The term "altitude" is the same as "height of a triangle". It is perpendicular to the base. Since we can rotate the triangle to have any side be horizontal, there are effectively 3 possible bases. Hence, there are 3 heights. It all depends how you look at it.
Let h1, h2, and h3 be the three altitudes or heights.
Without loss of generality, we'll focus on the first two heights h1 and h2. Their respective bases are b1 and b2.
If we use b1 as the base, then the area is...
area = 0.5*base*height = 0.5*b1*h1
Similarly, the other base gives the area of:
area = 0.5*b2*h2
------------------------
Since both formulas refer to the same area (because we're working with the same triangle), we can set the expressions equal to one another
0.5*b1*h1 = 0.5*b2*h2
b1*h1 = b2*h2
Let's see what happens when b1 = b2, so,
b1*h1 = b2*h2
b1*h1 = b1*h2
b1h1 - b1h2 = 0
b1(h1 - h2) = 0
b1 = 0 or h1 - h2 = 0
b1 = 0 or h1 = h2
If the bases b1 and b2 were equal, then either those bases must be 0 which isn't possible, or the altitudes must be equal. However, the initial premise is that the heights must be different from one another.
Therefore, the bases b1 and b2 can't be the same length.
We could follow the same steps and logic to conclude that if the altitudes h1 and h3 were different, then the bases b1 and b3 can't be the same. Similarly, we would conclude that b2 and b3 can't be the same. This is where the "without loss of generality" kicks in.
In other words, we only need to focus on one subcase to extend the logic to the other cases, without having to actually do every single step. That would be a bit tedious busywork.
In conclusion, we've shown that if the heights are different, then their respective bases must be different. This leads to wrapping up the proof that we have a scalene triangle.
Side note: I used an indirect proof or proof by contradiction. I assumed that a non-scalene triangle was possible and it led to a contradiction of h1 = h2.
Polynomial functions can model situations that change directions multiple times. what is a situation in which a polynomial model might make sense and why?
A polynomial function exists a function that contains only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc.
What are the two types of functions?The variety of function kinds can model circumstances in the real world. Properties or key components of functions create more or less suitable models, relating to the circumstances.
Linear: These relationships maintain constant rates of change. The distance traveled by a car moving at a constant speed over time exists linear.
Exponential: Connections that show a percent change exist exponential. The growth of a financial investment accumulating interest over time exists exponential.
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Please help!! 100 points
The graph shows a system of inequalities.
Which point is a solution to the system
(-1,6)
(0,22)
(2,9)
(8,2)
Answer: (2,9)
Step-by-step explanation:
The point lies in the region that is shaded by both inequalities.
Divide. Write the quotient in lowest terms.
4 2/3 / 7
The quotient in lowest term exists 2/3.
What is an improper fraction?An improper fraction exists as a fraction whose numerator exists equivalent to, larger than, or of equivalent or higher degree than the denominator.
Given the quotient: [tex]$4\frac{2}{3} \div 7[/tex]
Write [tex]$\frac{2}{3}[/tex] in improper fraction
[tex]$4\frac{2}{3}= \frac{14}{3}[/tex]
Dividing throughout by 7 on both sides of the equation, we get
[tex]$4\frac{2}{3} \div 7 = \frac{14}{3}\div 7[/tex]
Change the division sign to multiplication by taking the reciprocal of 7
[tex]$\frac{14}{3}\div 7 = \frac{14}{3} \times \frac{1}{7}[/tex]
Simplifying the above equation, we get
[tex]$\frac{14}{3}\div 7 = \frac{2}{3}[/tex]
Therefore, the quotient in lowest term exists 2/3.
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Graph a line that contains the point (-3, 5) and has a slope of -2/5.
Answer:
y=-\frac{2}{5}x+\frac{19}{5}y=−52x+519
Further explanation:
We have to find the equation of the line first to graph the line.
The general form of slope-intercept form of equation of line is:
y=mx+by=mx+b
Given
m=-\frac{2}{5}m=−52
Putting the value of slope in the equation
y=-\frac{2}{5}x+by=−52x+b
To find the value of b, putting the point (-3,5) in equation
\begin{gathered}5=-\frac{2}{5}(-3)+b\\5=\frac{6}{5}+b\\5-\frac{6}{5}+b\\b=\frac{25-6}{5}\\b=\frac{19}{5}\end{gathered}5=−52(−3)+b5=56+b5−56+bb=525−6b=519
Putting the values of b and m
y=-\frac{2}{5}x+\frac{19}{5}y=−52x+519
If a sample of n = 4 scores is obtained from a normal population with µ = 70 and σ = 12. What is the z-score corresponding to a sample mean of m = 69?
