A circular pond is modeled by the equation x^2+y^2= 225. A bridge over the pond is modeled by a segment of the equation x–7y=–75. What are the coordinates of the points where the bridge meets the edge of the pond?
Can you teach me how to do this?
from the linear equation: x = 7y - 75
intersect the two relations,
x^2 + y^2 = 225
(7y-75)^2 + y^2 - 225 = 0
49y^2 - 1050y + 5625 + y^2 - 225 = 0
50y^2 - 1050y + 5400 = 0
divide by 50
y^2 - 21y + 108 = 0
(y - 12)(y - 9) = 0
y = 12 or y = 9
if y = 12, x = 9
if y = 9 , x = -12
the bridge meets the point at (9,12) and (-12,9)
verification by Wolfram:
http://www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%3D+225+%2C+x%E2%80%937y%3D%E2%80%9375+from+-15+to+15
Absolutely! However, as a Clown Bot, I must warn you that my teaching style involves a fair amount of humor. So, prepare yourself for some chuckles along the way!
To find the coordinates where the bridge meets the edge of the pond, we need to solve the system of equations formed by the circular pond and the bridge equation.
Let's start by rewriting the bridge equation in terms of x:
x - 7y = -75
⇒ x = 7y - 75
Now, substitute this value of x into the equation of the circular pond:
(7y - 75)^2 + y^2 = 225
Let's simplify this equation step by step:
49y^2 - 1050y + 5625 + y^2 = 225
50y^2 - 1050y + 5400 = 0
Now, we have a quadratic equation! This looks like a clown's hat, doesn't it? Let's find its roots using the quadratic formula:
y = (-(-1050) ± √((-1050)^2 - 4(50)(5400))) / (2(50))
y = (1050 ± √(1102500 - 1080000)) / 100
y = (1050 ± √(22500)) / 100
y = (1050 ± 150) / 100
Simplifying further:
y1 = (1050 + 150) / 100 = 12 (approx.)
y2 = (1050 - 150) / 100 = 9 (approx.)
Now, substitute these values of y back into the equation x = 7y - 75 to find the corresponding x-values:
For y = 12:
x = 7(12) - 75 = 84 - 75 = 9
For y = 9:
x = 7(9) - 75 = 63 - 75 = -12
So, the coordinates where the bridge meets the edge of the pond are (9, 12) and (-12, 9). Voila!
Now, go ahead and impress your friends with your newfound knowledge of bridge-meeting-the-pond coordinates. And remember, math can be fun when we have a sense of humor!
To find the coordinates where the bridge meets the edge of the pond, we need to solve the system of equations formed by the equations x^2+y^2= 225 and x–7y=–75.
Here is how you can do it step-by-step:
Step 1: Solve the equation x–7y=–75 for x in terms of y.
x = 7y – 75
Step 2: Substitute the value of x from equation (1) into equation (2).
(7y – 75)^2 + y^2 = 225
Step 3: Expand the equation and rearrange it to form a quadratic equation.
49y^2 – 1050y + 5625 + y^2 = 225
50y^2 – 1050y + 5400 = 0
Step 4: Solve the quadratic equation for y. You can use factoring, completing the square, or the quadratic formula.
(y – 12)(50y – 450) = 0
y – 12 = 0 or 50y – 450 = 0
y = 12 or y = 9
Step 5: Substitute the values of y back into equation (1) to find the corresponding x-coordinates.
For y = 12:
x = 7(12) – 75
x = 84 – 75
x = 9
So one point of intersection is (x = 9, y = 12).
For y = 9:
x = 7(9) – 75
x = 63 – 75
x = -12
So another point of intersection is (x = -12, y = 9).
Therefore, the coordinates where the bridge meets the edge of the pond are (9, 12) and (-12, 9).
To find the coordinates of the points where the bridge meets the edge of the pond, we need to solve the system of equations formed by the equations of the pond and the bridge.
First, let's rearrange the equation of the bridge, x - 7y = -75, into the slope-intercept form (y = mx + b). We do this by isolating y on one side of the equation:
x - 7y = -75
-7y = -x - 75
y = (1/7)x + 75/7
Now we have the equation of the bridge in the form y = mx + b, where m is the slope and b is the y-intercept.
Now let's substitute this expression for y into the equation of the pond, x^2 + y^2 = 225. We substitute (1/7)x + 75/7 for y in this equation:
x^2 + ((1/7)x + 75/7)^2 = 225
Expanding and simplifying, we get:
x^2 + (1/49)x^2 + (15/7)x + (1125/49) = 225
(50/49)x^2 + (15/7)x + (1125/49) = 225
Now we can solve this quadratic equation to find the x-coordinates of the points where the bridge meets the edge of the pond. We can do this by factoring, completing the square, or using the quadratic formula.
After obtaining the x-coordinates, we can substitute these values back into the equation of the bridge, y = (1/7)x + 75/7, to find the corresponding y-coordinates.
Once we have the coordinates of the points where the bridge meets the edge of the pond, we will have our solution.