We will have that the expression that represents the problem is:
[tex]1200>(3+22)C+190\Rightarrow1200>25C+190[/tex][tex]\Rightarrow1010>25C\Rightarrow C<40.4[/tex]So, at most he will be able to carry 40 crates with him.
You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 8.4% compounded monthly.
The desired monthly yield at the retirement time will be equal to $565,714.28.
Compound Interest may be defined as the interest earned by the bank on the basis of principle and also accumulated interest which increases exponentially and not linearly with respect to time. In calculating compound interest, the amount earned at the end of first year becomes principle for the next year and so on. Compound interest can be calculated, annually, half-yearly or quarterly etc.
Time for which work is planned = 40 years, Principle = $4500 and APR = 8.4% = 0.084/12 = 0.007.
The value of n = 12 × 25 = 300
The amount can be calculated by the formula A = P/r [1 - (1 + r) ⁻ⁿ]
A = (4500/0.007) [1 - (1 + 0.007) ⁻³⁰⁰]
A = 642,857.14 [1 - 0.12]
A = 642,857.14 × 0.88
A = $565,714.28 which is required amount.
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Complete Question:
You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 8.4% compounded monthly. What monthly deposits are required to achieve the desired monthly yield at retirement?
4^5 x 4^3 x 4^2 x 4^3 / 4^2 x 4 x 4^2 simplify help
The simplification form of the given mathematical equation 4^5 x 4^3 x 4^2 x 4^3 / 4^2 x 4 x 4^2 is [tex]4^{8}[/tex]
In the given question, it is given
We know the simplification property of the division of numbers as,
Division in exponential form
[tex]\frac{a^{x} }{a^{y} }[/tex] = [tex]a^{x-y}[/tex] , and
Multiplication in exponential form
[tex]a^{x} . a^{y}[/tex] = [tex]a^{x+y}[/tex]
Similarly, we'll apply the same property to solve this question,
[tex]4^{5+3+2+3 + ( -2 -1 -2)}[/tex]
[tex]4^{5+3+2+3 - ( 2 + 1 + 2)}[/tex]
[tex]4^{13 - 5}[/tex]
[tex]4^{8}[/tex]
Hence, the simplification form of the given mathematical equation 4^5 x 4^3 x 4^2 x 4^3 / 4^2 x 4 x 4^2 is [tex]4^{8}[/tex]
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Which number line shows all the values of x that make the inequality - 3x +1 <7 true?A2-5-4--3-2-10123B.5in-4-3-2-1012.34С5-5-4-3-2-1012345D-4-3-2.12345
First let's solve the given inequality:-
[tex]\begin{gathered} -3x+1<7 \\ -3x<6 \\ x>-2 \end{gathered}[/tex]So the correct option is (D).
Simplify the expression by first transforming the radical to exponential form. Leave the answer in exact form as a radical or a power, not as a decimal approximation.
Answer:
[tex]\textsf{Radical form}: \quad \sqrt[4]{2}\\\\\textsf{Exponent form}: \quad 2^{\frac{1}{4}}[/tex]
Step-by-step explanation:
Given expression:
[tex]\sqrt{8} \div \sqrt[4]{32}[/tex]
[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies 8^{\frac{1}{2}} \div 32^{\frac{1}{4}}[/tex]
Rewrite 8 as 2³ and 32 as 2⁵:
[tex]\implies (2^3)^{\frac{1}{2}} \div (2^5)^{\frac{1}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 2^{\frac{3}{2}} \div 2^{\frac{5}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad a^b \div a^c=a^{b-c}:[/tex]
[tex]\implies 2^{\frac{3}{2}-\frac{5}{4}}[/tex]
[tex]\implies 2^{\frac{6}{4}-\frac{5}{4}}[/tex]
[tex]\implies 2^{\frac{1}{4}}[/tex]
[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}:[/tex]
[tex]\implies \sqrt[4]{2}[/tex]
Hi I need help with this! I completed most of it, but I need help with the last portion. Thank you!
