At a certain time, the length of a rectangle is 5 feet and its width is 3 feet. At that same moment, the length is decreasing at 0.5 feet per second and the widthis increasing at 0.4 feet per second.
What is the length of the diagonal at that time?
How fast is the length of the diagonal changing? Is this length increasing or decreasing?
Answers
[tex]\bf r^2=x^2+y^2\implies 2r\cfrac{dr}{dt}=2x\cfrac{dx}{dt}+2y\cfrac{dy}{dt}\implies \cfrac{dr}{dt}=\cfrac{x\frac{dx}{dt}+y\frac{dt}{dt}}{r} \\\\\\ \cfrac{dr}{dt}=\cfrac{(5\cdot -0.5)+(3\cdot 0.4)}{\sqrt{34}}[/tex]
if it's a negative value, thus a negative rate, thus is decreasing, if it is a positive value, then increasing.
d = (L^2 + W^2)^.5 = SQRT(34) or 34^.5
Taking the derivative of d:
d' = (1/2)(2LL' + 2WW')(L^2 + W^2)^(-.5)
Solving for d' given the L=5, L'=-.5, W=3, W'=+.4
yields d is decreasing at a rate of -2229 feet/sec.
Related Questions
The price of an item has been reduced by 15% . The original price was $51 .
Answers
85 - (60/100)*85
= 85 - 0.6 *85
= 85 - 51
= $34
The question is about calculating the new price of an item after a discount. The original price of the item was $51.00, and it was reduced by 15%, making the new price $43.35.
Explanation:The subject of this question is Mathematics and it is looking for a solution to a percentage price reduction problem. The item had an original price of $51.00 and its price has been reduced by 15%. To find the new price after the discount, we have to calculate the amount of the reduction and subtract it from the original price.
First, let's calculate the amount of the discount: 15/100 * 51 = $7.65.}
Now, we subtract this amount from the original price: 51 - 7.65 = $43.35.
Therefore, the new price of the item after a 15% discount is $43.35.
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A ship traveled at an average rate of 22 miles per hour going east. It then traveled at an average rate of 17 miles per hour heading north. If the ship traveled a total of 212 miles in 11 hours, how many miles were traveled heading east?
Answers
let's say the ship went East at 22mph, now, the ship travelled a total of 212 miles in 11 hours... ok... how many miles did it go East? well, let's say it went "d" miles, and it took "t" hours.
now, if the ship after that went North at a rate of 17mph, then it took the slack from the 11 hours total and "d", or it took going North " 11 - t ", and it covered a distance, of also the slack from 212 miles and "d", or " 212 - d ".
[tex]\bf \begin{array}{lccclll} &distance&rate&time\\ &-----&-----&-----\\ East&d&22&t\\ North&212-d&17&11-t \end{array} \\\\\\ \begin{cases} \boxed{d}=22t\\ 212-d=17(11-t)\\ ----------\\ 212-\boxed{22t}=17(11-t) \end{cases}[/tex]
solve for "t", to see how long it took the ship going East.
how many miles it covered? well d = 22t
What is the third step when factoring the trinomial ax^2+bx+c, after you have factored out a common factor in each term?
a.) Add the linear terms together
b.)Multiply the factors together to check
c.)Factor the simplified trinomial
d.) Distribute the common factor
Answers
After factored out a common factor in each term. Factor the simplified trinomial. Option c) is correct.
Step after the the third step when factoring the trinomial ax^2+bx+c to be determine.
Factors is are the sub multiples of the value.
Here,
After factored out a common factor in each term. The next step come is to factor the simplified term which implies taking common and kept in parenthesis.
Thus, after factored out a common factor in each term. Factor the simplified trinomial.
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Ruby is visiting a wildlfe center to gather information for he paper . The center has circular pond with a diameter if 20. What is the approximate area of the pond ?
Answers
area = PI x r^2
r = 20/2 = 10
3.14 x 10^2 = 314 square units
Definition 7.1.1 laplace transform let f be a function defined for t ≥ 0. then the integral {f(t)} = ∞ e−stf(t) dt 0 is said to be the laplace transform of f, provided that the integral converges. to find {f(t)}. f(t) = cos t, 0 ≤ t < π 0, t ≥ π
Answers
Given that
[tex]f(t)=\begin{cases}\cos t&\text{for }0\le t<\pi\\0&\text{for }t\ge\pi\end{cases}[/tex]
the Laplace transform of [tex]f(t)[/tex] is given by the definite integral
[tex]\displaystyle\int_{t=0}^{t\to\infty}f(t)e^{-st}\,\mathrm dt=\int_{t=0}^{t=\pi}\cos t\,e^{-st}\,\mathrm dt+\int_{t=\pi}^{t\to\infty}0\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^\pi\cos t\,e^{-st}\,\mathrm dt[/tex]
[tex]=\dfrac{(1-e^{-\pi s})s}{s^2+1}[/tex]
(which you can find by integrating by parts twice)
A package contains 3 cups of trail mix. A serving of trail mix is ⅓ cup. How many servings of trail mix is in the package?