The z-score corresponding to a sample mean of m = 69 is -0.167
In this problem, we have been given :
population mean (μ) = 70, standard deviation (σ) = 12, sample size (n) = 4, sample mean (m) = 69
We know that, the Z-score measures how many standard deviations the measure is from the mean.
Also, the formula when calculating the z-score of a sample with known population standard deviation is:
[tex]Z=\frac{m-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]
where z = standard score
μ = population mean
σ = population standard deviation
m = the sample mean
and [tex]\frac{\sigma}{\sqrt{n} }[/tex] is the Standard Error of the Mean for a Population
First we find the Standard Error of the Mean for a Population
σ /√n
= 12 / √4
= 12 / 2
= 6
So, the z-score would be,
⇒ [tex]Z=\frac{m-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]
⇒ [tex]Z=\frac{69-70}{6 }[/tex]
⇒ Z = -1/6
⇒ Z = -0.167
Therefore, the z-score corresponding to a sample mean of m = 69 is -0.167
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Factors to zero inverse operations
The zeros of the given equation are -5 and -7
Zeros of a quadratic equationQuadratic equations are equations that has a leading degree of 2. Given the factors of a quadratic equation as expressed below;
(-3x - 15)(x+7) = 0
The expressions -3x -15 and x + 7 are the factors of the equation. Equating both factors to zero
-3x - 15 = 0
Add 15 to both sides of the equation
-3x -15 + 15 = 0 + 15
-3x = 15
Divide both sides of the equation by -3
-3x/-3 = 15/-3
x = -5
Similarly;
x + 7 = 0
x = -7
Hence the zeros of the given equation are -5 and -7
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pls look at pic before help out confused
Answer:
Option 4
Step-by-step explanation:
By the Pythagorean identity, and the fact we are in the first quadrant,
[tex]\cos \theta=\frac{4}{5}[/tex]
Using the double angle formula for cosine,
[tex]\cos 2\theta=2\cos^{2} \theta-1=2\left(\frac{4}{5} \right)^2 -1=\frac{7}{25}[/tex]
The total mass of 2 similar clay pots and 2 similar metal pots was 13.2 kg. The mass of 1 such clay pot was 3 times the mass of a metal pot. What was the mass of a clay pot?
Answer:
mass of a clay pot = 4.95 kg
Kindly award branliest
Step-by-step explanation:
Let the mass of a clay pot be x
Let the mass of a metal pot be y
Thus; 2x + 2y = 13.2
And ;
x = 3 times y
x = 3y
2x + 2y = 13.2
2(3y) + 2y = 13.2
6y + 2y = 13.2
8y = 13.2
y = 13.2/8 = 1.65
x = 3y = 3(1.65) = 4.95
mass of a clay pot = 4.95 kg
To calculate the hourly revenue from the buffet after x $1 increases, multiply the price paid by each customer and the average number of customers per hour. Create an inequality in standard form that represents the restaurant owner’s desired revenue.
Type the correct answer in each box. Use numerals instead of words.
The inequality in the box has to be written as
x² + 2x - 80 ≤ - 65
How to solve the inequalityWe have
(10 + x)1 * (16-2x) ≥ 130
Next we would have to open the bracket
160 + 16x - 20x - 2x² ≥ 130
Then we would have to arrange the equation
- 2x² - 4x + 160 ≥ 130
Divide the equation by two
- x² - 2x + 80 ≥ 65
This is arranged as
x² + 2x - 80 ≤ - 65
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What is the step that comes after 3x(x+1)-5(x+1) when factoring by grouping?
Answer:
Separating the 3x and -5 apart from the (x+1)
Step-by-step explanation:
It would turn out to be (3x-5)(x+1) !
Factor out x+1 from the expression
(x+1) x (3x-5)
4/3 + -1/6 + 13/12. Please answer step by step if possible. Thanks.
Answer:
9/4
Step-by-step explanation:
We follow bodmas
4/3 +( -1/6 + 13/12)
( lcm = 12)
( -2+ 13/12)
( 11/ 12)
4/3 + ( 11/12)
4/3 + 11/12
lcm = 12 also
and that will equal to
=16 + 11/ 12
= 27/ 12
divide by 3 to simplest form
= 9/4
What is the sum of the geometric sequence -3, 18, -108,
if there are 8 terms?
