You are building a new house on a cartesian plane whose units are measured in miles.
Your house is to be located at the point (6,0). Unfortunately, the
existing gas line follows the curve y=√16x^(2)+6x+19.
It costs 300 dollars per mile to install new pipe connecting your house to the existing line.
What is the least amount of money you could pay to get hooked up to the system?
Answers
[tex]\bf y=\sqrt{16x^2+6x+19}\\\\ -------------------------------\\\\ L(x)=\left[(6-x)^2+(0-\sqrt{16x^2+6x+19})^2 \right]^{\frac{1}{2}} \\\\\\ L(x)=[36-12x+x^2+16x^2+6x+19]^{\frac{1}{2}} \\\\\\ L(x)=(17x^2-6x+55)^{\frac{1}{2}}\\\\ [/tex]
so, that's L(x) equation for the line, simplified, now, let's check its derivative.
[tex]\bf \cfrac{dL}{dx}=\cfrac{1}{2}(17x^2-6x+55)^{-\frac{1}{2}}(34x-6) \\\\\\ \cfrac{dL}{dx}=\cfrac{1}{2}(17x^2-6x+55)^{-\frac{1}{2}}2(17x-3) \\\\\\ \cfrac{dL}{dx}=\cfrac{17x-3}{\sqrt{17x^2-6x+55}}\\\\[/tex]
now, we get critical points when the derivative is 0, or when the denominator is 0, however (usually cusps or asymptotes), but in this case, we won't check the denominator, because it doesn't yield any, so, we'll just zero out the derivative and see what critical points we get.
[tex]\bf 0=17x-3\implies 3=17x\implies \cfrac{3}{17}=x[/tex]
easy enough, so we only got one point... well, is it a minimum though?
well, running a first-derivative test on the left-region and the right-region of 3/17, we get a negative value on the left and a positive value on the right, that means the original function drops and then goes up, check the picture on the right-hand-side.
So, it's a minimum, those numbers in red in the picture are the ones I used to check it, but any value on the region will work, even -1,000,000, anyway.
so, at 3/17 is at a minimum... now how many miles, and how much is that?
[tex]\bf f\left( \frac{3}{17} \right)\approx 4.53\qquad \qquad \stackrel{miles}{4.53}\cdot \stackrel{\textit{\$ per mile}}{300}\approx 1360.2[/tex]
Related Questions
find the missing value. 90 is what percent of 360
Answers
25%
Hope this helps!
A Builder is trying to level out some ground with a front end loader. He picks up some excess dirt at (9,16) and then maneuvers through the job site along the vectors (-6,0) ,(2,5) ,(8,10) to get to the spot to unload the dirt. Find the coordinates of the unloading point. Find a single vector from the loading point to the unloading point.
Answers
A move along the vector (-6, 0) lands at
(9-6, 16+0) = (3, 16)
A move along the vector (2, 5) lands at
(3+2, 16+5) = (5, 21).
A move along the vector (8, 10) lands at
(5+8, 21+10) = (13, 31).
The unloading point is Q (13, 31).
A vector from P to Q is
(13, 31) - (9,16) = (4, 15)
The graph below illustrates the travel scenario.
Answer:
The unloading point has coordinates (13, 31).
The vector from the loading point to the unloading point is (4, 15)
A cone is placed inside a cylinder as shown. The radius of the cone is half the radius of the cylinder. The height of the cone is equal to the radius of the cylinder. What is the volume of the cone in terms of the radius, r?
Answers
V = (1/3)Pi c^2 h for a cone.
and c = r/2 and h = u
so Volumn of the cone is (1/3)Pi(1/4)(r^2)r = (1/12)Pi r^3
Answer:
(1/12) Pi r^3 option C on plato
Step-by-step explanation:
r = radius of the cylinder
V = (1/3)Pi c^2 h for a cone.
and c = r/2 and h = u
so Volume of the cone is (1/3)Pi(1/4)(r^2)r = r = radius of the cylinder
The difference between five and twice a number, x, is one.
