Find the domain of the logarithmic function and then graph the function. y= In (4x - 5) Find the domain of the function.
Answers
Domain = ( 5/4 , ∞)
Explanations:The logarithm function is given as:
y = ln (4x - 5)
the domain of the functions is a set of all the values of x that makes the function true (defined)
Note that the natural logarithm of a negative number is undefined. This means that the natural logarithm (ln) is only defined for positive numbers
This means that in y = ln (4x - 5), 4x -5 has to be greater than zero for the function to be defined.
Set 4x - 5 > 0
4x > 5
x > 5 / 4
Therefore, the domain of the function is a set of values greater than 5/4
Domain = ( 5/4 , ∞)Related Questions
The graph of a piecewise function, f x( )is plotted in Figure 1. Discuss the continuity of f x( )at x =−1using the continuity test
Answers
the graph is not continuous at x=-1
[tex]\lim _{x\rightarrow-1^-}f(x)\ne\lim _{x\rightarrow-1^+}f(x)\ne f(-1)[/tex]The Weasley brothers estimate that they will need a thousand galleons just to set up their joke shop, Weasley's Wizarding Wheezes. The first products they are manufacturing are pygmy puffs, and they realize that the cost of making a single box of pygmy puffs is 3 galleons. The brothers would like to estimate the cost, in galleons, of setting up their shop and creating pygmy puffs, as a function of the number of boxes of pygmy puffs produced, p. Which of the following equations is the correct equation representing this scenario?
1. ()=3−1000
2. ()=3+1000
3. ()=1000−3
4. ()=1000+3
Answers
The most appropriate choice for linear equation will be given by -
Total cost of setting up their shop and creating pygmy puffs, as a function of the number of boxes of pygmy puffs produced, p
C(p) = 3p + 1000
Second option is correct
What is linear equation?
Equation shows the equality between two algebraic expressions by connecting the two algerbraic expressions by an equal to sign.
A one degree equation is known as linear equation.
Here,
Initial cost of manufacturing Weasley's Wizarding Wheezes by the Weasley brothers = 1000 galleons
Cost of producing one pygmy puff = 3 galleons
Cost of producing p boxes of pygmy puffs = 3p galleons
Total cost of setting up their shop and creating pygmy puffs, as a function of the number of boxes of pygmy puffs produced, p
C(p) = 3p + 1000
Second option is correct
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Complete Question
The Weasley brothers estimate that they will need a thousand galleons just to set up their joke shop, Weasley's Wizarding Wheezes. The first products they are manufacturing are pygmy puffs, and they realize that the cost of making a single box of pygmy puffs is 3 galleons. The brothers would like to estimate the cost, in galleons, of setting up their shop and creating pygmy puffs, as a function of the number of boxes of pygmy puffs produced, p. Which of the following equations is the correct equation representing this scenario?
1) C(p) = 3p - 1000
2) C(p) = 3p + 1000
3) C(p) = 1000p - 3
4) C(p) = 1000p + 3
We can find s , the slant height using Pythagorean theorem , and since this solid is made of parts of simple solids , we can combine the formulas to find surface area and volume
Answers
EXPLANATION
Given that the slant represents the diagonal side of a solid, we can use the Pythagorean Theorem as shown as follows:
Thus, if we know the height and the base, we can compute the slant height by the Pythagorean as explained above.
The Pythagorean Theorem says:
Therefore, we can substitute the radius and the height in the Pytagorean Equation to obtain the slant height.
[tex]\text{radius}^2+\text{height}^2=\text{slant height\textasciicircum{}2}[/tex][tex]r^2+h^2=s^2[/tex]Pluggin in the given values into the equation:
[tex]5^2+7^2=s^2[/tex]Isolating the slant height:
[tex]\sqrt[]{5^2+7^2}=s[/tex]Now, if we need to compute the Surface Area, we need to combine the formulas for all the solids that form the figure.