The equation of the axis of symmetry is x = 2
Explanation:The equation of the axis of symmetry is the value of x that divides the parabola into equal halves (mirror images).
The value of x which gives the mirror images is the value of x coordinate of the vertex. The vertex is at (2, 0). The x coordinate = 2
From the graph, we can also see it occurred at x = 2
The axis of symmetry: x = x-coordinate of the vertex
The equation of the axis of symmetry is x = 2
-3-23x=-14(x-21)-15(x+33)
Find (x)
Please I keep messing up somewhere so please show step by step
The value of (x) that satisfy the equation → - 3 - 23x = - 14(x - 21) - 15(x + 33) is x = - 33.
What is Equation?
An equation is a mathematical statement with an 'equal to' symbol between two expressions that have equal values.
Given is the following equation -
- 3 - 23x = - 14(x - 21) - 15(x + 33)
The given equation is -
- 3 - 23x = - 14(x - 21) - 15(x + 33)
Simplifying for (x), we get -
- 3 - 23x = - 14x + 294 - 15x - 495
- 23x + 14x + 15x = 3 + 294 - 495
6x = - 198
x = - 33
Therefore, the value of (x) that satisfy the equation → - 3 - 23x = - 14(x - 21) - 15(x + 33) is x = - 33.
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I need help with part b.
log ( √2.86668684 ) is value of logarithm 18 .
What is logarithm used for in math?
In order to get another number, a number must be raised to a certain power, which is known as a logarithm (see Section 3 of this Math Review for more about exponents). For instance, the base ten logarithm of 100 is 2, since ten multiplied by two yields the number 100: log 100 = 2.The answer is found using a logarithm (or log).[tex]log_{b} \sqrt{18}[/tex] = [tex]log_{b} = log_{b} \sqrt{15} + log_{b} \sqrt{3}[/tex]
= log( √15 + √3 )
= log ( √1.0986 * 1.6094 + √1.0986 )
= log ( √1.76808684 + √1.0986 )
= log ( √2.86668684 )
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write a point slope equation for the line that has a slope 5and passes the point (6,22).
Solution:
The general equation of a line of slope m passing through a point A is expressed as
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ \text{where} \\ (x_1,y_1)\text{ is the coordinate of the point A through which the line passes through} \end{gathered}[/tex]Given that the line has a slope of 5, and passes through the (6, 22), we have
[tex]\begin{gathered} m=5 \\ x_1=6 \\ y_1=22 \end{gathered}[/tex]thus,
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ \Rightarrow y-22=5(x-6) \end{gathered}[/tex]Hence, the point-slope equation for the line is expressed as
[tex]y-22=5(x-6)[/tex]
Helppppp!!!! Please!!
n > 39/4 is value of quartic equation.
What does quartic equation mean?
A fourth-degree equation, often known as a quartic equation, is one that reduces a quartic polynomial to zero and has the formula: where a 0. A quartic function's derivative is a cubic function.
For a quadratic ax² + bx +c , the sign of its determinant, given by
Δ = b²- 4ac
"determines" the nature of its roots. In particular, if Δ<0 , then the quadratic has two distinct non-real roots.
Now, we have
3z² - 9z = n - 3
3z² - 9z - (n - 3) = 0
with determinant
Δ = (-9)² - 4 .3( n - 3 ) = 117 - 12n
Solve for such that Δ < 0
Δ = 117 - 12n < 0 ⇒ 12n > 117
n > 117/12
n > 39/4
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I BEG YOU FOR HELP!!! Determine the relationship between the two triangles and whether or not they can be proven to be congruent.
Answer:
Step-by-step explanation:
The triangles are congruent because they follow the SSS triangle congruence postulate. As the SSS postulate says that all 3 sides of one triangle are congruent to another triangle's sides, these triangles shown have all three of their sides congruent to each other.
Answer: I am no sure but the only way to decide whether a pair of triangles are congruent would be to measure all of the sides and angles, and these triangles do not look the same so I would say that these tringles are not congruent.
