logx + log(3x-13) = 1
Answers
Condense the left side.
log x(3x-13)=1
Put a base 10 on each side to clear the log.
x(3x-13)=10
3x^2-13x-10=0
Factoring you get x=5 and x=-2/3. The domain for log is x>0 so the -2/3 is an extraneous solution.
The solutions for [tex]\log x + \log (3\cdot x - 13) = 1[/tex] are [tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex], respectively.
In this question, we are going to solve for [tex]x[/tex] with the help of Logarithm Properties, which are described in the image attached below.
[tex]\log x + \log (3\cdot x - 13) = 1[/tex]
[tex]\log [x\cdot (3\cdot x - 13)] = 1[/tex]
[tex]\log (3\cdot x^{2}-13\cdot x) = 1[/tex]
[tex]10^{\log(3\cdot x^{2}-13\cdot x)} = 10^{1}[/tex]
[tex]3\cdot x^{2}-13\cdot x = 10[/tex]
[tex]3\cdot x^{2}-13\cdot x -10 = 0[/tex]
This is a Second Order Polynomial, which can be solved by Quadratic Formula:
[tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex]
The solutions for [tex]\log x + \log (3\cdot x - 13) = 1[/tex] are [tex]x_{1} = 5[/tex] and [tex]x_{2} = -\frac{2}{3}[/tex], respectively.
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Related Questions
A company has started selling a new type of smartphone at the price of $120 - 0.1x where x is the number of smartphones manufactured per day. The parts for each smartphone cost $60 and the labor and overhead for running the plant cost $4000 per day. How many smartphones should the company manufacture and sell per day to maximize profit?
Answers
To maximize profit, derive the profit equation ((120 - 0.1x) * x - (60x + 4000)) and set it to zero to find the critical points. These points will dictate the number of smartphones the company needs to manufacture and sell each day.
Explanation:The subject of this question is the maximization of profit, which is a concept in economic mathematics. To maximize the profit, you'll need to determine the number of smartphones that the company should produce and sell each day.
The profit is the revenue (the money earned from selling goods) minus the costs. Revenue is given by the equation R = (120 - 0.1x) * x, and the total cost is given by the equation C = 60x + 4000, where x is the number of smartphones produced.
So, the profit P becomes P = R - C = (120 - 0.1x) * x - (60x + 4000). To maximize this profit, we'd take the derivative of P in terms of x and set it equal to zero to find the critical points. After finding the critical points, we then test these values back in the profit equation to find which gives the highest value and hence maximizes the profit.
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A cable 20 feet long connects the top of a flagpole to a point on the ground that is 16 feet from the base of the pole. How tall is the flagpole?
Answers
The flagpole is vertical and is at 90° to the ground. This makes a right-angle to the ground, which is flat.
Since we have two sides of the triangle already, and they are in a ratio of ?? : 16 : 20 = ?? : 4 : 5, that means that because the triangle is a right-angled triangle, the third and last side must be 3 to complete the ratio! (This is called 'Pythagoras')
Since the other sides were divided by 4 to get the simple ratio, we must multiply 3 to get the actual ratio! 3 × 4 = 12 ft tall flagpole.
The height of the flagpole is 12 feet.
To determine the height of the flagpole, we can use the Pythagorean theorem.
Given: The length of the cable (hypotenuse) is 20 feet.The distance from the point on the ground to the base of the flagpole (one leg) is 16 feet.Let the height of the flagpole be h (the other leg).Using the Pythagorean theorem:
c² = a² + b²
Here, c = 20 feet (length of cable), a = 16 feet (distance from flagpole base to ground point), and h = b (height of the flagpole).
Therefore, we have:
202 = 162 + h²
400 = 256 + h²
To find h, subtract 256 from 400:
400 - 256 = h²
144 = h²
h = √144 = 12.
A landscaper earns $30 for each lawn her company mows, but she pays $210 per day in salary to her employees. if her company made more than $150 profit from mowing lawns in a 7-day week, what are the possible numbers of lawns the company could have mowed?
Answers
The company could have mowed more than 54 lawns.
How to Solve linear inequality?Let the number of lawns mowed be x.
Then the total earning of landscaper is 30x.