Figure:
Thus, the surface area is:
[tex]Total\text{ Surface Area=}\frac{SurfaceArea_{\text{sphere}}}{2}+Surface\text{ Area of the Cone}-\text{ Surface Area of the Base}[/tex]Replacing terms:
[tex]=\frac{4\cdot\pi\cdot r^2}{2}+(\pi rs+\pi r^2)-\pi r^2[/tex]We can apply the same reasoning to the Volume:
[tex]Total\text{ Volume}=\frac{Volume\text{ Sphere}}{2}+Volume\text{ of the cone}[/tex][tex]=\frac{\frac{4}{3}\pi r^3}{2}+\frac{1}{3}\pi r^2h[/tex]Finally, just replacing the corresponding values, give us the appropiate surfaces and volumes.
please help me out quickly
Answers
Answer:
take the first 1 an flip it to the other side and with the second row put that 1 on the other side an switch the other two the same way
A small tabletop is 3/4 ft by 3 1/2 ft. Rita solves 3/4⋅3 1/2 to find the area of the tabletop in square feet. Choose numbers to complete Rita's calculations for the area.
Answers
Number which represents the area of table top which has measures of 3/4 ft by 3 1/2 ft is equal to 21/8 square feet.
As given,
Measures of a small tabletop is equal to 3/4 ft by 3 1/2 ft
length=3 1/2 ft
=7/2 ft
Width=3/4 ft
Formula used to measure area of the small tabletop is equal to
=Length ×width
Substitute the values in formula to get a number which represents the area of small tabletop :
Area=(3/4) × (7/2)
=21/8 square feet
Therefore, number which represents the area of table top which has measures of 3/4 ft by 3 1/2 ft is equal to 21/8 square feet.
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There are two parts ill send the second part after the first part is answered. Please!
Answers
∠A and ∠B are vertical angles. If m∠A = (7x-6)° and m∠B = (8x-27)°, then find the measure of ∠B
Answers
Vertically opposite angles are equal
Therefore,
[tex]\begin{gathered} \angle A=\angle B \\ m\angle A=7x-6 \\ m\angle B=8x-27 \\ 7x-6=8x-27 \\ 7x-8x=-27+6 \\ -x=-21 \\ x=21 \\ \angle B=8(21)-27=168-27=141\text{ degre}e \end{gathered}[/tex]I need help with questions 5 and 6 the one with the graph?
Answers
From the table it can be seen that when r=3, V(r=3) =113
On the other hand, when r=7, V(r=7)=1436 and when r=2, V(2)=33.5, therefore,
[tex]\begin{gathered} V(7)-V(2)=1436-33.5 \\ =1402.5 \end{gathered}[/tex]hi, I nee help with this, I don't know if my answer is correct! it would be helpful if you told me which one it is and than do your explaination if that's possible, it's just more convenient for me!
Answers
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion. In other words, similar triangles are the same shape, but not necessarily the same size.
therefore, in this case we can see that the triangles have an equal angle and their sides are similar.
I'm kind of confused on this one I got some help but I'm still struggling to figure it out
Answers
Explanation
Step 1
the standard form of a quadratic equation is given by
[tex]f(x)=ax^2+bx+c[/tex]then, by the table, we have:
[tex]y=f(x)[/tex][tex]\begin{gathered} f(x)=ax^2+bx+c \\ f(-4)=a(-4)^2+b(-4)+c \\ f(-4)\rightarrow30=16a-4b+c\rightarrow\text{Equation}(1) \\ f(-3)=a(-3)^2+b(-3)+c \\ f(-3)=9a+-3b+c \\ f(3)=0=9a+-3b+c\rightarrow\text{Equation}(2) \\ \end{gathered}[/tex]now, we have 2 equations, and 3 variables ( a, b and c, so we need one more equation)
when x= 2
[tex]\begin{gathered} f(x)=ax^2+bx+c \\ f(2)=0=a2^2+b\cdot2+c \\ f(2)=0=4a+2b+c\rightarrow Equation\text{ (3)} \end{gathered}[/tex]Step 3
solve the equations