Answers
3 * 3/1 =
9 <== the package contains 9 servings
The widrh of a rectangle is w yards and the length of a rectangle is (6w-4) yards. The perimeter of the rectangle is given by the algebraic expression 2w+2(6w-4). Simplify the algebraic expression 2w+2(6w-4) and determine the perimeter of a rectangle whose width w is 4 yards
Answers
p=2w+2(6w-4) can be simplified to:
p=2w+12w-8
p=14w-8
if w=4
p=14(4)-8
p=56-8 = 48 yards
check:
w = 4
length = 6w-4= 6(4)-4 = 24-4=20
perimeter = 4*2 + 20*2 = 8+40 = 48
it checks out, perimeter = 48 yards
The average winter snowfall in City A is 105 cm. City B usually gets 2.8 m of snow each winter. Compare the yearly snowfall in the two cities. Complete parts a and b. (A) the difference in one year is __ m. (B) the difference over two years is ___ cm
Answers
City A: 105 centimeters OR 1.05 meters
City B: 280 centimeters OR 2.8 meters
Now that we know the measurements, let's look at the first problem. (A) asks for the difference in one year in meters. The difference of course means subtraction. Now it's here where I personally am confused on what it asking for in terms of which city is it asking for us to subtract from, but using an educated guess, I'll say we're subtracting from city B because if we were to subtract from city A, we'd have a negative, and you can't have a negative amount of snow, only 0 snow.
So once again, for question (A), we subtract 1.05 from 2.8 (2.80 - 1.05) and we get 1.75 meters of snow.
Question (B) asks for the difference during two years in centimeters, so we multiply both measurements by 2 and use the centimeter measurements. (105 x 2 = 210) (280 x 2 = 560) Subtract 560 from 210 (560 - 210) and we get 350 centimeters.
So your answers are: (A) 1.75 meters (B) 350 centimeters
I hope this helps!
the longest runway at an airport has the shape of a rectangle with an area of 2181600sqft. this runway is 180 ft wide. how long is the runway
Answers
If 5x=17, what is the value of 15x-11
Answers
5x=17
x=17/5 = 3.4
15x-11 =
15(3.4)-11 = 51-11 = 40
A carpenter is assigned the job of expanding a rectangular deck where the width is one-fourth the length. The length of the deck is to be expanded by 6 feet, and the width by 2 feet. If the area of the new rectangular deck is 68 ft2 larger than the area of the original deck, find the dimensions of the original deck.
Answers
The area of the original deck is [tex]W*4W=4W^{2} [/tex]
The dimensions of the new deck are :
length = 4W+6
width=W+2
so the area of the new deck is :
[tex](4W+6)(W+2)= 4W^{2}+8W+6W+12= 4W^{2}+14W+12[/tex]
"the area of the new rectangular deck is 68 ft2 larger than the area of the original deck" means that we write the equation:
[tex]4W^{2}+14W+12=68+4W^{2}[/tex]
[tex]14W+12=68[/tex]
[tex]14W=68-12=56[/tex]
[tex]W= \frac{56}{14}= 4 [/tex]
the length is [tex]4W=4*4=16[/tex] ft
Answer: width: 4, length: 16
If 2^m = 4x and 2^w = 8x, what is m in terms of w?
Answers
Two consecutive odd integers have a sum of 44 . Find the integers.
Answers
x + x + 2 = 44
2x + 2 = 44
2x = 44 - 2
2x = 42
x = 42/2
x = 21
x + 2 = 21 + 2 = 23
ur numbers are 21 and 23
2n+2=44 subtract 2 from both sides
2n=42 divide both sides by 2
n=21
So n and n+2 are 21 and 23.
In the first 120 miles over 240 mile journey a truck driver maintained an average speed of 50 mph what was his average featuring the next 120 miles if the average speed of the entire trip with 60 mph
Answers
Next: determine the length of time required to drive the first 120 miles at 50 mph. It is (120 miles) / (50 mph), or 2.4 hours.