Step-by-step explanation:
jamals lawn is shaped like a square with an area of 224.9 ft2. which measurement is closest to the side length of his lawn in feet?
Given:
Area of the square shaped lawn = 224.9 ft²
A square has all four side lengths equal.
To find the side length of this lawn, use the formula for area of a square below:
[tex]\text{Area = }L^2[/tex]Take the square root of both sides to find the sile length L:
[tex]\begin{gathered} \sqrt[]{Area\text{ }}=\sqrt[]{L^2} \\ \\ \sqrt[]{Area}\text{ = L} \\ \\ \sqrt[]{224.9}\text{ = L} \\ \\ 14.99\text{ ft = L} \end{gathered}[/tex]Therefore, the measurement that is closest to the side length in feet is 15 ft
ANSWER:
15 ft
Answer the questions below.
(a) Amount of data entered
The amount of data entered affects his pay at the end of the week.(b) R
The independent variable is the input.(c) Average temperature
The average temperature depends on the city's distance from the equator.In reference to 3x + 2 ≥-4Is 0 a solution?Yes or No?
ANSWER :
yes
EXPLANATION :
From the problem, we have the inequality :
[tex]3x+2\ge-4[/tex]To check if a number is a solution, substitute it to the inequality and check the truthfulness.
Check if 0 is a solution :
[tex]\begin{gathered} 3(0)+2\ge-4 \\ 2\ge-4 \end{gathered}[/tex]Since 2 is greater than -4, therefore 0 is a solution
Write a two-column proof.
4. Given: AB EF
AC DF
Prove: ABC ~ FED
Please help
Explanation:
The following is a proof that ∆ABC ~ ∆FED.
Statement . . . . Reason1. AB║EF, AC║FD . . . . given
2. ∠BCA ≅ ∠EDF . . . . alternate exterior angles theorem
3. ∠ABC ≅ ∠FED . . . . alternate interior angles theorem
4. ∆ABC ~ ∆FED . . . . AA similarity postulate
It took 12 men 5 hours to build an airstrip. Working at the same rate, how many additional men could have been hired in order for the job to have taken 1 hour less?
A) Two
B) Three
C) Four
D) Six
Answer:
B
Step-by-step explanation:
hello the question is if it took 12 men 5 hours to build an age is working at the same rate how many additional men could have been hired in order for the job to have taken 1 hour less ok so we have to find that how many extra may be required to complete the job for a strip 1 hour less than five hours that is 4 hours ok bye have to complete the airstrip in 4 hours and they have to find that how many experiment we have to required for that we will assume that let extra number of number of extra man bhi X show the number of men when we are finishing in it in 4 hours would be 12 + X ok
would be the number of men now and time required would be equal to 1 hour less than 5 hours that is 4 hours ok no from the given data we can say that one cares if strip x 12 men and fibres ok so all vacancy job correct so vacancy job per man are would be 1 divided by 12 in 25 this is the job or the amount of a strip that is completed when
one man works for one hour ok so this is the amount of the job that is done for men power and we have we have this number of men that are not want working and the number of hours that their working for so for one job we will need for one job would be best job per man per hour into number of men into number of hour ok and we have 1 equal to number of Doberman per Rs 1 by 2 11 25 and number of men we have already know that 12 + 6 is the number of men that we will require 12 + X number
forces were less than 5 that is 44 4312 so it would give us 12 + X / 3515 1 to 15 of this site it would give us 12 + X equal to 15 which implies X is equals to 15 - 12 and X is equals to 3 significant required 3 more men to complete the job in Porus dancer is 3 which is given is be in the question make you
Solve the system of inequalities by graphing.
y ≥ 2
y < 4
Select a line to change it between solid and dotted. Select a region to shade it.
The area between the solid line (y=x+4) and the dotted line (y=-2x-2) represents the system of inequalities solution set.