The total amount given to the employees in a week is 210*7=$1470.
Then the profit of landscaper would be 30x-1470 which is greater than $150.
30x-1470 > 150
30x > 1620
x > 54
Therefore, possible numbers of lawns the company could have mowed are greater than 54 lawns.
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Between 61 and 80 lawns could have been mowed by the business in a week in order to turn a profit of more than $150. So options (d) and (e) are correct.
To determine the possible numbers of lawns the company could have mowed, we need to calculate the profit for a 7-day week and compare it with the given condition that the profit exceeds $150.
Step 1: Determine daily and weekly expenses
The daily salary expense is $210.
For 7 days, the total salary expense is:
[tex]\[ 210 \times 7 = 1470 \text{ dollars} \][/tex]
Step 2: Calculate profit per lawn
The company earns $30 for each lawn mowed.
Let ( n ) be the number of lawns mowed in a week.
Step 3: Calculate total earnings from mowing lawns
Total earnings for ( n ) lawns is:
[tex]\[ 30n \text{ dollars} \][/tex]
Step 4: Determine the profit
Profit is total earnings minus total expenses:
[tex]\[ \text{Profit} = 30n - 1470 \][/tex]
Step 5: Apply the condition that profit is more than $150
The inequality to solve is:
[tex]\[ 30n - 1470 > 150 \][/tex]
Step 6: Solve the inequality
Add 1470 to both sides:
[tex]\[ 30n > 1620 \][/tex]
Divide both sides by 30:
[tex]\[ n > 54 \][/tex]
Step 7: Identify the possible number of lawns mowed
The number of lawns must be greater than 54.
From the given options (12, 37, 54, 61, 80), the possible numbers are:
[tex]\[ 61 \text{ and } 80 \][/tex]
Thus, the two possible numbers of lawns the company could have mowed to make a profit of more than $150 in a week are 61 and 80.
Complete Question:
A landscaper earns $30 for each lawn her company mows, but she pays $210 per day in salary to her employees. If her company made more than $150 profit from mowing lawns in a 7-day week, what are the possible numbers of lawns the company could have mowed? Select two options. 12
37
54
61
80
The shop charges $60 per hour and 7% sales tax on parts (no tax on labor). The total cost of the repair is $180.80. A. True B. False The statement is False.
Answers
y<=x-5, and y<=-x-4 would be points (1,10)? or
(-1,10)
(10,1)
1,-10)
Answers
If a right circular cone intersects a plane that passes through one of its nappes but the plane is not parallel to an edge of the cone the resulting curve will be:
parabola
circle
ellipse
hyperbola
Answers
the edge of the cone of the resulting curve would be :
Parabola
Hope this helps
Answer:
D) Hyperbola.
Step-by-step explanation:
If a right circular cone intersects a plane that passes through one of its nappes, but the plane is not parallel to an edge of the cone, the resulting curve will be one arc of a hyperbola.
Therefore, the answer is Hyperpola.
Hope this will helpful.
Thank you.
How to write a real-world situation that could be modeled by the equation 3 2/3x=5/6x-7/8?
Answers
A real-world situation that can be modeled by the equation 3 2/3x = 5/6x - 7/8 is when you have a certain amount of money, and you are earning money at a certain rate while also incurring expenses.
Explanation:To write a real-world situation that could be modeled by the equation 3 2/3x=5/6x-7/8, let's consider a scenario where you have $3.67, and you are earning additional money at a rate of $0.833 per hour. On the other hand, you have expenses of $0.875 per hour. We can represent the money you have earned and the money you have spent using the equation 3 2/3x=5/6x-7/8.
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To write a real-world situation that could be modeled by the equation 3 2/3x=5/6x-7/8, we can consider a scenario where a store is selling two different types of products and the equation represents the difference in the number of sales for the two products.
Explanation:To write a real-world situation that could be modeled by the equation 3 2/3x=5/6x-7/8, we need to think about a scenario where two quantities are being compared or related. For example, let's consider a situation where a store is selling two different types of products. The equation could represent the difference in the number of sales for the two products over a certain period of time. The left side of the equation represents the number of sales for one product, and the right side represents the number of sales for the other product, with a difference of 7/8 units.
