1963 AMC 12 Problem 40

Attempt Problem 40 of the 1963 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1963 AMC 12 solutions, or check the answer key.

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40.

If

x

is a number satisfying the equation

\sqrt[3]{x+9}-\sqrt[3]{x-9}=3,

then

x^2

is between:

55

and

65

65

and

75

75

and

85

85

and

95

95

and

105

Answer: C

Concepts:radical sum and difference of cubes symmetry (algebra)

Difficulty rating: 2290

Small hint:

Set

u=\sqrt[3]{x+9}

and

v=\sqrt[3]{x-9}

Big hint:

Use both

u-v=3

and

u^3-v^3=18

to find

uv

Solution:

Let

u=\sqrt[3]{x+9}

and

v=\sqrt[3]{x-9}.

Then

u-v=3

and

\begin{aligned} 18&=u^3-v^3\\ &=(u-v)(u^2+uv+v^2), \end{aligned}

so

u^2+uv+v^2=6.

Comparing with

(u-v)^2=9

gives

uv=-1.

Hence

(u+v)^2=5.

Cubing

2u=3\pm\sqrt5

gives

x=\pm4\sqrt5,

and either way

x^2=80,

between

75

and

85.

Therefore, the correct answer is C.

Problem 40 in other years