Step-by-step explanation:
using gp formula
tn=ar^n-1
which our a which is the first term is = -3
our r which is t2/t1=-6
t8=(-3)(-6)^8-1
=(-3)(-6)^7
=(-3)(-279936)
=839808
14) Adelina is comparing prices for two brands of health and energy bars at the local grocery store.
She wants to get the best price for each bar. Feel Great energy bars are $18 for 12 bars. Super
Power bars cost $21.75 for 15 bars.
a) Write an equation to find the price for each bar of the Feel Great brand.
b) Write an equation to find the price of each bar for the Super Power brand.
c) Which bar should Adelina buy? Explain your reasoning.
The bar which Adelina should buy is super power brand because it cost less than feel great energy bar at $1.45 per bar
Unit priceFeel great energy bars = Total cost / number of bars
= $18 / 12
= $1.5 per bar
Super power brand = Total cost / number of bars
= $21.75 / 15
= $1.45 per bar
Therefore, the bar which Adelina should buy is super power brand because it cost less than feel great energy bar at $1.45 per bar
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Which number belongs to the set of rational numbers and the set of integers?
F. –5.5 H. –0.5
G. – 115 J. –15
The number which belongs to the set of rational numbers and the set of integers is -115 which is third option,-15 which is fourth option.
Given four options:
–5.5 –0.5– 115 –15We are required to find the number which is included in the set of rational numbers and the set of integers.
Rational numbers are those numbers which can be written in the form of p/q in which q cannot be equal to zero because if q becomes zero then the fraction becomes infinity.
-5.5 is not a rational number,
-0.5 is also not a rational number.
-115 is a rational number and also an integer.
-15 is a rationalnumber and also an integer.
Hence the number which belongs to the set of rational numbers and the set of integers is -115 which is third option,-15 which is fourth option.
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d. (x + y, 3x-2y) = (7,11)
Answer:
x = 5, y =2
Step-by-step explanation:
I guess the question is saying x+y = 7 and 3x-2y = 11?
then there are multiple ways but
I will multiply the first one by 2 so 2x+2y = 14
you add the equations to get 5x = 25 so x = 5 plug x into the first equation you get y = 2
if that isn't what the question means just comment and I'll change it
‼️‼️‼️‼️HELP‼️‼️‼️‼️‼️
A certain item is available at 7 stores. Three stores sell it for $20, two stores sell it for $15, one store sells it for $13, and one sells it for $16. What is the average (arithmetic mean) of the median price and the mode price?
The average median and mode price is $18. Option C is correct.
Given that certain item is available at 7 stores, three stores sell them for $20, two stores sell them for $15, one store sells them for $13, and one sells them for $16.
The mean is the average of the given numbers and is calculated by dividing the sum of the given numbers by the total number of numbers.
Firstly, we will find the median of the given items by arranging the given numbers in ascending order, we get
13,15,15,16,20,20,20
To find the median use the formula (n+1)/2, where n is the number of values in your dataset.
(7+1)/2=8/2=4
In the ascending order numbers 4th term is 16.
So, median is 16
Mode is the highest repeating term in the set or numbers.
So, here mode is 20
Now, we will calculate the average of median and mode, we get
Average=(median +mode)/2
Average=(16+20)/2
Average=18
Hence, the average (arithmetic mean) of the median price and the mode price where certain item is available at 7 stores, three stores sell it for $20, two stores sell it for $15, one store sells it for $13, and one sells it for $16 is $18.
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Let f(x) = [infinity] xn n2 n = 1. find the intervals of convergence for f. (enter your answers using interval notation. ) find the intervals of convergence for f '. find the intervals of convergence for f ''
Best guess for the function is
[tex]\displaystyle f(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}[/tex]
By the ratio test, the series converges for
[tex]\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{(n+1)^2} \cdot \frac{n^2}{x^n}\right| = |x| \lim_{n\to\infty} \frac{n^2}{(n+1)^2} = |x| < 1[/tex]
When [tex]x=1[/tex], [tex]f(x)[/tex] is a convergent [tex]p[/tex]-series.
When [tex]x=-1[/tex], [tex]f(x)[/tex] is a convergent alternating series.