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A water cooler can hold 50 pt of water. About how many liters of water can it hold? (1 L ≈ 1.06 qt)
23.6 L 26.5 L 94.3 L 106.0 L
Answers
1qt = 2pt
That means that 50 pt is:
50/2 = 25 qt
And now we convert qt to L
25/1.06 = 23,6 L
Answer is A) 23,6 liters
A classmate claims that having no slope and having a slope of 0 are the same. Is your classmate correct? Explain. ...?
Answers
what is 0.83 repeating as a fraction?
Answers
where is the opposite of 8 located on a number line?
Answers
How do you find the reciprocal of 3.6?? ...?
Answers
To find the reciprocal of 3.6, divide 1 by 3.6.
Explanation:To find the reciprocal of 3.6, you divide 1 by 3.6. The reciprocal of a number is obtained by flipping the number. So, the reciprocal of 3.6 is
1/3.6
or approximately 0.2778.
A stemplot is one way to display these batting averages, as shown below. In this type of display, the entry 0.32|9 denotes one occurrence of the batting average _____.
0.32 | 9
0.33 | 0 1 1 3 3 3 4 4 4 6 7 8 8 8
0.34 | 0 0 1 1 2 2 2 4 4 5 6 9
0.35 | 6 8
0.36 | 6 ...?
Answers
Answer:
0.356
Step-by-step explanation:
Based on the batting averages denoted in the stemplot, the entry of 0.32|9 shows an occurrence of 0.329.
What average is shown by 0.32|9?In a stemplot, the numbers that are stems are on the left side and will be the first numbers that should be denoted.
The numbers on the right are leaf numbers and come after the stem. This means that a stemplot showing 0.32|9 is a way of writing 0.329.
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Vicki works 19 hours per week. Her take home pay is $17.30 per hour. If Vicki is able to save all of her earnings, how long will it take her to save at least $4,000?
a.
12 weeks
b.
13 weeks
c.
14 weeks
d.
15 weeks
Answers
1 hour is $17.30
so 1week=19hours is $328.7
x=? <...................$4000
12.16 weeks
the answer is
a.
12 weeks
Answer: B
Step-by-step explanation:
on edg
how do you plot y-5= 3/2 (x-2)
Answers
Add 5 to each side of the equation.
(y - 5) + 5 = (3/2(x - 2)) + 5
Now the equation will look like y = 3/2(x - 2) + 5
We can also distribute the 3/2, but we'll leave it like that.
Now, to plot a line, you must find two points and run a line through the two points! That's it!
To find a point, enter some values for x. I'll do x = 0
Substitute: y = 3/2(0 - 2) + 5
After doing the math, y equals to -2.
So, the first point will be (0, 2)
Now for the second point. x = 1
y = 3/2(0 - 1) + 5
After doing the math, y equals to 6.5
Now out second point is (1, 6.5)
Now after plotting these two points, run a line through them.
Sure Fire Auto Supplies marked down its entire stock of imported tires 30% on Wednesday only. The sale price of all tires was $79. What will be the price to the nearest cent, of each tire on Thursday when the tires are marked back to their original price?
Answers
The answer is $112.86
Why is it important to learn, in algebra, the proper order in which to read algebraic expressions and solve algebraic equations?
Answers
Learning the proper order in which to read algebraic expressions and solve algebraic equations is important in algebra for understanding the order of operations, developing logical reasoning, and ensuring consistency in mathematical communication.
Explanation:It is important to learn the proper order in which to read algebraic expressions and solve algebraic equations in algebra for several reasons:
Order of operations: Understanding the order of operations (PEMDAS) is crucial to correctly evaluating algebraic expressions. This ensures that the operations are performed in the correct order and leads to the correct result. For example, without following the order of operations, the expression 8 + 2 x 3 could be evaluated as 10 x 3 = 30 instead of 8 + 6 = 14.Logical reasoning: Solving algebraic equations requires logical reasoning and following a step-by-step process. Learning the proper order helps develop this logical reasoning and helps students approach problems systematically. It allows them to break down complex problems into simpler steps and solve them more efficiently.Consistency and communication: Following the proper order in algebra ensures consistency in mathematical communication. It allows students and mathematicians to communicate their solutions and reasoning effectively. When everyone understands and follows the same order, they can accurately communicate and discuss algebraic expressions and equations.The graph of f(x) = 6(0.25)x and its reflection across the y-axis, g(x), are shown.