Next, find the length of time required to drive the second 120 miles. It is 4 hours less 2.4 hours, or 1.6 hours.
Lastly, find the average speed over which the second 120 miles were covered in 1.6 hours:
average speed (2nd 120 miles) = (120 miles) / (1.6 hours) = 75 mph.
The Center of the Circle is at the origin on a coordinate grid. The vertex of a Parabola that opens upward is at (0,9). If the Circle intersects the parabola at the parabola's vertex, which Statement must be true?
Answers
The parabola and the circle have the same axis of symmetry, and can intersect at one point only.
The statement that must be true is; The maximum number of solution is one
Reason:
The given parameters are;
Location of the center of the circle = The origin (0, 0)
Location of the vertex of the parabola opening upwards = (0, 9)
Point where the circle intersects the parabola = The vertex
Required:
The statement that must be true
Solution;
The equation of the circle is x² + y² = r²
The vertex (0, 9) is a point on the circle, therefore;
0² + 9² = r²
The radius, r = 9
The highest point on the circle is the point with the maximum vertical
distance from the center, which is the point (0, 9), which is also the lowest
point on the parabola.
Therefore, the parabola and the circle can intersect at only the point (0, 9),
which gives;
The maximum number of solution is one.
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The altitude of a triangle is increasing at a rate of 1 cm/ min while the area of the triangle is increasing at a rate of 2 cm2 / min. at what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2 ?
Answers
The rate of change of the base of a triangle, given an increase of the altitude at 1 cm/min and an increase in area at 2 cm2/min, when the altitude is 10 cm and the area is 100 cm2, is 4 cm/min.
Explanation:The subject of this question is related to the field of calculus, specifically dealing with determining the rate of change, or the derivative, of a function. We're asked to determine the rate at which the base of the triangle is changing when the altitude is 10 cm and the area is 100 cm2, given that the altitude of the triangle is increasing at a rate of 1 cm/ min and the area of the triangle is increasing at a rate of 2 cm2 / min.
We know that the area of a triangle is given by the formula 1/2 * base * height. When it comes to rates, we can differentiate this with respect to time t to get dA/dt = 1/2 * (base * dh/dt + height * db/dt) where dA/dt is the rate of change of the area, dh/dt is the rate of change of the height, and db/dt is the rate of change of the base.
Given that dA/dt = 2 cm2/min and dh/dt = 1 cm/min, and we are finding db/dt when the height is 10 cm and the area is 100 cm2, we substitute these values to solve for db/dt. This simplifies to find that the base is increasing at a rate of 4 cm/min.
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is the graph of y=sin(x^6) increasing or decreasing when x=12
Answers
y=sin(x^6)
is
y'=dy/dx=cos(x^6)*(6x^5)
Substitute x=12 (radians)
y'(12)=cos(12^6)*(6*12^5)
=1492992cos(2985984)
=1128592.5 >0
Therefore y(x) is increasing at x=12 (radians).
The house shown is a composite of more than one shape. Which of these methods would you use to find the volume of the house?
Answers
The method that can be used to find the volume of the house is:
Add the volume of a rectangular prism to the volume of the triangular prism.
Step-by-step explanation:In order to find the volume of the house we need to find the volume of the bottom part of the house which in the shape of a rectangular prism or cuboid and volume of the top of the house which is in the shape of a triangular prism.
Hence, the total volume of the house is:
Volume of rectangular prism+Volume of triangular prism.
Derek and Mia place two green marbles and one yellow marble in a bag. Somebody picks a marble out of the bag without looking and records its color (G for green and Y for yellow). They replace the marble and then pick another marble. If the two marbles picked have the same color, Derek loses 1 point and Mia gains 1 point. If they are different colors, Mia loses 1 point and Derek gains 1 point. What is the expected value of the points for Derek and Mia?
Answers
Answer:
Thus, the expected value of points for Derek and Mia are [tex]\dfrac{-1}{9}[/tex] and [tex]\dfrac{1}{9}[/tex] respectively.
Step-by-step explanation:
Number of green marbles = 2 and Number of Yellow marbles = 1
Then, total number of marbles = 2+1 = 3
A person selects two marbles one after another after replacing them.