So, the interval of convergence for [tex]f(x)[/tex] is the closed interval [tex]\boxed{-1 \le x \le 1}[/tex].
The derivative of [tex]f[/tex] is the series
[tex]\displaystyle f'(x) = \sum_{n=1}^\infty \frac{nx^{n-1}}{n^2} = \frac1x \sum_{n=1}^\infty \frac{x^n}n[/tex]
which also converges for [tex]|x|<1[/tex] by the ratio test:
[tex]\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{n+1} \cdot \frac n{x^n}\right| = |x| \lim_{n\to\infty} \frac{n}{n+1} = |x| < 1[/tex]
When [tex]x=1[/tex], [tex]f'(x)[/tex] becomes the divergent harmonic series.
When [tex]x=-1[/tex], [tex]f'(x)[/tex] is a convergent alternating series.
The interval of convergence for [tex]f'(x)[/tex] is then the closed-open interval [tex]\boxed{-1 \le x < 1}[/tex].
Differentiating [tex]f[/tex] once more gives the series
[tex]\displaystyle f''(x) = \sum_{n=1}^\infty \frac{n(n-1)x^{n-2}}{n^2} = \frac1{x^2} \sum_{n=1}^\infty \frac{(n-1)x^n}{n} = \frac1{x^2} \left(\sum_{n=1}^\infty x^n - \sum_{n=1}^\infty \frac{x^n}n\right)[/tex]
The first series is geometric and converges for [tex]|x|<1[/tex], endpoints not included.
The second series is [tex]f'(x)[/tex], which we know converges for [tex]-1\le x<1[/tex].
Putting these intervals together, we see that [tex]f''(x)[/tex] converges only on the open interval [tex]\boxed{-1 < x < 1}[/tex].
an alloy is made with 3 gram of silver 18 gram of copper 6 gram of aluminium and three Gram of zinc find what part of the total is used for each metal?
Answer:
see explanation
Step-by-step explanation:
total parts = 3 + 18 + 6 + 3 = 30
3 grams of silver = [tex]\frac{3}{30}[/tex] = [tex]\frac{1}{10}[/tex]
18 grams of copper = [tex]\frac{18}{30}[/tex] = [tex]\frac{3}{5}[/tex]
6 grams of aluminium = [tex]\frac{6}{30}[/tex] = [tex]\frac{1}{5}[/tex]
3 grams of zinc = [tex]\frac{3}{30}[/tex] = [tex]\frac{1}{10}[/tex]
someone help me out please
Answer:
[tex]\displaystyle x=\frac{5}{4},\;\;1\frac{1}{4}, \;\; or \;\; 1.25[/tex]
Step-by-step explanation:
To solve for x, we need to isolate the x variable.
Given:
[tex]\displaystyle x+\frac{1}{2} =\frac{7}{4}[/tex]
Subtract [tex]\frac{1}{2}[/tex] from both sides of the equation:
[tex]\displaystyle (x+\frac{1}{2})-\frac{1}{2} =(\frac{7}{4})-\frac{1}{2}[/tex]
[tex]\displaystyle x=\frac{7}{4}-\frac{1}{2}[/tex]
Now, we will create common denominators to simplify.
[tex]\displaystyle x=\frac{7}{4}-\frac{2}{4}[/tex]
[tex]\displaystyle x=\frac{5}{4}[/tex]
Slope=
Help me please thanks
Answer:
0
Step-by-step explanation:
In cartesian graphs, the slope or a gradient of a fully horizontal line will always be 0.
20 POINTS
The following are the ages of 15 music teachers in a school district. 24, 26, 27, 29, 29, 32, 37, 40, 40, 41, 45, 52, 56, 56, 58. Notice that the ages are ordered from least to greatest. Make a box-and-whisker plot for the data.
The box and whisker plot is plotted with minimum value 24, maximum value 58, first quartile 29, third quartile 52 and median 40.
Given that, the ages of 15 music teachers in a school district are 24, 26, 27, 29, 29, 32, 37, 40, 40, 41, 45, 52, 56, 56, 58.
A box and whisker plot also called a box plot displays the five-number summary of a set of data. The five-number summary is the minimum, first quartile, median, third quartile, and maximum. In a box plot, we draw a box from the first quartile to the third quartile. A vertical line goes through the box at the median.