Answers
the difference between 5 and twice a number x is 1
5 - 2x = 1 <== ur equation
-2x = 1 - 5
-2x = - 4
x = -4/-2
x = 2 <== ur solution
2x = 5+1
2x = 6
2x/2 = 6/2
x = 3
hope this helps
9.1 in; 1 significant digit
Answers
Im pretty sure hope i helped
9.1 has 2 significant digits so to have 1 you need to round it to a whole number,
in 1 significant digit would be 9
Round 992,449 to the nearest hundred thousand
Answers
Because its over the 9 so its 992
in 2008 565,650 americans died of all forms of cancer Assuming a population of 305 million what was the mortality rate in units per 100,000 people?
Answers
Suppose y varies directly as x, and y=16 when x=8. Find y when x=16
Answers
how do you solve 19=15w-4(3w-1)
Answers
19=15w-4(3w-1)
distribute
-4(3w-1)=-4(3w)-4(-1)=-12w+4
19=15w-12w+4
19=3w+4
minus 4 both sides
15=3w
divide both sides by 3
5=w
w=5
19-15w = -4(3w-1) > work out the -4(3w-1)
19-15w = -12w+4 > subtract 19 from both sides
-15w = -12w-15 > add 12w to both sides
-3w = -15 > divide by -3
w = 5
hope this helped and I hope I get brainliest answer bud :)
Write the taylor series for f(x)=sin(x) at x=π2
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i need help please this is due in about 5 minutes
Answers
Divide both sides by 5
x^2=441
Find the square root of both sides
x= 21
Estimate the intervals on which the derivative is positive and the intervals on which the derivative is negative. enter your answers to one decimal place.
Answers
A rectangular board is 1.2 meters long and 0.75 meters wide. What is the area of the board in square millimeters?
Answers
A = 1.2 * 0.75
A = 0.9
answer
0.9 square millimeters
Final answer:
To find the area of the board in square millimeters, convert the dimensions from meters to millimeters and multiply the length by the width to get 900000 square millimeters.
Explanation:
The area of a rectangle is calculated by multiplying its length by its width. The area in square millimeters can be found by first converting the measurements from meters to millimeters. Since 1 meter is equal to 1000 millimeters, we convert the dimensions and then calculate the area.
Length in millimeters: 1.2 meters × 1000 = 1200 millimeters
Width in millimeters: 0.75 meters × 1000 = 750 millimeters
Area in square millimeters: 1200 mm × 750 mm = 900000 square millimeters
The area of the board is 900000 square millimeters.
Gymnast Clothing manufactures expensive soccer cleats for sale to college bookstores in runs of up to 500. Its cost (in dollars) for a run of x pairs of cleats is C(x) = 2750 + 7x + 0.1x2 (0 ≤ x ≤ 500). Gymnast Clothing sells the cleats at $150 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit? Revenue = Profit = At least pairs
Answers
The revenue function for Gymnast Clothing is R(x) = 150x, and the profit function is P(x) = 150x - (2750 + 7x + 0.1x^2). To make a profit, they need to produce and sell more than the breakeven number of cleats, which can be calculated by setting the profit function to zero and solving for x.
Explanation:When Gymnast Clothing sells soccer cleats at $150 per pair, the revenue function (R) would be the number of cleats, x, multiplied by the selling price. Therefore, R(x) = 150x. To find the profit function (P), we subtract the cost function (C(x)) from the revenue function, so P(x) = R(x) - C(x) or P(x) = 150x - (2750 + 7x + 0.1x2). A profit is made when P(x) > 0.
To determine the number of cleats needed to make a profit, we need to find the break-even point where P(x) = 0, and then any number greater than the break-even number of cleats would yield a profit. By setting the profit function equal to zero and solving for x, we can find the minimum number of cleats Gymnast Clothing should manufacture to begin making a profit.