So, the probabilities of selecting different combinations of colors are,
[tex]1.\ P(GG)=P(G)\times P(G)\\\\P(GG)=\dfrac{2}{3}\times \dfrac{2}{3}\\\\P(GG)=\dfrac{4}{9}[/tex]
[tex]2.\ P(GY)=P(G)\times P(Y)\\\\P(GY)=\dfrac{2}{3}\times \dfrac{1}{3}\\\\P(GY)=\dfrac{2}{9}[/tex]
[tex]3.\ P(YG)=P(Y)\times P(G)\\\\P(YG)=\dfrac{1}{3}\times \dfrac{2}{3}\\\\P(YG)=\dfrac{2}{9}[/tex]
[tex]4.\ P(YY)=P(Y)\times P(Y)\\\\P(YY)=\dfrac{1}{3}\times \dfrac{1}{3}\\\\P(YY)=\dfrac{1}{9}[/tex]
Now, we have that,
If two marbles are of same color, then Mia gains 1 point and Derek loses 1 point.
If two marbles are of different color, then Derek gains 1 point and Mia loses 1 point.
Also, the expected value of a random variable X is [tex]E(X)=\sum_{i=1}^{n} x_i\times P(x_i)[/tex].Then, the expected value of points for Derek is,
[tex]E(D)= (-1)\times \dfrac{4}{9}+1\times \dfrac{2}{9}+1\times \dfrac{2}{9}+(-1)\times \dfrac{1}{9}\\\\E(D)= \dfrac{-5}{9}+\dfrac{4}{9}\\\\E(D)=\dfrac{-1}{9}[/tex]
And the expected value of points for Mia is,
[tex]E(M)= 1\times \dfrac{4}{9}+(-1)\times \dfrac{2}{9}+(-1)\times \dfrac{2}{9}+1\times \dfrac{1}{9}\\\\E(M)= \dfrac{5}{9}-\dfrac{4}{9}\\\\E(M)=\dfrac{1}{9}[/tex].
Thus, the expected value of points for Derek and Mia are [tex]\dfrac{-1}{9}[/tex] and [tex]\dfrac{1}{9}[/tex] respectively.
Answer: P(GG)= 4/9
P(GY)= 2/9
P(YG)= 2/9
P(YY)= 1/9
Derek, E(X) = -1/9
Mia, E(X) = 1/9
Step-by-step explanation: just did it on edge
If f(x) = 3/x+2 - √x-3, complete the following statement (round to the nearest hundredth) f(7)= PLEASE HELP ME
Answers
f(7)=3/9-sqrt(4)
f(7)=1/3-2 = 1/3-6/3 = -5/3 = -1.67
The value of given function f(7) is -1.8.
What is a function?A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and codomain or range.
According to the given problem,
f(x) = [tex]\frac{3}{x + 5}- \sqrt{x - 3}[/tex]
At x = 7,
⇒ f(7) = [tex]\frac{3}{7 + 5} - \sqrt{7-3}[/tex]
⇒ f(7) = [tex]\frac{1}{4} - 2[/tex]
⇒ f(7) = [tex]-\frac{7}{4}[/tex]
⇒ f(7) = - 1.75
≈ -1.8
Hence, we can conclude, the value of function f(7) is -1.8.
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F(x,y)=eâ8xâx2+8yây2. find and classify all critical points of the function. if there are more blanks than critical points, leave the remaining entries blank.
Answers
[tex]F(x,y)e^{8x-x^2}+8y-y^2[/tex]
We have
[tex]\dfrac{\partial F}{\partial x}=(8-2x)e^{8x-x^2}[/tex]
[tex]\dfrac{\partial F}{\partial y}=8-2y[/tex]
Both partial derivatives vanish when
[tex](8-2x)e^{8x-x^2}=0\implies 8-2x=0\implies x=4[/tex]
[tex]8-2y=0\implies y=4[/tex]
so there is only one critical point [tex](4,4)[/tex]. The Hessian matrix for [tex]F(x,y)[/tex] is
[tex]\mathbf H(x,y)=\begin{bmatrix}\dfrac{\partial^2F}{\partial x^2}&\dfrac{\partial^2F}{\partial x\partial y}\\\\\dfrac{\partial^2F}{\partial y\partial x}&\dfrac{\partial^2F}{\partial y^2}\end{bmatrix}=\begin{bmatrix}e^{8x-x^2}(62-32x+4x^2)&0\\0&-2\end{bmatrix}[/tex]
At the critical point, we have
[tex]\det\mathbf H(4,4)=4e^{16}>0[/tex]
[tex]\dfrac{\partial^2F}{\partial x^2}\bigg|_{(x,y)=(4,4)}=-2e^{16}<0[/tex]
which indicates that a relative maximum occurs at [tex](4,4)[/tex], and the function takes on a maximum value of [tex]F(4,4)=16+e^{16}[/tex].