What is the domain of g(x)?
all real numbers
all real numbers less than 0
all real numbers greater than 0
all real numbers greater than or equal to 0
Answers
The domain of this function is all the real numbers.
Reflection accross the y-axis => the argument is multiplied by - 1. and g(x) is 6(0.25)^(-x).
The graph of the original function flips around the y-axis, but the domain keeps being all the real numbers.
Answer: all real numbers
The correct answer is the first option: All real numbers.
I just did the quiz and got it right.
For all the non BrainlyPlus members
Which of the following has a graph that is a straight line?
Equation 1: y = 4x^3 + 6
Equation 2: y = 5x − 4.5
Equation 3: y^2 = x − 1
Equation 4: y = 2x^2 + 6
Answers
if no indices - that's straight line
Answer:
y=5x - 4.5
Step-by-step explanation:
I took the test
A small business purchases a
piece of equipment for $875. After 5 years the equipment will be outdated, having no value.
(a) Write a linear equation giving the value of the equipment y in terms of the time
(b) Find the value of the equipment when X=2
(c) Estimate (to two-decimal-place accuracy) the time when the value of the equipment is $200. ...?
Answers
The rate at which the equipment loses value is $ 875 / 5 years =$ 175 / year.
a) y = 875 - 175x
b) x = 2 => y = $875 - [S175/ year] (2year) = $875 - $350 = $ 525
c) y = 200 = 875 - 175x => x = [875 - 200] / 175 = 3.86 years
Using the straight-line depreciation method, the linear equation for the equipment's value over time is y = -175t + 875. After 2 years, the equipment's value is $525. To find when the equipment's value is $200, solve for t and get approximately 3.86 years.
To calculate the depreciation of the equipment using a linear equation, we'll use the straight-line depreciation method. The business purchases the equipment for $875 and it will be worthless after 5 years, so it depreciates $875 over 5 years.
Part A
To write the linear equation representing the value of the equipment y over time t, we need to find the slope (m) of the line which in this case is the annual depreciation. The slope is the change in value divided by the time, thus m = -$175/year. The initial value (intercept) is $875. The equation is:
y = -175t + 875
Part B
Substitute t = 2 into the equation to find the value of the equipment at year 2:
y = -175(2) + 875
y = -350 + 875
y = $525
Part C
We want to find the time t when the value y is $200. Substitute y = $200 into the equation:
200 = -175t + 875
-175t = 200 - 875
-175t = -675
t = [tex]\frac{-675}{-175}[/tex]
Mass is the ________(1)________, where as depth is the ________(2)________.
Answers
Brandon is on one side of a river that is 50 m wide and wants to reach a point 300 m downstream on the opposite side as quickly as possible by swimming diagonally across the river and then running the rest of the way. find the minimum amount of time if brandon can swim at 2 m/s and run at 5 m/s
Answers
Brandon can reach a point 300m downstream on the opposite side of the river in a minimum of 60 seconds by swimming diagonally across the river and then running the rest of the way.
Explanation:In order to determine the minimum amount of time for Brandon to reach a point 300m downstream on the opposite side of a river, we can use the concept of relative velocities.
First, we need to find the velocity of the current. We can use the formula v = d/t, where v is the velocity, d is the distance, and t is the time.
The distance Brandon needs to cross the river is 50m, and he wants to reach a point 300m downstream, so the total distance he needs to travel horizontally is 50m + 300m = 350m.
Since the current is perpendicular to Brandon's swimming direction, it doesn't affect the time it takes for him to swim across the river. Therefore, we only need to consider his running speed.
The time it takes for Brandon to run the remaining distance of 300m is equal to the distance divided by his running speed, which is 300m / 5m/s = 60s.
So, the minimum amount of time for Brandon to reach the point 300m downstream is 60 seconds.
To find the minimum amount of time for Brandon to reach the point across the river, divide the problem into two components: swimming and running. Use the Pythagorean theorem to find the resultant velocity of Brandon's swimming motion. To find the total minimum time, add the swimming time and the running time.
