Question

We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $x$ -axis, the line $x=4,$ and the parabola $y=x^2$ is:

$24$

$35$

$34$

$30$

not finite

Answer: B

Concepts:lattice point parabola counting integers in a range

Difficulty rating: 1550

Small Hint:

Only the integer $x$ -coordinates $0,$ $1,$ $2,$ $3,$ and $4$ occur

Big Hint:

For a fixed integer $x,$ count the integer $y$ -values from $0$ through $x^2$

Solution:

For each $x$ among $0,$ $1,$ $2,$ $3,$ and $4,$ there are $x^2+1$ integer values from $y=0$ through $y=x^2.$ Therefore the number of lattice points is $\begin{aligned} &(0^2+1)+(1^2+1)+(2^2+1)\\ &\quad +(3^2+1)+(4^2+1)\\ &=1+2+5+10+17\\ &=35. \end{aligned}$

Thus, the correct answer is B.

1957 AMC 12 Problem 43

Answer: B

Difficulty rating: 1550

Small Hint:

Big Hint:

Solution:

Problem 43 in Other Years