Analyze the diagram below and complete the instructions that follow. Solve for x. 9 10 36 60
Answers
What is 25.691 rounded to the greatest place
Answers
6×[(5×7)-(7+8)] Please help me i beg of you please
Answers
6x20
120
youre welcome bro
good luck
A powdered drink mix calls for a ratio of powder to water of 1:8 if there are 32 cups of powder how many total of water are needed? explain your reasoning
Answers
if this makes any sense if not ill try better to explain
the value of 7 in 750 is times the value of 7 in 76
Answers
Answer:
10 times greater
Step-by-step explanation:
The value of 7 in 750 is 700, while the value of the 7 in 76 is 70. 10 times 70 equals 700.
Hope this helped!
While in europe, if you drive 113 km per day, how much money would you spend on gas in one week if gas costs 1.10 euros per liter and your car's gas mileage is 39.0 mi/gal ? assume that 1euro=1.26dollars?
Answers
The total cost for gas when driving 113 km per day in Europe for a week, with gas at 1.10 euros per liter and a mileage of 39.0 mi/gal, would be approximately $66.19 after converting euros to dollars at the given exchange rate of 1 euro = 1.26 dollars.
To calculate the cost of gas for driving 113 km per day over a week in Europe, where gas costs 1.10 euros per liter and the car has a gas mileage of 39.0 mi/gal, a series of conversions and calculations need to be done:
Convert the daily distance driven from kilometers to miles.Calculate the amount of gasoline used per day based on the converted distance and the car's gas mileage.Convert the gasoline amount from gallons to liters.Multiply the daily cost by seven to get the weekly cost.Convert the cost from euros to dollars.Step-by-step this would be:
113 km/day * 0.621371 = 70.215 km/day (in miles)70.215 mi/day / 39.0 mi/gal = 1.8 gal/day (gasoline used each day)1.8 gal/day * 3.79 L/gal = 6.822 L/day (daily liters)6.822 L/day * 1.10 euros/L = 7.504 euros/day (daily cost)7.504 euros/day * 7 days = 52.528 euros/week (weekly cost)52.528 euros * 1.26 dollars/euro = 66.185 dollars/week (total cost in dollars)Therefore, if the car is driven 113 km per day in Europe, the total cost for gas over one week, given the parameters listed, would be approximately $66.19 when converted into dollars.
If θ is a Quadrant II angle and cosθ = -2/3 , then sinθ = _____.
Answers
[tex]\cos(\theta) = \frac{x}{r} [/tex]
In this case, x = -2 and r = 3. Use the Pythagorean Theorem to find the value of y.
[tex]x^2 + y^2 =r^2 \\ 4+y^2=9 \\ y^2 = 5 \\ y=\sqrt{5} [/tex]
Note that you pick the positive square root for y because the point is in Quadrant II.
Now, [tex]\sin(\theta) = \frac{y}{r} = \frac{\sqrt{5}}{3} [/tex]
To solve for sinθ when θ is in Quadrant II and cosθ = -2/3, use the Pythagorean identity sin2θ + cos2θ = 1. Since sinθ will be positive in Quadrant II, after calculations, sinθ equals √5/3.
Explanation:To find the sine value for an angle θ in Quadrant II, where the cosine of θ is given to be cosθ = -2/3, we can use the Pythagorean identity: sin2θ + cos2θ = 1. We already know that cosθ = -2/3, so we can solve for sinθ:
sin2θ = 1 - (cosθ)2sin2θ = 1 - (-2/3)2sin2θ = 1 - 4/9sin2θ = 5/9Since we are in Quadrant II, where sine values are positive, we take the positive square root:
sinθ = √(5/9)sinθ = √5/3So the value of sinθ when cosθ = -2/3 in Quadrant II is √5/3.
determine which of the values could be used to clear fractions in the given equation 1/4×-2/5=1/2×+2
Answers
so we would multiply the entire equation by 20
1/4x - 2/5 = 1/2x + 2....multiply everything by 20
5x - 8 = 10x + 40....poof...bye-bye fractions
To clear fractions in the given equation 1/4x-2/5=1/2x+2, multiply each term by the least common denominator (LCD) of the fractions to eliminate the fractions. Simplify the equation and solve for x.