Explanation:To find the minimum amount of time for Brandon to reach the point across the river, we can use the concept of vector addition. Since the river is flowing downstream, Brandon needs to swim at an angle with respect to the direction of the river current. Let's break down the problem into two components: swimming and running.
Swimming:
Using the Pythagorean theorem, we can find the resultant velocity of Brandon's swimming motion. The width of the river is the base, the downstream distance is the height, and the resultant velocity is the hypotenuse. By substituting the given values, we can find the resultant velocity of Brandon's swimming motion as 5.12 m/s. This is the velocity he will need to swim at an angle to counteract the river current.
Running:
After crossing the river, Brandon needs to run the remaining distance of 300 m at a speed of 5 m/s. The time it takes to run this distance is 300 m / 5 m/s = 60 seconds.
To find the time it takes for Brandon to swim across the river, we can use the formula:
time = distance / velocity
. The distance is 50 m, and the velocity is 5.12 m/s. Therefore, the time it takes for Brandon to swim across the river is 50 m / 5.12 m/s = 9.77 seconds.
Finally, we can add the swimming time and the running time to find the total minimum time it takes for Brandon to reach the point across the river: 9.77 seconds + 60 seconds = 69.77 seconds.
solve the exponential equation using the method of relating the bases by first rewriting the equation in the form of e^u=e^v
1/e ^6x =√e ÷e ^6−x ...?
Answers
General admission tickets to the fair cost 3.50 per person. Ride passes cost an additional 5.50 per person. Parking costs 6 dollars for the family. the total costs for ride passes and parking was 51 dollars. How many people in the family attend the fair
Answers
a golf ball travels a distance of 600 feet as measured along the ground and reaches an altitude of 200 feet. If the origin represents the tee and the ball travels along a parabolic path that opens downward, find an equation for the path of the golf ball.
...?
Answers
y0 = 0
g ≈ - 32 ft / s^2
(1) y = V0y * t - 16t^2
(2) x = V0x * t
Use (1) and the maximum heigth formula to determine V0y
y max = (V0y)^2 / (2g) = 200 => (V0y)^2 = 2*32*200 = 12,800 => V0y ≈ 113.14 ft/s
y = 113.14 t - 16t^2
for y = 200 => 200 = 113t - 16t^2 => -16t^2 + 113.14t - 200 = 0
Solve that equation using the quadratic formula and you will ge t = 3.54 s
The total time is 3.54 s * 2 = 7.08 s
Then use that time in (2) to find V0x
600 = V0x * t => V0x = 600 / 7.08 s = V0x = 84.7 ft/s
Then, x = 84.7 t.
Now solve for t and replace it in y = 113.14 t - 16t^2 :
t = x / 84.7
y = 113.14 (x/84.7) - 16(x^2 / 84.7 ^2) = 1.336 x - 0.00223x^2
You can check that for, except for a small difference due to the approximations, for x = 600, y = 0 and for x =300 y = 200
Answer: y = 1.336 x - 0.00223x^2
Answer:
y=1.32x-0.0022[tex]x^{2}[/tex]
Step-by-step explanation:
Let the standard equation of parabola be y=a[tex]x^{2}[/tex]+bx+c ....... (1)
Let y be the altitude of the ball and x be the distance traveled by the ball.
Now by putting the points one by one, we get the required equation.
first point = (0,0)
golf ball travels a distance of 600 feet along the ground. Therefore point = (600,0)
At altitude of 200, it travels the distance of 300 feet(mid point) = (300,200)
By putting y=0 and x=0 in (1),
we get c=0,
By putting y=0 and x=600 we get
0 = 360000a+600b ..............(2)
By putting y=200 and x=300, we get
200=90000a+300b ................(3)
Solving (2) and(3) we get, a=-0.0022 and b = 1.32
So the required equation is :
y= =0.0022[tex]x^{2}[/tex]+1.32x
The axis of symmetry for a quadratic equation can be found using the formula x = -b/2a , where a and b are coefficients in the quadratic equation and x represents the values along a vertical line on the coordinate plane.
When the equation is solved for a, such that b is the numerator of the resulting fraction, what is the denominator of the fraction?
