÷ (-2 × √(5) × √(13))) ≈ 120°
cos(C) = (13 - (26 + 5))÷ (-2 × √(5) × √(26))
∠C = arccos((13 - (26 + 5))÷ (-2 × √(5) × √(26))) ≈ 38°
The transformation from ΔABC to ΔA'B'C', obtained from a similar question online, is a translation 5 units to the right and 2 units up
The transformation from ΔA'B'C' to ΔA''B''C'' is a reflection across the line y = -x
The transformation from ΔA''B''C'' to ΔA'''B'''C''' is a counterclockwise rotation by 270° about the origin
The translation, reflection, and rotation transformations are rigid transformations, such that the shape and sizes of the pre-images are preserved, and we get;
[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\Delta ABC &Measure & \Delta A'B'C' &Measure & \Delta A''B''C'' & Measure & \Delta A'''B'''C''' & Measure \\\angle ABC & 120^{\circ} & \angle A'B'C' & 120^{\circ} &\angle A''B''C'' &120^{\circ} &\angle A'''B'''C''' &120^{\circ} \\\angle ACB &38^{\circ} & \angle A'C'B' & 38^{\circ} &\angle A''C''B'' &38^{\circ} &\angle A'''C'''B''' &38^{\circ} \\\\end{bmatrix}[/tex]
[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\Delta ABC &Measure & \Delta A'B'C' &Measure & \Delta A''B''C'' & Measure & \Delta A'''B'''C''' & Measure \\\angle CAB & 22^{\circ} &\angle C'A'B' & 22^{\circ} &\angle C''A''B'' &22^{\circ} &\angle C'''A'''B''' &22^{\circ} \end{bmatrix}[/tex]
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Write a polynomial to this
The polynomial of degree 3 in factored form is calculated and of the form: y = x(x + 3)(x - 10) + 210
How to write the polynomialInformation given in the question include
degree 3
leading coefficient 1
x intercept (10, 0) and (-3, 0)
y intercept (0, -210)
The polynomial is written starting from the x intercept
x = 10, (x - 10)
x = -3, (x + 3)
(x + 10)(x - 10)
y = (x + 3)(x - 10)
for a polynomial of degree three and leading coefficient 1
y = x(x + 3)(x - 10)
for a y intercept (0, -210)
y = x(x + 3)(x - 10)
- 210 = 0
y = x(x + 3)(x - 10) + 210
The factored form of the polynomial is y = x(x + 3)(x - 10) + 210
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What do the following equations represent in R3?
Match the two sets of letters:
a. a vertical plane
b. a horizontal plane
c. a plane which is neither vertical nor horizontal
A. −5x+3y=−5
B. x=3
C. y=−9
D. z=−10
A. −5x+3y=−5=a vertical plane
B. x=3=a =a vertical plane
C. y=−9=a=a vertical plane
D. z=−10=b=a horizontal plane
What is a vertical plane?A plane that crosses a vertical line: a perspective plane that is parallel to both the ground plane and the image and passes through the point of sight.
What is a horizontal plane?The body is divided into superior and inferior halves by an anatomical plane called the transverse plane, also referred to as the horizontal plane, axial plane, and transaxial plane. Its angle with the coronal and sagittal planes is perpendicular.
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I need help with my math homework
Answer: for the first one, the one that says 7 x 5 = 5 x 7, as the blank is 7.
for the second one, 10 x 11 = 11 x 10, the blank is 10
for the third one, which is Julia's candy, draw 6 pieces of candy in each box and do 4 x 6, which is 24 pieces of candy [4 being the amount of bags].
for the fourth one, draw 5 pieces of candy in each box, then do 5 x 6 = 30.
for the fifth one, it is 4 x 6 = 6 x 4
for the final box, it is 5 x 6 = 6 x 5
Step-by-step explanation:
math
Answer: Hope this helps..
Step-by-step explanation:
7 * 5 - 5 = 30
10 * 11 - 11 = 99
For the candy questions the answers are:
Julia has 24 pieces of candy.
