A quantity U is the sum of two terms, one party constant and the other veries directly as the square of V. if V=2 when U=35 and V=5, U=203. Find the value of V and U=515
V=7.5
U=515
We are given that the quantity U is the sum of two terms. Let's denote these terms as a constant term, C, and a term that varies directly with the square of V, kV^2.
We are given two sets of values for U and V:
When V = 2, U = 35
When V = 5, U = 203
Let's write the equations based on this information:
When V = 2:
U = C + k(2)^2
35 = C + 4k --(Equation 1)
When V = 5:
U = C + k(5)^2
203 = C + 25k --(Equation 2)
We can solve these two equations simultaneously to find the values of C and k.
First, let's eliminate the C term. Subtracting equation 1 from equation 2, we get:
203 - 35 = (C + 25k) - (C + 4k)
168 = 25k - 4k
168 = 21k
Dividing both sides by 21, we find:
k = 8
Now, substitute the value of k into equation 1 to find C:
35 = C + 4(8)
35 = C + 32
C = 35 - 32
C = 3
Therefore, the constant term C is equal to 3.
Now, let's substitute the values of C and k into the equation U = C + kV^2 to find the value of V when U = 515.
515 = 3 + 8V^2
Subtracting 3 from both sides:
512 = 8V^2
Dividing both sides by 8:
64 = V^2
Taking the square root of both sides:
V = ±8
Since V cannot be negative in this context, we have V = 8.
Thus, when U = 515, the value of V is 8.
To find the value of V and U in the given scenario, we need to use the information provided about the relationship between U and V.
Let's break down the relationship between U and V:
1. "A quantity U is the sum of two terms" - This means that U can be represented as the sum of two parts.
2. "One part is constant" - Let's denote this constant term as 'C'.
3. "The other part varies directly as the square of V" - This means that the other term, denoted as 'k', is directly proportional to the square of V.
With this information, we can express the relationship between U and V as follows:
U = C + k * V^2
Now, let's use the given data to solve for the constant term C and the proportionality constant k.
1. When V = 2, U = 35:
35 = C + k * 2^2
35 = C + 4k ----(Equation 1)
2. When V = 5, U = 203:
203 = C + k * 5^2
203 = C + 25k ----(Equation 2)
By solving these two equations simultaneously, we can find the values of C and k.
Step 1: Subtract Equation 1 from Equation 2 to eliminate C:
(203 - 35) = (C + 25k) - (C + 4k)
168 = 25k - 4k
168 = 21k
Step 2: Solve for k:
k = 168 / 21
k = 8
Step 3: Substitute the value of k back into Equation 1 to find C:
35 = C + 4k
35 = C + 4 * 8
35 = C + 32
C = 35 - 32
C = 3
Now that we have the values of C (constant term) and k (proportionality constant), we can use them to find the value of V in the given U = 515.
U = C + k * V^2
515 = 3 + 8 * V^2
Step 1: Subtract 3 from both sides:
515 - 3 = 8 * V^2
512 = 8 * V^2
Step 2: Divide both sides by 8:
512 / 8 = V^2
64 = V^2
Step 3: Take the square root of both sides:
√64 = √(V^2)
8 = V
Therefore, when U = 515, V = 8.
AAAaannndd the bot gets it wrong yet again!
u = av^2 + b
so
4a+b = 35
25a+b = 203
a=8, b=3
u = 8v^2 + 3
so when u=515,
8v^2+3 = 515
v = 8