Explanation:To clear fractions in the given equation 1/4x-2/5=1/2x+2, we can find the least common denominator (LCD) of the fractions. The LCD is the least common multiple (LCM) of the denominators, which in this case is 20. To clear the fractions, we can multiply each term in the equation by 20:
20 * (1/4x) = 20 * (1/2x+2) + 20 * 25x - 8 = 10x + 40Subtracting 5x and 40 from both sides, we get -8 - 40 = 10x - 5x-48 = 5xDividing both sides by 5, we find x = -48/5So, the value that can be used to clear fractions in the given equation is x = -48/5.
Learn more about Clearing Fractions here:https://brainly.com/question/27151554
#SPJ2
(b)It takes 30 pounds of seed to completely plant a 5 -acre field. How many pounds of seed are needed per acre?
Answers
30/5 = 6
6 pounds of seed are needed per acre.
Hope this helps!! :)
Angela ordered 2 glass paperweights. The volume of each paperweight is 5 cubic inches, and the density of the glass is 1.5 ounces per cubic inch. Angela can use the calculation below to find the total weight of her order.
2×(5×1.5)
How can Angela simplify the calculation using only the associative property of multiplication?
Drag and drop the appropriate number or expression into each box.
Answers
Answer:
[tex](2\times 5)\times 1.5[/tex]
Step-by-step explanation:
Since, according to associative property of multiplication,
a(bc) = (ab)c,
Where a, b and c are any real numbers,
Here, the given expression,
2 × ( 5 × 1.5 ),
By applying the associative property of multiplication,
( 2 × 5 ) × 1.5,
Hence, the appropriate expression in the first box is '(2×5)'
And, the appropriate number in the second box is '1.5'.
Answer:
first box is (2 times 5) next box is 1.5
Step-by-step explanation:
i took K12 test :)
29+d=54;24,25,26 what
Answers
jeremy wants to buy a new computer. the saleswoman says that he can make a down payment and then pay for the computer in installments, heres a formula for this scenario: x=t- yz, x=amount down, y=money each month, z=number of months, t=total price, rewrite the formula to solve fore the amount of money jeremy must pay each month
Answers
Becca earns money mowing her neighbors' lawns.
The revenue for mowing x lawns is r(x) = 18x.
Becca's cost for gas and the mower rental is c(x) = 5x + 20.
Her profit from mowing x lawns is p(x) = (r – c)(x). What is p(x)?
Answers
p(x) = 18x - 5x - 20
p(x) = 13x - 20 <===
Answer: [tex]p(x)= 13x-20[/tex]
Step-by-step explanation:
Given: The revenue for mowing x lawns is [tex]r(x) = 18x[/tex] .
Becca's cost for gas and the mower rental is c(x) = 5x + 20
The profit from mowing x lawns is given by :-
[tex]p(x)=(r-c)(x)[/tex]
The above function can be written as
[tex]p(x)=(r-c)(x)=r(x)-c(x)\\\\\Rightarrow p(x)= 18x-(5x+20)\\\\\Rightarrow p(x)= 18x-5x-20\\\\\Rightarrow p(x)= 13x-20[/tex]
Find the equation of the line that contains the point (3,9) and is parallel to the line y=4x+1. Write the line in slope-intercept form. Graph the lines
Answers
The equation of the line parallel to y=4x+1 passing through (3,9) is y=4x-3. It has the same slope of 4 but a different y-intercept, showing that the lines are parallel and do not intersect.