Two linear inequalities system on a coordinate plane. The first has a solid line graphed with a negative slope of one, a negative y-intercept, and a shaded origin area. The area encompassing the origin is shaded, and the second is a dashed vertical line 3 units to the left of the origin.
x ≥ –3; y ≥ x – 2
x > –3; 5y ≥ –4x – 10
x > –3; y ≥ –x + 1
x > –2; y ≥ –x – 1
The solution set for the system of inequalities is represented by the region between the solid line (y=x+4) and the dotted line (y=-2x-2).
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Which of the expressions are equivalent to the one below? Check all that apply.
( 15 • 3) - 20
Answer:
I don't know your answer choices but...
Step-by-step explanation:
( 15 • 3) - 20 is equal to:
45-20
25
(5)(3)(3)-20
2^3= 8 is equivalent to log, C = D.cand D
we have
2^3= 8
Applying log both sides
log(2^3)=log(8)
Apply property of log
3log(2)=log(8)
therefore
C=3 and D=log(8)
Suppose you buy two sweaters. Each sweater cost the same amount. The amount of money you spend varies directly with the number of sweaters you buy. If you spent $37.50 for two sweaters, what is the constant of variation? A. 2.00 B. 37.50 C. 0.053 D. 18.75
Answer:
D
Step-by-step explanation:
37.50 ÷ 2 = 18.75. jdjdjfbffcv
equati 1. v = 3.1 + 14 y = -1
y= 3x +14 (a)
y=-4x (b)
Replace the value of y (b) on equation (a), and solve for x
-4x = 3x+14
-4x-3x = 14
-7x = 14
x= 14/-7
x = -2
Replace the value of x on any initial equation and solve for y:
y= -4 (-2) = 8
y=8
Answer:
1*v=3,0.1+14y=-1 : v=3,y=-0.07857
[1.v=3}
[0.1+14y=-1]
v=3,y=-0.07857
Step-by-step explanation:
I need help with problem 7.Use the figure to find the values of x, y, and z that makes triangle DEF similar to triangle GHF.
ANSWER
• x = 12
,• y = 16
,• z = 7
EXPLANATION
Because the triangles are similar, we have that:
• The ratio between corresponding sides is constant:
[tex]\frac{DE}{GH}=\frac{EF}{GF}=\frac{DF}{HF}[/tex]• Corresponding angles are congruent:
[tex]\begin{gathered} \angle D\cong\angle H \\ \angle E\cong\angle G \\ \angle F\cong\angle F \end{gathered}[/tex]We know that the measure of angle E is 16°, so the measure of angle G must be the same because they are congruent,
[tex]16\degree=2(x-4)\degree[/tex]With this equation, we can find x. First, divide both sides by 2,
[tex]\begin{gathered} \frac{16}{2}=\frac{2(x-4)}{2} \\ \\ 8=x-4 \end{gathered}[/tex]And then, add 4 to both sides,
[tex]\begin{gathered} 8+4=x-4+4 \\ \\ 12=x \end{gathered}[/tex]Hence, x = 12.
Now we know that the length of side EF is,
[tex]EF=x-5=12-5=7[/tex]To find y and z, we will use the proportions we got at the top of this explanation,
[tex]\frac{DE}{GH}=\frac{EF}{GF}=\frac{DF}{HF}[/tex]Replace with the known values and the expressions with y and z,
[tex]\frac{25}{6z+8}=\frac{7}{14}=\frac{24}{3y}[/tex]With the first two, we can find z,
[tex]\frac{25}{6z+8}=\frac{7}{14}[/tex]Simplify the right side,
[tex]\frac{25}{6z+8}=\frac{1}{2}[/tex]Rise both sides to the exponent -1 - i.e. flip both sides of the equation,
[tex]\frac{6z+8}{25}=2[/tex]Multiply both sides by 25,
[tex]\begin{gathered} 25\cdot\frac{(6z+8)}{25}=2\cdot25 \\ \\ 6z+8=50 \end{gathered}[/tex]Subtract 8 from both sides,
[tex]\begin{gathered} 6z+8-8=50-8 \\ 6z=42 \end{gathered}[/tex]And divide both sides by 6,
[tex]\begin{gathered} \frac{6z}{6}=\frac{42}{6} \\ \\ z=7 \end{gathered}[/tex]Hence, z = 7.
