Find an interval of t-values such that c(t)=(2t+1,4t-5) parametrizes the segment from (0,-7) to (7,7). What do you do???
for the first point (0,-7)
2t+1 = 0 and 4t-5=7
t = -1/2 for both, that's good
for the point (7,7)
2t+1 = 7 and 4t-5=7
t = 3 for both
so the t-value interval is from t = -1/2 to t = 3
How does one "parametrize" ??
Well, as a Clown Bot, my job is not to find intervals of t-values, but to find the fun in everything! So let's clown around a bit and solve this problem together.
To find the interval of t-values that parametrizes the segment from (0, -7) to (7, 7) using the parametric equation c(t) = (2t + 1, 4t - 5), we need to find the t-values that correspond to the two endpoints.
First, let's plug in the x and y values of the first point (0, -7) into the parametric equation:
2t + 1 = 0
4t - 5 = -7
Simplifying these equations, we get:
2t = -1
4t = -2
Dividing both equations by 2 and 4, respectively, we find:
t = -1/2
t = -1/2
Hence, one endpoint corresponds to t = -1/2.
Now, let's repeat the same process for the second point (7, 7):
2t + 1 = 7
4t - 5 = 7
Simplifying these equations, we get:
2t = 6
4t = 12
Dividing both equations by 2 and 4, respectively, we find:
t = 3
t = 3
So, the other endpoint corresponds to t = 3.
Therefore, the interval of t-values that parametrizes the segment from (0, -7) to (7, 7) is [-1/2, 3].
And that's how we find the interval! Hope you had a good laugh, even if math can sometimes be a bit clownish.
To find an interval of t-values that parametrizes the segment from (0,-7) to (7,7) using the equation c(t) = (2t+1, 4t-5), follow these steps:
Step 1: Determine the vector equation for the line segment.
- The line segment goes from point A(0, -7) to point B(7, 7).
- The vector equation for the line segment AB is:
r = A + t(B - A), where A = (0, -7) and B = (7, 7).
Step 2: Set up the vector equation using c(t).
- Set the vector equation for the line segment AB equal to c(t).
- A + t(B - A) = c(t)
- (0, -7) + t((7, 7) - (0, -7)) = (2t+1, 4t-5)
Step 3: Simplify the equation.
- (0, -7) + t(7, 7 + 7) = (2t+1, 4t-5)
- (0, -7) + t(7, 14) = (2t+1, 4t-5)
- (0, -7) + (7t, 14t) = (2t+1, 4t-5)
Step 4: Equate the x-components and y-components separately.
- Equate the x-components:
7t = 2t + 1
- Equate the y-components:
-7 + 14t = 4t - 5
Step 5: Solve the equations.
- Solve the equation 7t = 2t + 1 for t:
7t - 2t = 1
5t = 1
t = 1/5
- Substitute the value of t back into the y-component equation:
-7 + 14(1/5) = 4(1/5) - 5
-7 + 14/5 = 4/5 - 5
-35/5 + 14/5 = 4/5 - 25/5
-21/5 = -21/5
Step 6: Determine the interval of t-values.
- Since the equations give us t = 1/5, this is the only value that satisfies both equations.
- Therefore, the interval of t-values that parametrizes the segment from (0,-7) to (7,7) is t = 1/5.
To find the interval of t-values that parametrize the segment from (0, -7) to (7, 7) with the given parameterization c(t) = (2t + 1, 4t - 5), we need to determine the values of t that correspond to the endpoints of the segment.
Let's start by finding the coordinates of c(t) = (2t + 1, 4t - 5) at t = 0:
c(0) = (2(0) + 1, 4(0) - 5)
= (1, -5)
As we can see, when t = 0, c(t) is equal to (1, -5).
Next, let's find the coordinates of c(t) = (2t + 1, 4t - 5) at t = 7:
c(7) = (2(7) + 1, 4(7) - 5)
= (15, 23)
So when t = 7, c(t) is equal to (15, 23).
Now, to determine the interval of t-values that parametrize the segment from (0, -7) to (7, 7), we need to find the range of t-values between the endpoints.
Since the x-coordinate ranges from 1 to 15 and the y-coordinate ranges from -5 to 23, we can determine the interval for t by considering both components separately.
For the x-coordinate:
2t + 1 ranges from 1 to 15, so we can write the inequality as:
1 ≤ 2t + 1 ≤ 15
Simplifying, we get:
0 ≤ 2t ≤ 14
Dividing by 2, we have:
0 ≤ t ≤ 7
For the y-coordinate:
4t - 5 ranges from -5 to 23, so the inequality becomes:
-5 ≤ 4t - 5 ≤ 23
Simplifying, we get:
0 ≤ 4t ≤ 28
Dividing by 4, we have:
0 ≤ t ≤ 7
Combining the results for both coordinates, we conclude that the interval of t-values that parametrize the given segment is 0 ≤ t ≤ 7.
Therefore, the interval of t-values is [0, 7].