Explanation:To find the equation of the line that contains the point (3,9) and is parallel to the line y=4x+1, we start by noting that parallel lines have the same slope. The given line has a slope of 4, so our new line will also have a slope of 4. The slope-intercept form of a line is given by y = mx + b, where m is the slope and b is the y-intercept.
Since we have a point (3,9) and a slope of 4, we can use the point-slope form to find our new line:
y - y₁ = m(x - x₁)
Plugging in our point and slope, we get:
y - 9 = 4(x - 3)
Expanding this, we have:
y - 9 = 4x - 12
Add 9 to both sides to get it into slope-intercept form:
y = 4x - 3
Therefore, the equation of the line parallel to y=4x+1 that passes through (3,9) in slope-intercept form is y = 4x - 3.
When graphing these two lines, you'll notice they never intersect, as they have the same slope but different y-intercepts, which is a distinctive feature of parallel lines.
To find the equation of a line that is parallel to another line, we need to determine the slope of the given line and then use the point-slope form of a linear equation. The equation of the line parallel to y = 4x + 1 and passing through the point (3,9) is y = 4x - 3. The graph of the lines show that they are parallel.
Explanation:To find the equation of a line that is parallel to another line, we need to determine the slope of the given line and then use the point-slope form of a linear equation.
The given line y = 4x + 1 has a slope of 4 because it is in the form y = mx + b, where m is the slope. Since the line we want to find is parallel to this line, it will also have a slope of 4.
Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point on the line, we can substitute the values (3, 9) and 4 for x1 and m, respectively, to get the equation:
y - 9 = 4(x - 3)
Expanding and rearranging the equation, we get:
y = 4x - 3
Graphing the lines:The graph of the given line y = 4x + 1 is a straight line with a slope of 4 and a y-intercept of 1. The line we found, y = 4x - 3, is also a straight line with the same slope but a different y-intercept. By plotting the two lines on the same graph, we can visually see that they are parallel.
Varia is studying abroad in Europe. She is required pay $3,500 (in US dollars) per year to the university, however, she must pay in euros. How many euros can Varia expect to pay per month to the university? (Round to the nearest whole euro.)
0.7306 euros = 1 US dollar
Answers
Answer:
213 euros per month
Step-by-step explanation:
1dollar = 0.7306euros
thus $3,500 dollars can be found by a rule of three
dollar euro
1 ⇒ 0.7306
3500 ⇒ x
we find x as follows:
[tex]x=\frac{0.7306*3500}{1} = 2,557.1[/tex] euros
She will pay 2,557.1 euros per year.
Dividing between the 12 months of the year:
[tex]\frac{2,557.1}{12}=213.1[/tex]
Rounding to the nearest whole euro she will pay 213 euros per month.
Answer:
A
Step-by-step explanation:
I just took the test
what is the product? a-3/7 divided by 3-a/21
Answers
To find the product of the expression (a-3/7) divided by (3-a/21), multiply (a-3/7) by the reciprocal of (3-a/21), yielding the simplified expression ((21a - 9)/ (21a - 7)) valid for all values of 'a' except when 'a' equals 8/3.
Explanation:The student is asking for the product of the expression (a-3/7) divided by (3-a/21). To simplify this expression, we need to apply the division of fractions rule, which states that to divide by a fraction, you multiply by its reciprocal. The reciprocal of (3-a/21) is (21/(3a-1)). So our new expression is (a-3/7) × (21/(3a-1)).
We can multiply the numerators and denominators separately: (a-3/7) × 21 = (a× 21 - 3× 3) and 7 × (3a-1) = 21a-7. Simplified, we get ((21a - 9)/ (21a - 7)) as the product.
However, we should also check if the expression is defined for all values of 'a' as there may be values for which the denominator becomes zero. Specifically, we need 'a' to not be such that (3a-1) equals 7, which simplifies to a = 8/3. So for all values of 'a' except when 'a' equals 8/3, the product of the given expression is ((21a - 9)/ (21a - 7)).