Finally, with the last two proportions, we can find y,
[tex]\frac{7}{14}=\frac{24}{3y}[/tex]The first two steps are the same we did to find z: simplify the left side and flip both sides,
[tex]2=\frac{3y}{24}[/tex]Multiply both sides by 24,
[tex]\begin{gathered} 24\cdot2=24\cdot\frac{3y}{24} \\ \\ 48=3y \end{gathered}[/tex]And divide both sides by 3,
[tex]\begin{gathered} \frac{48}{3}=\frac{3y}{3} \\ \\ 16=y \end{gathered}[/tex]Hence, y = 16.
HELP!!
Triangle ABC is shown with exterior ∠z.
triangle ABC with angle A labeled 58 degrees, angle B labeled 44 degrees, and side AC extended with angle z labeled as exterior angle to angle C
Determine m∠z.
136°
102°
78°
58
The sum of all interior angles of a triangle is 180° thus the measure of the exterior angle m∠K is 102° so option (B) is correct.
What is a triangle?A triangle is a 3-sided shape that is occasionally referred to as a triangle. There are 3 sides and three angles in every triangle, some of which may be the same.
Triangle is a very common figure to deal with in our daily life.
It is known that the sum of all three angles inside a triangle will be 180°.
So, m∠A + m∠B + m∠C = 180°
m∠C = 180° - 58° - 44°
m∠C = 78°.
The exterior angle of C = 180- 78 = 102°.
Hence "The sum of all interior angles of a triangle is 180° thus the measure of the exterior angle to m∠K is 102°".
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Answer:
136
Using the exterior angle theorem,
Step-by-step explanation:
A true-false test contains 11 questions. In how many different ways can this test be completed. (Assume we don't care about our scores.)Your answer is :
Let's suppose that 1 = TRUE and 0 = FALSE, we want to find how many combinations we can do with 11 zeros and ones, in fact, it's:
[tex]\begin{gathered} \text{ 000 0000 0000} \\ \text{ 000 0000 0001} \\ \text{ 000 0000 0010} \\ \text{ 000 0000 0011} \\ ... \\ \text{ 111 1111 1111} \end{gathered}[/tex]To evaluate the number of combinations we can do:
[tex]C=2^{11}[/tex]2 because we can pick 2 different options (true or false) and 11 because it's the number of questions, then
[tex]\begin{gathered} C=2^{11} \\ \\ C=2048 \end{gathered}[/tex]We have 2048 different ways that this test can be completed.
please help with this practice question thank you
Answer:
Formula for slope of line is given as y2-y1 ÷ x2-x1 or y1-y2 ÷ x1-x2, where x and y are the coordinates of the points.
First, identify the coordinates of the two points shown on the graph.
First coordinate is (0,-1) and second coordinate is (3,1).
After that, find the slope of the line using the formula.