[tex]\begin{gathered} 30=16a-4b+c\rightarrow\text{Equation}(1) \\ 0=9a-3b+c\rightarrow\text{Equation}(2) \\ 0=4a+2b+c\rightarrow Equation\text{ (3)} \end{gathered}[/tex][tex]\begin{gathered} 0(eq2)=0(eq3) \\ \text{9a-}3b+c=4a+2b+c \\ \text{subtract c in both sides} \\ \text{9a-}3b+c-c=4a+2b+c-c \\ \text{9a-}3b=4a+2b \\ \text{subtract 4a in both sides} \\ \text{9a-}3b-4a=4a+2b-4a \\ 5a-3b=2b \\ \text{add 3b in both sides} \\ 5a-3b+3b=2b+3b \\ 5a=5b \\ \text{divide both sides by 5} \\ \frac{5a}{5}=\frac{5b}{5} \\ a=b \end{gathered}[/tex]Now, replace a=b in equation (1) and equation (2)
[tex]\begin{gathered} 30=16a-4b+c\rightarrow\text{Equation}(1) \\ 30=16a-4a+c \\ 30=12a+c\rightarrow\rightarrow equation\text{ (4)} \\ 0=9a-3b+c\rightarrow\text{Equation}(2) \\ 0=9a-3a+c \\ 0=6a+c\rightarrow\rightarrow equation(5) \end{gathered}[/tex]Step 3
use equations (4) and (5) to find a and c
[tex]\begin{gathered} 30=12a+c\rightarrow\rightarrow equation\text{ (4)} \\ 0=6a+c\rightarrow\rightarrow equation(5) \end{gathered}[/tex]a)isolate c in both equations and equal the expressions to find a
[tex]\begin{gathered} 30=12a+c\rightarrow\rightarrow equation\text{ (4)} \\ 30-12a=c \\ c=30-12a \\ 0=6a+c\rightarrow\rightarrow equation(5) \\ -6a=c \\ c=c \\ 30-12a=-6a \\ \text{add 6a in both sides} \\ 30-12a+6a=-6a+6a \\ 30-6a=0 \\ \text{subtract 30 in both sides} \\ 30-6a-30=0-30 \\ -6a=-30 \\ \text{divide both sides by -6} \\ \frac{-6a}{-6}=\frac{-30}{-6} \\ a=5 \end{gathered}[/tex]we have a= 5,
now, replace in equation (4) to find x
[tex]\begin{gathered} 30=12a+c\rightarrow\rightarrow equation\text{ (4)} \\ 30=12\cdot5+c \\ 30=60+c \\ \text{subtrac 60 in both sides } \\ 30-60=60+c-60 \\ -30=c \\ c=-30 \end{gathered}[/tex]Therefore we have
a=5 b=5 c=-30
Step 4
finally, rewrite the equation
[tex]\begin{gathered} f(x)=ax^2+bx+c \\ f(x)=5x^2+5x-30 \end{gathered}[/tex]I hope this helps you
help me please
thank you
Answers
Answer:
Domain: A, [tex][1, \infty)[/tex]
Range: A, [tex][-4, \infty)[/tex]
Step-by-step explanation:
The domain is the set of x-values and the range is the set of y-values.
There are three groups in a community. Their demand curves for public television in hours of programming, T, are given respectively by Wy = 200-T W2 = 240-T W3 = = 320-T Suppose public television is a pure public good that can be produced at a constant marginal cost of $200 per hour. What is the efficient number of hours of public television?
Answers
In such a competitive private market, 80 hours of programming would be available.
What is defined as the Marginal Social Benefit?The marginal social benefit is the difference with in benefits of consuming one more unit of a product or service. It is determined by the amount of money individuals are inclined to pay for an extra unit of a good or service. When marginal benefit corresponds marginal cost for all groups, the efficient amount of hours of public television is reached.For the given question;
Marginal social benefit = Addition of all groups with in community.
MSB = W1 + W2 + W3
MSB = ($200 - T) + ($240 - 2T) + ($320 - 2T)
MSB = $200 + $240 + $320 - T - 2T - 2T
MSB = $760 - 5T
Calculate efficient number of hours for the public television:
MSB = MC
$760 - 5T = $200
$760 - $200 = 5T
5T = $560
T = 112 hours for the programming.
b. For $200 per hour,
Demand for group 1:
Private marginal benefit = Private marginal cost
$200 - T = $200
T = 0 hours for the programming.
Demand for group 2:
$240 - 2T = $200
$240 - $200 = 2T
$40 = 2T
T = 20 hours for the programming.
Demand for group 3:
$320 - 2T = $200
$320 - $200 = 2T
$120 = 2T
T = 60 hours for the programming.
Hours of television = group 1 + group 2 + group 3
Hours of television = 0 + 20 + 60
Hours of television = 80 hours
Thus, in such a competitive private market, 80 hours of programming would be available.
