Given that \( y = 37 \) at \( x = 4 \), find \( y \) in terms of \( x \), giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 2

Substituting ( x = 4 ) and ( y = 37 ):

$37 = 2(4)^3 + \frac{1}{3}(4) + \frac{2}{3}(4)^{3/2} + c$

Calculating the terms:

• ( 4^3 = 64 ) → ( 2 \cdot 64 = 128 )
• ( \frac{1}{3} \cdot 4 = \frac{4}{3} )
• ( (4)^{3/2} = 8 ) so ( \frac{2}{3} \cdot 8 = \frac{16}{3} )

Thus, we have:

$37 = 128 + \frac{4}{3} + \frac{16}{3} + c$

Combining the terms gives:

$37 = 128 + \frac{20}{3} + c$

To isolate ( c ), we convert 37 to a fraction:

$37 = \frac{111}{3},$

therefore:

$c = \frac{111}{3} - 128 = \frac{111}{3} - \frac{384}{3} = -\frac{273}{3}$

Substituting ( c ) back into the equation for ( y ):

$y = 2x^3 + \frac{1}{3}x + \frac{2}{3}x^{3/2} - \frac{273}{3}$

This gives the final simplified form of ( y ) in terms of ( x ).