Slope = (1-(-1))÷(3-0)
= 2/3
Trying to find the length of the robot arm given the xy point along with angle theta and angle theta r
8a. The length of the robot arm, and the angle in radian and degrees are as follows
length of the robot arm = 30 unitsangle in radian = 7π/6 radian (anticlockwise)angle in degrees = 210 degrees (anticlockwise)8b. The length of the robot arm, and the angle in radian and degrees are as follows
length of the robot arm = 60 unitsangle in radian = π/6 radian (clockwise)angle in degrees = 210 degrees (clockwise)How to find the length of the robot arm and the angles8a
i. let the length of the robot arm be r
The length of the lever arm using the points is done by
r^2 = ( -26 )^2 + ( -15 )^2
r^2 = 676 + 225
r^2 = 900
r = √ 900
r = 30
ii. theta in radian
tan θ = opposite / adjacent
tan θ = y direction / x direction
tan θ = -15 / -26
tan θ = 15/26
θ = Arc tan ( 15/26 )
θ = 29.9816 degrees
θ ≈ 30 degrees
θ ≈ 30 degrees + 180 = 210 degrees
In radian
π = 180 degrees
? = 210 degrees
we cross multiply to get
? * 180 degrees = π * 210 degrees
? = π * 210 degrees / 180 degrees
= 7π/6 radian
8b
i. let the length of the lever arm be r
The length of the lever arm using the points is done by
r^2 = ( -30 )^2 + ( 52 )^2
r^2 = 900 + 2704
r^2 = 3604
r = √ 3604
r = 60.033
r ≈ 60
ii. theta in radian
tan θ = opposite / adjacent
tan θ = y direction / x direction
tan θ = -30 / 52
tan θ = -30/52
θ = Arc tan ( -30/52 )
θ = -29.9816 degrees
θ ≈ -30 degrees (clockwise direction)
In radian
π = 180 degrees
? = 30 degrees
we cross multiply to get
? * 180 degrees = π * 30 degrees
? = π * 30 degrees / 180 degrees
= π/6 radian
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Use y = 3x2 + 18x - 2 to answer the following question(1, 19) is a point on the graph. What point is the reflection of (1, 19) across the axis of symmetry of the parabola?
Since y is a parabolla, there will be two values for y = 19. We already know that x = 1 is one value, to find the other, we can substitute y = 19 on the equation and solve for x to get the following:
[tex]\begin{gathered} 19=3x^2+18x-2 \\ \Rightarrow3x^2+18x-2-19=0 \\ \Rightarrow3x^2+18x-21=0 \\ \Rightarrow3(x^2+6x-7)=0 \\ \Rightarrow3(x-1)(x+7)=0 \end{gathered}[/tex]the solutions of the equation are x = 1 and x = -7. Since we already have that (1,19) is a point on the graph, then we have that the other point is (-7,19)
Determine if the line passing through A(7,5) and B(-14, -9) is parallel, or perpendicular to the line passing through C(0,1) and D(4, -5).
To solve this problem, we will use the two pair of points to find the slope of the equation of each line. Then, by comparing these slopes, we can determine either if they are perpendicular or parallel.
Slope calculations
To calcula the slopes, given the pairs of points, we are going to use the following formula: Given points (a,b) and (c,d) the slope of the line that passes through them is given by the formula
[tex]m=\frac{d\text{ - b}}{c\text{ -a}}=\frac{b\text{ - d}}{a\text{ -c}}[/tex]Let us calculate first the slope of the line that passes through the points (7,5) and (-14,-9). In this case, we have a=7,b=5,c=-14 and d=-9. So we get
[tex]m=\frac{5\text{ - (-9)}}{7\text{ - (-14)}}=\frac{14}{21}=\frac{2\cdot7}{3\cdot7}=\frac{2}{3}[/tex]Now, let us calculate the slope of the line that passes through the points (0,1) and (4,-5). In this case, we have a=0,b=1,c=4 and d=-5. So we get
[tex]m=\frac{1\text{ -(-5)}}{0\text{ - 4}}=\frac{6}{\text{ -4}}=\text{ -}\frac{3\cdot2}{2\cdot2}=\text{ -}\frac{3}{2}[/tex]Slope comparison
Now, we compare the slopes to determine if the lines are perpendicular or parallel. Recall that two lines are parallel if they have the same slope and they are perpendicular if the product of their slopes is -1. From our calculations, we can see that the slopes are not equal. Let us confirm that they are perpendicular. To do so, we multiply both slopes. So we get
[tex]\frac{2}{3}\cdot(\text{ -}\frac{3}{2})=\text{ -1}[/tex]Since their product is -1, this confirms that both lines are perpendicular.
mr. Morales mix for 4 4/5 pound of macaroni and cheese and brings it to the 5th grade party. the kids ate 3/4 of the total amount that mr. Morales brought. he took the rest home then gave 3/4 of a pound of the macaroni and cheese to Mr. kang the next day. how many pounds of macaroni and cheese is left over for mr. Morales to eat
Convert the Mixed number to an Improper fraction:
- Multiply the Whole number by the denominator.