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i have a hard time in school can some one pls tell me what this is
Answers
Answer: 17 1/3
Step-by-step explanation: hope this helps!
What is the equation of the line that passes through the point (-6,5) and has a slope of -\frac{1}{6}
?
HELP
Answers
The equation of the line with point (-6,5) and slope as -1/6 in slope intercept form is y=-1x/6+6
Given, the equation of the line passes through the point (-6,5)
and has a slope of -1/6.
point (-6,5) = (x₁,y₁)
slope (m)= -1/6
equation of the line is :
y-y₁=m(x-x₁)
substitute the values in the above formula:
y-5=(-1/6)(x-(-6)
y-5=-1x/6+6/6
y-5=-1x/6+1
y=-1x/6+1+5
y=-1x/6+6
Hence the equation of the line is y=-1x/6+6.
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Find the next two terms in this
sequence.
20, 32, 42, 50, 56, [ ? ], [ ? ]
Answers
The next two digits in the sequence are 60 and 62 and the sequence looks like this: 20, 32, 42, 50, 56, 60, 62
What do we mean by sequence?Sequences are ordered collections of numbers, also known as "terms," such as 2,5,8. There are some sequences that adhere to a particular pattern that allows for an endless extension. For instance, 2,5,8 adheres to the "add 3" pattern, so we can now continue the sequence. There are formulas for sequences that show us where to find any given term. You should be familiar with the following four main categories of sequences: arithmetic sequences, geometric sequences, quadratic sequences, and special sequences.So, the next to numbers will be:
20, 32, 42, 50, 56, [ ? ], [ ? ]We can observe that:
20 + 12 = 3232 + 10 = 4242 + 8 = 5050 + 6 = 56Similarly,
56 + 4 = 6060 + 2 = 62Therefore, the next two digits in the sequence are 60 and 62 and the sequence looks like this: 20, 32, 42, 50, 56, 60, 62
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find AB 15 8 A-3 and - 1:03:36:13 : C12
Answers
Given :
[tex]\begin{gathered} A=\begin{bmatrix}1 & {2} & {} \\ {3} & {4} & \end{bmatrix} \\ B=\begin{bmatrix}{6} & {7} & {} \\ {5} & {8} & \end{bmatrix} \end{gathered}[/tex]So,
[tex]AB=\begin{bmatrix}{1} & {2} & {} \\ {3} & {4} & {}\end{bmatrix}\begin{bmatrix}{6} & {7} \\ {5} & {8}\end{bmatrix}=\begin{bmatrix}{c11} & {c12} \\ {c21} & {c22}\end{bmatrix}[/tex]c11 = row 1 column 1 = 1 * 6 + 2 * 5 = 6 + 10 = 16
c12= row 1 column 2 = 1 * 7 + 2 * 8 = 7 + 16 = 23
c21 = row 2 column 1 = 3 * 6 + 4 * 5 = 18 + 20 = 38
c22 = row 2 column 2 = 3 * 7 + 4 * 8 = 21 + 32 = 53
V is the midpoint of UW. If UV = 2x and VW = x + 8,what is UV?
Please explain and show work
Answers
Based on the definition of the midpoint of a line segment, the length of UV is determined as: 16 units.
What is the Midpoint Definition of a Segment?The midpoint of a segment is defined as the point on a line segment where the line segment is bisected into two equal smaller segments. The length of the smaller segments formed will be equal to each other.
Given that V is the midpoint of segment UW, it means that V divides segment UW into two equal smaller segments, UV and VW.
UV = 2x
VW = x + 8
Therefore:
UV = VW
Substitute
2x = x + 8
2x - x = x + 8 - x [subtraction property of equality]
x = 8
Find the length of UV:
UV = 2x
Plug in the value of x
UV = 2(8)
UV = 16 units.
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Patrick rolls a die. What is the probability that he rolls an 1, 3 or 5?
Answers
The total possible outcomes of a dice are: 1,2,3,4,5 and 6.
There are 6 possible outcomes, all with equal probability.
Then, as all the outcomes have the same probability, we have to calculate what is the probability of having one ot three specific outcomesout of the possible 6 outcomes.