To find and classify critical points of a two-variable function, calculate and set the first partial derivatives to zero to find critical points. Then, use the second derivatives to classify these points. The determinant of the Hessian matrix, made up of the second derivatives, contributes to this classification.
Explanation:To find the critical points of the function F(x,y)=e^8x - x^2 + 8y - y^2, you first need to find the partial derivatives F_x and F_y and set them both equal to zero.
F_x = 8e^8x - 2x and F_y = 8 - 2y. By setting these equal to zero and solving for x and y, you will find the critical points.
Once the critical points are found, we classify them using the second derivative test. This involves computing the second partial derivatives F_xx, F_yy, and F_xy, and evaluating them at the critical points.
Finally, we calculate the determinant D of the Hessian matrix, composed of the second derivatives, at the critical points. The signs and values of these results and the determinants help classifying the critical points.
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What is greater: a half dozen dozen pair of shirts or a half of two dozen dozen shirts
Answers
A half dozen dozen pair of shirts is:
6 dozen pairs:
6 * 12 pairs
72 pairs
PAIRS.
So, for every 1 shirt in this group, there is another that goes with it. It must be multiplied by two.
72 * 2 = 144
There is a total of 144 shirts in this group.
Now, the second group...
Half of two dozen dozen shirts:
Half of two dozen = 12
12 dozen shirts.
12 * 12
144!
See? The amounts are exactly the same!
Sounds more like a riddle than math homework to me...
Hope I could help you out! If my math is incorrect, or I provided an answer you were not looking for, please let me know. However, if my math is explained well and correct, please consider marking my answer as Brainliest! :)
Have a good one.
God bless!
From the computation, a half of two dozen shirts will be greater.
A dozen = 12
It should be noted that a half dozen pair of shirts will be:
= 1/2 × 12
= 6 shirts
A half of two dozen shirts will be:
= 1/2 × (2 × 12)
= 12 shirts
Therefore, a half of two dozen shirts will be greater.
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solve for m
2m = -6n -5; n = 1, 2 ,3
Answers
at n=1:
2m=-6(1)-5=-6-5=-11
m = -11/2
at n=2:
2m=-6(2)-5=-12-5=-17
m = -17/2
at n=3:
2m=-6(3)-5=-18-5=-23
m = -23/2
What is the standard form of 8 hundreds + 2 hundreds
Answers
The standard form of 8 hundred + 2 hundred will be 1000.
What is the standard form of the number?A number can be expressed in a fashion that adheres to specific standards by using its standard form. Standard form refers to any number that may be expressed as a decimal number between 1.0 and 10.0 when multiplied by a power of 10.
Given that the number is 8 hundred + 2 hundred the standard form of the number will be:-
Standard form = 8 hundreds + 2 hundreds
Standard form = 800 + 200
Standard form = 1000
Therefore, the standard form of 8 hundred + 2 hundred will be 1000.
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Show that the series is convergent. how many terms of the series do we need to add in order to find the sum to the indicated accuracy? sum_(n=1)^(infinity) (-1)^(n+1)/( n^7)text( ) \(|text(error)| < 0.00005 \)
Answers
[tex]|S-S_k|\le|a_{k+1}|[/tex]
We have
[tex]a_n=\dfrac1{n^7}[/tex]
so in order to have an error within 0.00005 of the sum's actual value, we need [tex]k[/tex] terms such that
[tex]\left|S-S_k\right|\le\left|\dfrac{(-1)^{k+2}}{(k+1)^7}\right|=\dfrac1{(k+1)^7}<0.00005[/tex]
[tex]\implies (k+1)^7<\dfrac1{0.00005}=20000[/tex]
[tex]\implies k+1<20000^{1/7}\approx4.1156[/tex]
[tex]\implies k<3.1156[/tex]
which suggests that we require [tex]k=4[/tex] terms at the least to approximate the series within the given accuracy.
The series is convergent, but the number of terms needed to find the sum to a specific accuracy cannot be determined.
Explanation:To determine the convergence of the series sum_(n=1)^(infinity) (-1)^(n+1)/( n^7), we can use the Alternating Series Test. The Alternating Series Test states that if the terms of a series alternate in sign and decrease in absolute value, then the series is convergent. In this case, the terms of the series alternate in sign and decrease as n increases, so the series is convergent.
