Alright, let's break down the problem step-by-step to find the necessary equations and values.
1. We know that the total price [tex]\( P \)[/tex] is the sum of two commodities.
- One commodity varies directly as [tex]\( q^2 \)[/tex].
- The other varies inversely as [tex]\( q \)[/tex].
Let:
- The first commodity be [tex]\( k \cdot q^2 \)[/tex] (where [tex]\( k \)[/tex] is a constant)
- The second commodity be [tex]\( \frac{m}{q} \)[/tex] (where [tex]\( m \)[/tex] is another constant)
Therefore, the total price [tex]\( P \)[/tex] can be given by the equation:
[tex]\[ P = k \cdot q^2 + \frac{m}{q} \][/tex]
We are given two conditions:
1. [tex]\( P = 32 \)[/tex] when [tex]\( q = 2 \)[/tex]
2. [tex]\( P = 86 \)[/tex] when [tex]\( q = 4 \)[/tex]
Using these conditions, we can set up two equations to solve for [tex]\( k \)[/tex] and [tex]\( m \)[/tex].
For [tex]\( q = 2 \)[/tex]:
[tex]\[ 32 = k \cdot (2^2) + \frac{m}{2} \][/tex]
[tex]\[ 32 = 4k + \frac{m}{2} \][/tex]
For [tex]\( q = 4 \)[/tex]:
[tex]\[ 86 = k \cdot (4^2) + \frac{m}{4} \][/tex]
[tex]\[ 86 = 16k + \frac{m}{4} \][/tex]
Now let's solve these equations simultaneously:
Step 1: Multiply the first equation by 2 to make the coefficients of [tex]\( \frac{m}{q} \)[/tex] equal:
[tex]\[ 64 = 8k + m \][/tex]
Step 2: Subtract the equation [tex]\( 64 = 8k + m \)[/tex] from the second equation [tex]\( 86 = 16k + \frac{m}{4} \)[/tex]:
[tex]\[ 86 = 16k + \frac{m}{4} \][/tex]
[tex]\[ \text{Simplify the second equation:} \][/tex]
[tex]\[ 86 = 16k + \frac{m}{4} \][/tex]
[tex]\[ 86 \cdot 4 = 16k \cdot 4 + m \cdot 4 \cdot \frac{1}{4} \][/tex]
[tex]\[ 86 - 64 = 16k - 8k \][/tex]
[tex]\[ 22 = 8k \][/tex]
[tex]\[ k = \frac{22}{8} = 2.75 \][/tex]
Step 3: Substitute [tex]\( k = 2.75 \)[/tex] back into the equation [tex]\( 64 = 8k + m \)[/tex]:
[tex]\[ 64 = 8 \cdot 2.75 + m \][/tex]
[tex]\[ 64 = 22 + m \][/tex]
[tex]\[ m = 64 - 22 \][/tex]
[tex]\[ m = 42 \][/tex]
So, the constants are:
[tex]\[ k = 2.75 \][/tex]
[tex]\[ m = 42 \][/tex]
a) Equation for [tex]\( P \)[/tex] in terms of [tex]\( q \)[/tex]
So, the equation for [tex]\( P \)[/tex] in terms of [tex]\( q \)[/tex] is:
[tex]\[ P = 2.75q^2 + \frac{42}{q} \][/tex]
b) Value of [tex]\( P \)[/tex] when [tex]\( q = 3 \)[/tex]
Now, substituting [tex]\( q = 3 \)[/tex] into the equation:
[tex]\[ P = 2.75 \cdot 3^2 + \frac{42}{3} \][/tex]
[tex]\[ P = 2.75 \cdot 9 + 14 \][/tex]
[tex]\[ P = 24.75 + 14 \][/tex]
[tex]\[ P = 38.75 \][/tex]
So, the value of [tex]\( P \)[/tex] when [tex]\( q = 3 \)[/tex] is:
[tex]\[ P = 38.75 \][/tex]
Therefore:
- (a) The equation for [tex]\( P \)[/tex] in terms of [tex]\( q \)[/tex] is [tex]\( P = 2.75q^2 + \frac{42}{q} \)[/tex]
- (b) The value of [tex]\( P \)[/tex] when [tex]\( q = 3 \)[/tex] is [tex]\( 38.75 \)[/tex]