We can calculate this as:
[tex]P=\frac{\#\text{success outcomes}}{\#\text{possible outcomes}}=\frac{3}{6}=\frac{1}{2}=0.5=50\text{ \%}[/tex]Answer: P(1,3, or 5) = 1/2 = 0.5 = 50%
Which number line shows the solution of 5x-25>-15
Answers
The solution to the inequality 5x - 25 > -15 in inequality and interval notation form are x > 2 and (2,∞)
How to solve linear inequality problems?A linear inequality is simply an expression where two values are compared by the inequality symbols <, >, ≤ or ≥.
Given the data in the question;
5x - 25 > -15
To solve for x, bring all terms containing the variable to the left side of the equation and the terms without variables to the right side of the equation.
5x - 25 > -15
Add 25 to both sides
5x - 25 + 25 > -15 + 25
5x > -15 + 25
5x > 10
Divide each term in 5x > 10 by 5 and simplify.
5x > 10
5x/5 > 10/5
x > 2
Therefore, the solution to inequality is x > 2.
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no solution one solution infinitely many solutions 3x + 7 = 14 + 3x 22x + 11 = 4x - 7 3(x + 2) = 3x + 6
Answers
no solution one solution infinitely many solutions
3x + 7 = 14 + 3x
22x + 11 = 4x - 7
3(x + 2) = 3x + 6
Part 1
we have
3x + 7 = 14 + 3x
solve for x
3x-3x=14-7
0=7 -----> is not true
that means
No solution
Part 2
we have
22x + 11 = 4x - 7
solve for x
22x-4x=-7-11
18x=-18
x=-1
so
One solution
Part 3
we have
3(x + 2) = 3x + 6
solve for x
Divide by 3 both sides
x+2=x+2 -----> its a identity
that means
infinitely many solutions
Oliver puts 600.00 into an account to use for school expenses the account earns 3% interest compounded annually how much will be in the account after 6 years
Answers
$600.00
3% interest = 0.03
compund anually
How much in 6 years
P = 600
r = 0.03
n = 1
t = 6
[tex]A\text{ = 600(1 + }\frac{0.03}{1})^{(1)(6)}=600(1.03)^{6\text{ }}\text{ = 600(1.1940) = 716.40}[/tex]Answer:
$716.40
Rachel's garden is square in shape. The area of the garden is36 ft? What is the length of one side of the garden?
Answers
1) If Rachel's garden is a square and the area is mentioned, let's use the area of the square.
S= l²
36 =l²
Flipping it:
l²=36
l=√36
l=6
2) So the length of Rachel's garden is 6 feet.
Find the height of a cylinder with a volume of 36pie cm with an exponent of 3 and a base with a radius of 3 cm.h = ???cmv \: is \: 36\pi \: cm ^{3}vis36πcm3r \: is \: 3cmris3cm
Answers
Data
Volume = 36 cm3
radius = 3 cm
Formula
Volume of a cylinder = pi x r2 x h
Solve for h
h = Volume / (pi x r2)
Substitution
h = 36 / (3.1416 x (3)2)
Simplification
h = 36 / (3.1416 x 9)
h = 36 / 28.2744
Result
height = 1.27 cm
A group of 45 people attended a ball game. There were twice as many children as adults in the group, Set up a system of equations that represents the numbers of adults and children who attended the game and solve the system to find the number of children who were in the Group
Answers
Answer:
There are 15 adults and 30 children
Step-by-step explanation:
Hey there!
Setting up the system of equations
----------------------------------------------------------------------------------------------------------
Lets first set a variable for the children and adults
Children - x
Adults - y
x + y = 45
x = 2y
----------------------------------------------------------------------------------------------------------
Solving the system of equations
----------------------------------------------------------------------------------------------------------
We can use substitution
(2y) + y = 45
3y = 45
y = 15
This means, there are 15 adults
Now we plug in this information
x = 2y
x = 2(15)
x = 30
This means, there are 30 children
----------------------------------------------------------------------------------------------------------
So, there are 15 adults and 30 children
Mateo says that 6⁸÷3⁴= 64 or 1,296 by applying the Quotient of Powers Law. Is Mateo correct? Explain.