To find the number of terms needed to achieve a sum with an error less than 0.00005, we need to use the Remainder Estimation Theorem. However, this theorem requires that the terms of the series decrease in absolute value, which is not the case in this series. Therefore, we cannot determine the number of terms needed to reach the desired accuracy.
Overall, the series is convergent, but we cannot determine the number of terms needed to reach a specific accuracy.
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Jane is going to walk once around the edge of a rectangular park. The park is 300 yards long and 200 feet wide. How far will Jane walk?
Answers
Jane will walk 733.34 yards around the edge of the rectangular park after converting the width from feet to yards and calculating the perimeter.
Jane is going to walk once around the edge of a rectangular park. The park is 300 yards long and 200 feet wide. To determine how far Jane will walk, we need to calculate the perimeter of the rectangle. First, let's convert all measurements to the same unit. Since the length is given in yards and the width in feet, we can convert the width to yards (1 yard = 3 feet).
Width in yards: 200 feet \/ 3 feet per yard = 66.67 yards.
Now that both measurements are in yards, we can calculate the perimeter:
Perimeter = 2 ×(length + width) = 2 × (300 yards + 66.67 yards) = 2 ×366.67 yards = 733.34 yards.
Therefore, Jane will walk 733.34 yards around the edge of the park.
A music company executive must decide the order in which to present 6 selections on a compact disk. how many choices does she have
Answers
What expression is equivalent to 10x2y+25x2
Answers
Effective rate (APY) is: Never related to compound table Interest for one year divided by annual rate Interest for one year divided by principal for 2 years Interest for one year divided by principal None of these
Answers
is the rate we get for depositing an amount for a year after taking into account compound interest.
Therefore it is the interest for one year divided b the principal.
standard form of the equation of a hyperbola that has vertices at (-10, -15) and (70, -15) and one of its foci at (-11, -15).
Answers
Final Answer:
The standard form of the equation of the hyperbola with vertices at (-10, -15) and (70, -15) and one of its foci at (-11, -15) is (x - 30)^2 / 1600 - (y + 15)^2 / 81 = 1
Explanation:
To write the standard form of the equation of a hyperbola with the given vertices and a focus, we'll follow these steps:
1. Determine the center of the hyperbola.
2. Calculate the distance between the vertices and the center to find the length of the transverse axis (2a).
3. Calculate the distance between a focus and the center to find the focal distance (c).
4. Use the relationship c^2 = a^2 + b^2 to determine the length of the conjugate axis (2b).
5. Write the standard form equation based on the orientation of the hyperbola.
Step 1: Determine the center of the hyperbola.
The center of the hyperbola is the midpoint of the line segment joining the two vertices. Since the vertices are at (-10, -15) and (70, -15), the center (h, k) can be found as follows:
h = (-10 + 70) / 2 = 60 / 2 = 30
k = (-15 + (-15)) / 2 = -30 / 2 = -15
So, the center of the hyperbola is at (30, -15).
Step 2: Calculate the length of the transverse axis (2a).
The distance between the vertices is the length of the transverse axis. The vertices are 80 units apart because they are at (-10, -15) and (70, -15). This means:
2a = 80
a = 40
Therefore, the length of the semi-transverse axis a is 40 units.
Step 3: Calculate the focal distance (c).
The focal distance is the distance between the center and one of the foci. We were given one focus at (-11, -15). Since the center is at (30, -15), the focal distance c is:
c = |30 - (-11)| = |30 + 11| = 41
Step 4: Use the relationship c^2 = a^2 + b^2 to determine b.
We know that a = 40 and c = 41. Plugging these values into the relationship gives us:
41^2 = 40^2 + b^2
1681 = 1600 + b^2
b^2 = 1681 - 1600
b^2 = 81
b = 9
Therefore, the length of the semi-conjugate axis b is 9 units.
Step 5: Write the standard form equation.
Since the hyperbola is horizontal (the vertices have the same y-coordinate), the standard form of its equation is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Plugging in the values for h, k, a, and b, we get:
(x - 30)^2 / 40^2 - (y + 15)^2 / 9^2 = 1
Simplify further by squaring the values of a and b:
(x - 30)^2 / 1600 - (y + 15)^2 / 81 = 1
This is the standard form of the equation of the hyperbola with vertices at (-10, -15) and (70, -15) and one of its foci at (-11, -15).