A quantity u is the sum of two terms , one partly constant and the other varies directly as the square of v . If v =2 when u=203. Find the value of v when u=515
Let the constant term be represented by a constant "c" and the term that varies directly as the square of v be represented by "kv²", where k is a constant.
Given that u = c + kv².
When v = 2, u = 203:
203 = c + k(2)²
203 = c + 4k ----- (Equation 1)
Now we need to find the value of v when u = 515.
515 = c + kv² ----- (Equation 2)
To solve for the value of v, we need to eliminate the constant term "c" and solve for k using the given information.
From Equation 1, we can express c in terms of k:
c = 203 - 4k
Substituting this value for c in Equation 2:
515 = (203 - 4k) + kv²
515 = 203 - 4k + kv²
312 = kv² - 4k
Rearranging the equation:
kv² - 4k = 312
Factoring out k:
k(v² - 4) = 312
Divide both sides by (v² - 4):
k = 312 / (v² - 4)
Now we can substitute this value of k back into Equation 1 to solve for c:
203 = c + 4k
203 = c + 4(312 / (v² - 4))
203 = c + 1248 / (v² - 4)
Multiply both sides by (v² - 4) to eliminate the fraction:
203(v² - 4) = c(v² - 4) + 1248
203v² - 812 = cv² - 4c + 1248
Since the first term of the equation has a v² term and the constant term is a number, we can equate the coefficients:
203 = c
-812 = -4c + 1248
Solving the second equation for c:
-4c + 1248 = -812
-4c = -812 - 1248
-4c = -2060
c = -2060 / -4
c = 515
Therefore, the constant term "c" is 515.
Substituting this back into Equation 1:
203 = c + 4k
203 = 515 + 4k
Solving for k:
4k = 203 - 515
4k = -312
k = -312 / 4
k = -78
Now we have the values of "c" and "k" which we can substitute into Equation 2 to solve for v when u = 515:
515 = c + kv²
515 = 515 + (-78)v²
0 = -78v²
This equation has no real solutions.
Therefore, there is no value of v for which u = 515.
An octagon has interior angle of 120, 110, 130, 114 , 90.if the remaining angles are equal . what is the size of the equal angles
To find the size of the equal angles in the octagon, we need to first calculate the sum of the known interior angles.
Given that the octagon has interior angles measuring 120°, 110°, 130°, 114°, and 90°, we can add these angles:
120° + 110° + 130° + 114° + 90° = 564°.
The sum of the known interior angles is 564°.
Since an octagon has 8 interior angles, we subtract the sum of the known angles from the sum of all the interior angles of the octagon to determine the measure of the remaining equal angles:
Sum of all interior angles of an octagon = (8 - 2) * 180° = 1080°.
1080° - 564° = 516°.
We now have the measure of the remaining equal angles as 516°.
Since an octagon has 8 equal interior angles, we divide the total measure of the equal angles by 8 to find the measure of each equal angle:
516° / 8 = 64.5°.
Therefore, the size of the equal angles in the octagon is 64.5°.
Let's break down the problem step by step:
Step 1: Assign variables to the given information.
Let u be the quantity we need to find, v be the variable that varies directly as the square, and C be the constant term.
Step 2: Write the equation based on the given information.
We can write the equation: u = C + k(v^2), where k is the constant of proportionality.
Step 3: Use the given information to find the values of C and k.
Given that v = 2 when u = 203, we can substitute these values into the equation:
203 = C + k(2^2)
203 = C + 4k
Step 4: Solve for C.
To isolate C, subtract 4k from both sides:
203 - 4k = C
Step 5: Use the value of C to find the value of k.
Substitute the value of C from Step 4 into the equation:
203 = (203 - 4k) + 4k
203 = 203
This equation is always true, meaning k can have any value.
Step 6: Find the value of v when u = 515.
Substitute the values of C and k into the equation from Step 2:
515 = (203 - 4k) + k(2^2)
515 = 203 - 4k + 4k
515 = 203
Step 7: Conclusion
The equation obtained in Step 6 is not true. Therefore, there may be an error in the problem statement or there could be missing information. Please double-check the problem statement or provide additional information if necessary.
To solve this problem, we need to first determine the relationship between the variables u and v. The problem states that u is the sum of two terms, one of which is partly constant, and the other varies directly as the square of v.
Let's represent the constant term as "a" and the term that varies directly as the square of v as "bv^2". Therefore, we have the following equation:
u = a + bv^2
We are given one data point in the problem, where v = 2 and u = 203. Plugging these values into our equation, we get:
203 = a + b(2^2)
203 = a + 4b ---(eq. 1)
Now, we need to find the value of v when u = 515. Let's plug this into our equation:
515 = a + bv^2 ---(eq. 2)
To find the value of v, we need to eliminate the constant term "a" from equations 1 and 2. To do this, we subtract equation 1 from equation 2:
515 - 203 = (a + bv^2) - (a + 4b)
312 = bv^2 - 4b
Now, we can factor out the common term "b" from the right side:
312 = b(v^2 - 4)
To determine the value of v, we need to isolate it. Let's divide both sides of the equation by "b":
312/b = v^2 - 4
Next, let's add 4 to both sides of the equation:
312/b + 4 = v^2
Now, take the square root of both sides to solve for v:
sqrt(312/b + 4) = v
Since we don't know the exact value of "b" yet, we can't evaluate this expression directly. However, we can find the value of "b" by substituting our known data point into equation 1:
203 = a + 4b
We know that v = 2 and u = 203, so we can substitute these values into equation 1:
203 = a + 4b
203 = a + 4(2^2)
203 = a + 4(4)
203 = a + 16
a = 203 - 16
a = 187
Now that we know the value of the constant term, we can substitute it back into equation 1 to solve for "b":
203 = 187 + 4b
203 - 187 = 4b
16 = 4b
b = 16/4
b = 4
Now we can substitute this value of "b" into our expression for v:
v = sqrt(312/b + 4)
v = sqrt(312/4 + 4)
v = sqrt(78 + 4)
v = sqrt(82)
v ≈ 9.06
Therefore, the value of v when u = 515 is approximately 9.06.