Minimum value = 24
Maximum value = 58
First quartile = 29
Third quartile = 52
Median = 40
Therefore, the box and whisker plot is plotted with minimum value 24, maximum value 58, first quartile 29, third quartile 52 and median 40.
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For a population with = 100 and = 20, what is the x value corresponding to z = 1. 50?
The x value or observed value corresponding to z-score, z = 1.50 is 130.
According to the question.
For a population with µ = 100 and σ = 20.
Since, we know that
The z-score is a statistical evaluation of a value's correlation to the mean of a collection of values, expressed in terms of standard deviation.
And it is given by
z = (x - μ) / σ
Where,
x is the observed value.
μ is the mean.
and, σ is the standard deviation.
Therefore, the x value or observed value corresponding to z = 1.50 is given by
[tex]1.50 = \frac{x -100}{20}[/tex]
⇒ 1.50 × 20 = x - 100
⇒ 30 = x - 100
⇒ x = 30 + 100
⇒ x = 130
Hence, the x value or observed value corresponding to z-score, z = 1.50 is 130.
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Suppose that a, b, c, d are positive real numbers such that a/b < c/d . What can you say about the fraction (a+c)/(b+d) ? Is it always, sometimes, or never between the fractions a/b and c/d ? Provide evidence of your claim. (If always or never, give an algebraic proof, otherwise, give two quadruples values of a, b, c, d in which one quadruple has (a+c)/(b+d) between a/b and c/d and the other does not.)
Answer: (a−c)(b−c)>0
Step-by-step explanation:
ab>1 and ac<01. a>0 if c<0 and also b>02. a<0 if c>0 and also b<0
how i did it:
At the vert first, write the inequality as an equation.
Solve the provided equation for one or more values.
Now, display all the values obtained in the number line.
Use open circles to show the excluded values on the number line.
Find the interim.
At the moment, take any random value from the interval and substitute it in the inequality equation to check whether the values reassure the inequality equation.
Intervals that reassure the inequality equation are the solutions of the given inequality equation.
Use the laplace transform to solve the given initial-value problem. y' + y = (t − 1), y(0) = 5
Using the Laplace transform, the value of y' + y = (t − 1), y(0) = 5 is y(t) = 5e ^ -t + u (t - 1)e^(1-t)
Laplace rework is an critical rework approach that is in particular useful in fixing linear normal equations. It unearths very huge applications in regions of physics, electrical engineering, control optics, arithmetic and sign processing.
y' + y = (t − 1)
y (0) = 5
Taking the Laplace transformation of the differential equation
⇒sY(s) - y (0) + Y(s) = e-s
⇒(s + 1)Y(s) = (5+ e^-s)/s + 1
⇒y(t) = L^-1{5/s+1} + {e ^-s/s + 1}
⇒y(t) = 5 e^-t + u(t -1)e^1-t
The Laplace remodel method, the feature within the time area is transformed to a Laplace characteristic within the frequency domain. This Laplace feature will be inside the shape of an algebraic equation and it can be solved easily.
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How many nonzero terms of the maclaurin series for ln(1 x) do you need to use to estimate ln(1. 4) to within 0. 0001?
We need at least 7 terms of the Maclaurin series for ln(1 + x) to estimate ln 1.4 to within 0.0001
For given question,
We have been given a function f(x) = ln(1 + x)
We need to find the estimate of In(1.4) within 0.001 by applying the function of the Maclaurin series for f(x) = In (1 + x)
The expansion of ln(1 + x) about zero is:
[tex]ln(1+x)=x-\frac{x^2}{2} + \frac{x^3}{3} -\frac{x^4}{4} +\frac{x^5}{5} -\frac{x^6}{6} +.~.~.[/tex]
where -1 ≤ x ≤ 1
To estimate the value of In(1.4), let's replace x with 0.4
[tex]\Rightarrow ln(1+0.4)=0.4-\frac{0.4^2}{2} + \frac{0.4^3}{3} -\frac{0.4^4}{4} +\frac{0.4^5}{5} -\frac{0.4^6}{6} +.~.~.[/tex]
From the above calculations, we will realize that the value of [tex]\frac{0.4^5}{5}=0.002048[/tex] and [tex]\frac{0.4^6}{6}=0.000683[/tex] which are approximately equal to 0.001
Hence, the estimate of In(1.4) to the term [tex]\frac{0.4^6}{6}[/tex] is enough to justify our claim.