Tommy has 30 pieces of candy.
The equation for Julia is: 4 * 6 = 24
The equation for Tommy is: 5 * 6 = 30
You are ordering shirts for the math club at your school. Short-sleeved shirts cost $8 each. Long-sleeved shirts cost $16 each. You have a budget of $300 for the shirts. The equation $8x+16y=160$ models the total cost, where $x$ is the number of short-sleeved shirts and $y$ is the number of long-sleeved shirts. a. Graph the equation. Interpret the intercepts
The graph equation has been plotted on the graph which crosses the points (0, 10) and (20, 0) on the y and x axis.
What is a graph equation?Write the equation in the form y = MX + b to determine the slope m and the y-intercept.
This will allow you to graph the equation using the slope and y-intercept (0, b).
The point-slope form, standard form, and slope-intercept form are the three main types of linear equations.
So, we have the equation:
8x+16y=160
Where:
x represents the short sleeves.
y represents the long sleeves.
Now, we need to plot the equation as follows:
(Refer to the graph attached below)
So, the equation crosses the points (0, 10) and (20, 0) on the y and x axis.
Therefore, the graph equation has been plotted on the graph which crosses the points (0, 10) and (20, 0) on the y and x axis.
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i need to find da slope
Answer:
Slope is -1
Step-by-step explanation:
-5/5= -1
I am not sure if you want an explanation or just the answer, but I am willing to explain it if needed.
Please. Help quickly
Answer: it is a
Step-by-step explanation:
Elyse is going to buy an around-the-world plane ticket. She will get to see just one city from each continent she will visit. The following table below shows the continents on her trip and how many cities she can choose from for each continent.
Continent Number of cities
Europe 101010
Asia 888
Africa 444
Australia 333
How many different combinations of cities does Elyse have to choose from
Elyse has the option of combination of 960 cities, or another to choose from. If Elyse is going to buy an around-the-world plane ticket. She will get to see just one city from each continent she will visit.
Define permutation and combination.Combination and permutation are two alternative strategies in mathematics to divide up a collection of components into subsets. The subset's components can be listed in any order when combined. The components of the subset are listed in a permutation in a certain order.
Given,
Elyse is going to buy an around-the-world plane ticket. She will get to see just one city from each continent she will visit. The following table below shows the continents on her trip and how many cities she can choose from for each continent.
To determine how many options Elyse will have, we use permutations and combinations.
We can say that she has a choice of any of the eight cities in Asia for every ten cities in Europe. Similar to how she can select any of the 4 cities in Africa for every 8 cities in Asia. The same is true for Australia and Africa.
Elyse has a choice of a total:
10×8×4×3
960
Elyse has the option of combination of 960 cities, or another.
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4 - (2x + 4) = 5
Solve for x
Shayla’s family room measures 20 ft by 17 ft. There is an area rug centered in the room, 3 ft away from each wall. Shayla wants to pay a company to steam clean just the area outside the rug.
If they charge $10 per square foot, how much will Shayla have to pay?
Shyla pay $2380 to clean the area outside the rug.
What is steam clean?
To use a device that generates hot steam to clean (something).
Given:
Shayla’s family room measures 20 ft by 17 ft.
There is an area rug centered in the room, 3 ft away from each wall.
So, the remaining area of the room will be 17 ft by 14 ft.
Shayla wants to pay a company to steam clean just the area outside the rug.
The total area of the room outside the rug is
17 ft x 14 ft = 238 square feet.
The charge for one square foot is $10.
So, the charge for 238 square foot will be
238 x $10 = $2380.
Hence, Shyla pay $2380 to clean the area outside the rug.
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please help (question above)
The solution of the line is, (3, 2), (-1, 1), (1, 3/2), and the equation of the line is y = 1 / 4 x + 5 / 4, so options A, B, C, E, and F are correct.
What is a line?A line is a shape with an infinite length but no width, depth, or curvature. Lines are one-dimensional objects even though they can exist in two, three, or higher-dimensional contexts.