- Add the product to the numerator.
- Use the same denominator.
Then:
[tex]4\frac{4}{5}=\frac{(4)(5)+4}{5}=\frac{20+4}{5}=\frac{24}{5}[/tex]Then, the total amount of macaroni and cheese Mr. Morales brought was:
[tex]\frac{24}{5}lb[/tex]After the kids ate macaronis and cheese, the amount he took home was:
[tex]\frac{24}{5}lb-(\frac{24}{5}lb)(\frac{3}{4})=\frac{6}{5}lb[/tex]After he gave some macaroni and cheese to Mr. Kang the next day, the amount of macaroni and cheese (in pounds) left for mr. Morales to eat, is the following:
[tex]\frac{6}{5}lb-(\frac{6}{5}lb)(\frac{3}{4})=\frac{3}{10}lb[/tex]The answer is:
[tex]\frac{3}{10}lb[/tex]Graph the y = 2x ^ 2 - 12x + 15 Plot five points on the parabolathe vertextwo points to the left of the vertex, and two points to the right of the vertexThen click on the graph-a-function button
Explanation:
Given the function:
[tex]y=2x^2-12x+15[/tex]First, we find the vertex of the parabola.
Vertex
The equation of the axis of symmetry is calculated using the formula:
[tex]x=-\frac{b}{2a}[/tex]From the function: a=2, b=-12
[tex]\implies x=-\frac{-12}{2(2)}=\frac{12}{4}=3[/tex]Substitute x=3 into y to find the y-coordinate at the vertex.
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(3)^2-12(3)+15 \\ =18-36+15 \\ =-3 \end{gathered}[/tex]The vertex is at (3, -3).
Two points to the left of the vertex
When x=2
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(2)^2-12(2)+15=8-24+15=-1 \\ \implies(2,-1) \end{gathered}[/tex]When x=1
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(1)^2-12(1)+15=2-12+15=5 \\ \implies(1,5) \end{gathered}[/tex]Two points to the right of the vertex
When x=4
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(4)^2-12(4)+15=32-48+15=-1 \\ \implies(4,-1) \end{gathered}[/tex]When x=5
[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(5)^2-12(5)+15=50-60+15=5 \\ \implies(5,5) \end{gathered}[/tex]Answer:
Plot these points on the graph: (3, -3), (2,-1), (1,5), (4,-1), and (5,5).
Please help me with my calculus homework, only question 3****
I would start by stating the Fundamental Theorem of Calculus which states that;
If a function f is continuous on a closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then
[tex]\int ^b_af(x)dx=F(b)-F(a)\text{ }[/tex]Let
[tex]\begin{gathered} f(x)=x^3-6x^{} \\ F(x)=\int f(x)dx=\int (x^3-6x)dx \end{gathered}[/tex]Recall that;
[tex]\int x^n=\frac{x^{n+1}}{n+1},n\ne-1[/tex]That implies that,
[tex]F(x)=\int (x^3-6x)dx=\int x^3dx-\int 6xdx=\frac{x^4}{4}-6(\frac{x^2}{2})=\frac{x^4}{4}-3x^2+C[/tex]Applying the Fundamental Theorem of Calculus, where a=0, b=3
[tex]\begin{gathered} \int ^3_0(x^3-6x)dx=F(3)-F(0) \\ F(3)=\frac{3^4}{4}-3(3)^2+C=\frac{81}{4}-27+C=-\frac{27}{4}+C \\ F(0)=\frac{0^4}{4}-3(0)^2+C=C \\ \Rightarrow\int ^3_0(x^3-6x)dx=-\frac{27}{4}+C-C=-\frac{27}{4} \end{gathered}[/tex]So the answer is -27/4