Therefore, we need at least 7 terms of the Maclaurin series for function ln(1 + x) to estimate ln 1.4 to within 0.0001
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Which of the following is true with respect to the following functions:
The option that is true with regard to the following functions is Option B. "The domain g(x) and h(x) include all real number while the domain of i(x) and h(x) are restricted"
What is the explanation for the above?Lets examine f(x) = 3x + 14Note that this function is indicative of a straight-line. See the attached graph for function 1. Note that it doesn't have any end points. That is, it is Asymptote.
Let us examine h(x) = 3ˣ + 1This represents an exponential graph. Just like the function above it doesn't have any end point. It however has an asymptote:
y = 0
Let us look at F'(x) = Log₃ (x = 1).
This is indicative of logarithm graph. It doesn't have any end but has an asymptote x == 0
Let us take a look at g(x) = X⁴ + 3x² - 14Notice that in the mid point there is an end point given as (0, -14). Thus, it is correct to state that the function in Option B is the only one that exhibits end behavior and as such is restricted.
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Given AQRS-AXYZ, what is the value of tan(Q)?
A) 3/5
B) 3/4
C) 4/5
D) 4/3
The answer is B.
Since ΔQRS ~ ΔXYZ, the value of tan(Q) is :
∠Q = ∠Xtan(Q) = tan(X)tan(X) = 3/4tan(Q) = 3/4Krissy ran three miles one morning she ran the first mile in 11. 74 minutes the second mile in 11. 26 minutes in the third mile in 12.12 minute rounded to the nearest hundredth what is the total number of minutes that it took krissy to run these three miles?
Answer:
Step-by-step explanation:
Givens
Time 1 = 11.74
Time 2 = 11.26
Time 3 = 12.12 Add
Solution
11.74 + 11.26 + 12.12 =
Total Time = 35.12
35.12 miles is the total number of minutes that it took krissy to run these three miles.
What is a simple definition of time?
The measured or measurable period during which an action, process, or condition exists or continues : duration. b : a nonspatial continuum that is measured in terms of events which succeed one another from past through present to future.
Krissy ran three miles one morning she ran the first mile in time 1 = 11.74
Krissy ran three miles one morning she ran the first mile in time 2 = 11.26
Krissy ran three miles one morning she ran the first mile in time 3 = 12.12
To get total time , we have to add all the time 1,2,3
Total time = Time1 + Time2 + Time3
= 11.74 + 11.26 + 12.12
Total Time = 35.12 miles
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Burger Barn makes a dipping sauce by mixing 4 spoonfuls of honey with 1 spoonful of mustard. Sandwich Town makes a dipping sauce by mixing 8 spoonfuls of honey with 2 spoonfuls of mustard
Which dipping sauce has a stronger mustard flavor?
The dipping sauce which has a stronger mustard flavor between burger barn and be sandwich town is burger barn
RatioBurger bun:
Honey = 4 spoonfulsMustard = 2 spoonfulsMustard : honey
= 2 : 4
= 2/4
= 1/2
= 0.5
Sandwich:
Honey = 8 spoonfulsMustard = 2 spoonfulsMustard : honey
= 2 : 8
= 2/8
= 1/4
= 0.25
Therefore, burger barn has a more stronger mustard flavor of dipping sauce between burger barn and be sandwich town.
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Answer:
Step-by-step explanation:
The two dipping sauce have same taste.
Approximate the area under the
function between a and b using a
left-hand sum with the given
number of intervals.
f(x) = x³
a=0
b=3
3 Intervals
Split up the interval [0, 3] into 3 equally spaced subintervals of length [tex]\Delta x = \frac{3-0}3 = 1[/tex]. So we have the partition
[0, 1] U [1, 2] U [2, 3]
The left endpoint of the [tex]i[/tex]-th subinterval is
[tex]\ell_i = i - 1[/tex]
where [tex]i\in\{1,2,3\}[/tex].
Then the area is given by the definite integral and approximated by the left-hand Riemann sum
[tex]\displaystyle \int_0^3 f(x) \, dx \approx \sum_{i=1}^3 f(\ell_i) \Delta x \\\\ ~~~~~~~~~~ = \sum_{i=1}^3 (i-1)^3 \\\\ ~~~~~~~~~~ = \sum_{i=0}^2 i^3 \\\\ ~~~~~~~~~~ = 0^3 + 1^3 + 2^3 = \boxed{9}[/tex]