Answers
Answer:
6⁸÷3⁴ = 20736
Step-by-step explanation:
[tex]6^8\div3^4[/tex][tex]= (2\times 3)^8\div3^4[/tex][tex]= 2^8 \times 3^8\div3^4[/tex][tex]= 2^8 \times 3^{8-4}[/tex][tex]= 2^8 \times 3^{4}[/tex][tex]= 256 \times 81[/tex][tex]6^8\div3^4= 20,736[/tex]So, 6⁸÷3⁴ is neither equal to 64 nor 1296. It is equal to 20,736.If f(x)=sqrt x-x and g(x)+2^3-sqrt x-x, find f(x)-g(x)
Answers
Answer:
A
Step-by-step explanation:
f(x) - g(x)
= [tex]\sqrt{x}[/tex] - x - (2x³ - [tex]\sqrt{x}[/tex] - x) ← distribute parenthesis by - 1
= [tex]\sqrt{x}[/tex] - x - 2x³ + [tex]\sqrt{x}[/tex] + x ← collect like terms
= - 2x³ + 2[tex]\sqrt{x}[/tex]
A few days later, the American tourist went to a bank in Plymouth and exchanged 150 American dollars for British pounds. How many pounds did she receive? (Round your answer to two decimal places.)
Answers
The amount of money that the American tourist will receive will be 132.74 pounds.
What is money?It should be noted that money is the medium of exchange. In this case, it should be noted that 1 pound = 1.13 dollar
The American tourist went to a bank in Plymouth and exchanged 150 American dollars for British pounds, the pound received will be:
= Amount / rate
= 150 / 1.13
= 132.74 pounds.
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The indoor climbing gym is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 8 vans, v, and 3 buses, b, with 242 students. High School B rented and filled 2 vans and 6 buses with 260 students. Every van had the same number of students in it as did the buses. How many students can a van carry? How many students can a bus carry?
Answers
Solution:
We would start by analyzing the statements;
High School A rented and filled 8 vans, v, and 3 buses, b, with 242 students. Mathematically;
[tex]8v+3b=242.........equation1[/tex]High School B rented and filled 2 vans, v, and 6 buses, b, with 260 students. Mathematically;
[tex]2v+6b=260...............equation2[/tex]Now, we would solve equation 1 and equation 2 simultaneously.
From equation 2;
[tex]\begin{gathered} 2v+6b=260 \\ \\ \text{ Divide through by 2;} \\ \frac{2v}{2}+\frac{6b}{2}=\frac{260}{2} \\ \\ v+3b=130 \\ \\ v=130-3b..........equation3 \end{gathered}[/tex]Substitute equation 3 in equation 1;
[tex]\begin{gathered} 8(130-3b)+3b=242 \\ \\ 1040-24b+3b=242 \\ \\ 1040-242=24b-3b \\ \\ 798=21b \\ \\ b=\frac{798}{21} \\ \\ b=38 \end{gathered}[/tex]Substitute the value of b in equation3;
[tex]\begin{gathered} v=130-3(38) \\ \\ v=16 \end{gathered}[/tex]ANSWERS:
[tex]\begin{gathered} \text{ High School A: }8v+3b=242 \\ \\ \text{H}\imaginaryI\text{gh School B: 2}v+6b=260 \\ \\ \text{ Van: \$}16 \\ \\ \text{ Bus: \$}38 \end{gathered}[/tex]A sprinkler waters a circular region in Jason's yard that has a radius of 15 feet. Rounded to the nearest square foot, what is the area of the circular region that is watered by the sprinkler?
Answers
A sprinkler waters a circular region in Jason's yard that has a radius of 15 feet. Rounded to the nearest square foot, what is the area of the circular region that is watered by the sprinkler?
the area of the circular region is
A=pi*r^2
we have
pi=3.14
r=15 ft
substitute
A=3.14*(15^2)
A=707 ft2Devaugn’s age is three times Sydney’s age. The sum of their ages is 52. What is Sydney’s age?Solve by using linear equation.
Answers
ANSWER:
13 years old
STEP-BY-STEP EXPLANATION:
Let x be Devaugn's age and y be Sydney's age.
We establish the following linear equation system:
[tex]\begin{gathered} x=3y \\ x+y=52 \end{gathered}[/tex]We solve the first equation in the second equation:
[tex]\begin{gathered} 3y+y=52 \\ 4y=52 \\ y=\frac{52}{4} \\ y=13 \end{gathered}[/tex]Which means Sydney's age is 13 years old.