Given:
The graph of linear equation,
As you can see from the graph the line has coordinates of (3, 2), (-1, 1), (1, 3/2).
Calculate the equation of the line by using the coordinates as shown below,
[tex]y - y_1 = [y_2 - y_1 / x_2 - x_1] (x- x_1)[/tex]
y - 1 = [1 - 2 / -1 - 3] [x-(-1)]
y - 1 = -1 / (-4) (x + 1)
4y - 4 = x + 1
y = 1 / 4 x + 5 / 4
And line can have many solutions, as it can have more than two values.
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Which shape DOES NOT have the same perimeter as the other shapes?
The diameters of ball bearings are distributed normally . The mean diameter is 81 millimeters and the variance is 16. Find the probability that the diameter of a selected bearing is greater than 85 millimeters . Round your answer to four decimal places .
The required probability is approximately 0.0228 that the diameter is greater than 85 millimeters.
The probability that the diameter of a selected bearing is greater than 85 millimeters can be found by calculating the z-score for 85 millimeters and then using the standard normal table to find the corresponding probability.
First, we need to standardize the value 85 millimeters by subtracting the mean diameter and dividing by the standard deviation:
z = (85 - 81) / (4 / sqrt(16)) = 4 / 2 = 2
Next, we can use the standard normal table to find the probability that a randomly selected bearing has a diameter greater than 85 millimeters.
Since the distribution is symmetric, we can find the probability that the diameter is less than 85 millimeters and then subtract this probability from 1 to find the probability that the diameter is greater than 85 millimeters. Looking up the z-score 2 in the standard normal table, we find that the probability that the diameter is less than 85 millimeters is approximately 0.9772.
Therefore, the probability that the diameter is greater than 85 millimeters is approximately 1 - 0.9772 = 0.0228.
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Jesus and Isabella spent $3043.51 to buy a new computer together. Jesus spent $438.47 more than Isabella. How much did Jesus spend and how much did Kara spend?
Answer:
Isabella spent $1302.52 and Jesus spent $1740.99.
Step-by-step explanation:
In order to figure out how much each individual person spent, we need to write a mathematical equation to express this situation. An equation in math is a statement that relates two expressions that are equal in value, and can be used to solve for variables. A linear equation with one variable is often presented in is standard form, [tex]ax+b=0[/tex].
In this case, we can represent the amount that Isabella spent with the variable x. Since Jesus spent $438.47 more than Isabella, this amount can be represented with x + 438.47. We can set up our equation as follows:
[tex]x+(438.47+x)=3043.51[/tex]
*Note that the use of parentheses is not necessary but may be useful in making your work clearer!
To solve for x, we should combine like terms and then move the variables to one side and constants to the other. The variable can be isolated by subtracting 438.47 from 3043.51:
[tex]2x+438.47=3043.51[/tex][tex]2x=2605.04[/tex]Now that we have the variable by itself, we can divide by 2 on both sides to solve for x.
[tex]x=1302.52[/tex]As we defined, x equals the amount of money that Isabella spent, which is $1302.52. We can plug that number into x + 438.47 to get the amount that Jesus spent, which we calculate to be $1740.99.
Thus, Isabella spent $1302.52 and Jesus spent $1740.99.
To check our work, we can simply add up the two amounts to see if it totals to $3043.51:
$1302.52 + $1740.99 = $3043.51Learn more about solving equations with a variable here:
https://brainly.com/question/10839449
the function f(x) = x(x+2)^3 has the first derivative f'(x) = (4x+2)(x+2)^2 and second derivative f"(x) = 12(x+1)(x+2).
A) Find the intervals where f(x) is increasing or decreasing.
B) Find the values of x where f(x) has local maximum and local minimum values (if there are any).
C) Find the intervals of concavity and the (x,y) coordinates of inflection points (if there are any).
D) Use the info from parts a - c to sketch the graph.