2x
–2x
1/2x
–1/2x
Answers
Solving for a,
a = -b/2x
If b is the numerator, the denominator is -2x
The second option is correct.
The denominator of the fraction is -2x.
The correct answer is option (b)
What is equation?"It is a mathematical statement which consists of equal symbol between two algebraic expressions."
What is numerator?"In a fraction, the value placed above the horizontal line."
What is denominator?"In a fraction, the value placed below the horizontal line."
For given question,
The axis of symmetry for a quadratic equation can be found using the formula [tex]x=-\frac{b}{2a}[/tex]
where a and b are coefficients in the quadratic equation and x represents the values along a vertical line on the coordinate plane.
[tex]\Rightarrow x=\frac{-b}{2a}\\\\ \Rightarrow \frac{a}{x} \times x=\frac{-b}{2a}\times \frac{a}{x}\\\\\Rightarrow a=\frac{b}{-2x}[/tex]
For the fraction [tex]\frac{b}{-2x}[/tex] the numerator is b and denominator is -2x.
Therefore, the denominator of the fraction is -2x.
The correct answer is option (b)
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a table that originally cost $196 is on sale for $160.00. What is the percent of decrease, rounded to the nearest tenth?
Answers
Start by dividing 160 by 196 (which gives you 0.816... or 81.6%) which is the percent you're still paying, thus if you subtract that from 1, it should give you the percent off (1-0.816=0.186... or 100-81.6=18.4)
A total of 376 tickets were sold for the school play. they were either adult tickets or student tickets. the number of student tickets sold was three times the number of adult tickets sold. how many adult tickets were sold?
Answers
Final answer:
Using algebra, we set up equations based on the information given: the number of student tickets is three times the number of adult tickets, and the total number of tickets is 376. Solving for the number of adult tickets, we find that 94 adult tickets were sold.
Explanation:
The question asks us to figure out how many adult tickets were sold for the school play when a total of 376 tickets were sold, and student tickets sold were three times the number of adult tickets.
Let's denote the number of adult tickets sold as A and the number of student tickets sold as S. According to the problem, S = 3A. We are also told that the total number of tickets sold is 376, so A + S = 376.
Substituting the first equation into the second one, we get A + 3A = 376. Simplifying this equation, 4A = 376. Dividing both sides by 4, A = 94. Therefore, 94 adult tickets were sold.
Final answer:
As per the values, 94 adult tickets were sold.
Explanation:
The question asks us to determine how many adult tickets were sold for a school play, given a total number of tickets and a ratio of student tickets to adult tickets.
To solve this problem, let's define adult tickets as x and student tickets as 3x, since there are three times as many student tickets as adult tickets.
The total tickets sold is 376, therefore the equation we need to solve is:
x + 3x = 376
Combining like terms, we get:
4x = 376
Dividing both sides by 4, we find that:
x = 94
Therefore, 94 adult tickets were sold.
at how many points does the graph of the function y=16x^2-8x+1 intersect the x axis
Answers
D = b^2 - 4ac
in this case, a = 16, b = -8 and c = 1
Plug these values into the formula to get
D = b^2 - 4ac
D = (-8)^2 - 4(16)(1)
D = 64 - 64
D = 0
Since the discriminant is 0, this means that there is exactly 1 solution. This 1 solution corresponds to exactly 1 root or x intercept
So this parabola touches the x axis at exactly one point
Answer:aAPEX1
Step-by-step explanation:
Find y when x equals 10 if y equals eight what x equals 20
Answers
Assuming y is directly proportional to x
[tex]y=kx[/tex] where k is constant of proportionality
[tex]8=20k[/tex]
[tex]k=\frac{8}{20}=\frac{2}{5}[/tex]
[tex]y=\frac{2}{5}x[/tex]
[tex]y=\frac{2}{5}*10[/tex]
[tex]y=4[/tex]
Assuming y is indirectly proportional to x
[tex]y=\frac{k}{x}[/tex] where k is constant of proportionality
[tex]8=\frac{k}{20}[/tex]
[tex]k=160[/tex]
[tex]y=\frac{160}{x}[/tex]
[tex]y=\frac{160}{10}[/tex]
[tex]y=16[/tex]