Answer:
A) Increasing: (-0.5, ∞)
Decreasing: (-∞, -2) ∪ (-2, -0.5)
B) x = -¹/₂
C) Concave up: (-∞, -2) ∪ (-1, ∞)
Concave down: (-2, -1)
Points of inflection: (-2, 0) and (-1, -1)
D) See attachments.
Step-by-step explanation:
Given function and derivatives:
[tex]f(x) = x(x+2)^3[/tex]
[tex]f'(x) = (4x+2)(x+2)^2[/tex]
[tex]f''(x) = 12(x+1)(x+2)[/tex]
Part A[tex]\textsf{A function is \textbf{increasing} when the \underline{gradient is positive}}\implies f'(x) > 0[/tex]
[tex]\textsf{A function is \textbf{decreasing} when the \underline{gradient is negative}} \implies f'(x) < 0[/tex]
Increasing function
[tex]\begin{aligned}f'(x) & > 0\\(4x+2)(x+2)^2 & > 0\\2(2x+1)(x+2)^2 & > 0\\(2x+1)(x+2)^2 & > 0\end{aligned}[/tex]
Therefore, the function is increasing on the Interval:
[tex]\left(-\dfrac{1}{2}, \infty \right)[/tex]
Decreasing function
[tex]\begin{aligned}f'(x) & < 0\\(4x+2)(x+2)^2 & < 0\\2(2x+1)(x+2)^2 & < 0\\(2x+1)(x+2)^2 & < 0\end{aligned}[/tex]
Therefore, the function is decreasing on the Interval:
[tex](- \infty, -2) \cup \left(-2,-\dfrac{1}{2} \right)[/tex]
Part BLocal minimum/maximum points occur when f'(x) = 0.
[tex]\begin{aligned}f'(x) &= 0\\(4x+2)(x+2)^2 & = 0\\2(2x+1)(x+2)^2 & = 0\\(2x+1)(x+2)^2 &= 0\\\\2x+1&=0 \implies x=-\dfrac{1}{2}\\(x+2)^2&=0 \implies x=-2\end{aligned}[/tex]
[tex]f''\left(-\dfrac{1}{2}\right) = 12\left(-\dfrac{1}{2}+1 \right)\left(-\dfrac{1}{2}+2 \right)=9 > 0\implies \textsf{minimum}[/tex]
[tex]f''\left(-2\right) = 12\left(-2+1 \right)\left(-2+2 \right)=0\implies \textsf{point of inflection}[/tex]
Therefore, the value of x where f(x) has a local minimum is x = -¹/₂.
Part CAt a point of inflection, f''(x) = 0.
[tex]\begin{aligned}f''(x) &=0\\12(x+1)(x+2)&=0\\(x+1)(x+2)&=0\\\\x+1&=0 \implies x=-1\\x+2&=0 \implies x=-2\end{aligned}[/tex]
Substitute the found values of x into the original function to the find the y-coordinates of the points of inflection:
[tex]\implies f(-1) = -1(-1+2)^3=-1[/tex]
[tex]\implies f(-2) = -2(-2+2)^3=0[/tex]
Therefore, the inflection points are:
(-2, 0) and (-1, -1)A curve y = f(x) is concave up if f''(x) > 0 for all values of x.
A curve y = f(x) is concave down if f''(x) < 0 for all values of x.
Concave up
[tex]\implies f''(x) > 0[/tex]
[tex]\implies 12(x+1)(x+2) > 0[/tex]
[tex]\implies (x+1)(x+2) > 0[/tex]
Therefore, the function is concave up at:
[tex]\left(-\infty,-2\right) \cup \left(-1,\infty \right)[/tex]
Concave down
[tex]\implies f''(x) < 0[/tex]
[tex]\implies 12(x+1)(x+2) < 0[/tex]
[tex]\implies (x+1)(x+2) < 0[/tex]
Therefore, the function is concave down at:
[tex](-2,-1)[/tex]
Part DSee attachment 1 for how to sketch the graph using the information from parts A-C.
See attachment 2 for the final sketch of the graph.