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Difference between revisions of "1957 AHSME Problems/Problem 43"

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<math>\boxed{\textbf{(B) }35}</math>.
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We want to find the number of lattice points in and on the boundary of the shaded region in the diagram. To do this, we will look at the integer values of <math>x</math> from <math>0</math> to <math>4</math>. At a given value of <math>x</math>, the amount of lattice points in the region is <math>x^2+1</math>, because all of the integers from <math>0</math> up to and including <math>x^2</math> are in the region. Thus, evaluating this expression at at <math>x=0,1,2,3,</math> and <math>4</math> and adding the results together, we see that the number of lattice points is <math>1+2+5+10+17=35</math>, so our answer is <math>\boxed{\textbf{(B) }35}</math>.
  
 
== See Also ==
 
== See Also ==

Revision as of 09:47, 27 July 2024

Problem

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:

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 35\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ \infty$

Solution

[asy] path p = (0,0){right}..(1,1)..(2,4)..(3,9)..(4,16); // Shaded Region fill(p--(4,0)--cycle,lightred); // x-Axis draw((-4,0)--(16,0), arrow=Arrows); label("$x$",(18,0)); // y-Axis draw((0,-4)--(0,16), arrow=Arrows); label("$y$",(0,18)); // y=x^2 draw(p); // x=4 draw((4,-5)--(4,20), arrow=Arrows(TeXHead)); [/asy]

We want to find the number of lattice points in and on the boundary of the shaded region in the diagram. To do this, we will look at the integer values of $x$ from $0$ to $4$. At a given value of $x$, the amount of lattice points in the region is $x^2+1$, because all of the integers from $0$ up to and including $x^2$ are in the region. Thus, evaluating this expression at at $x=0,1,2,3,$ and $4$ and adding the results together, we see that the number of lattice points is $1+2+5+10+17=35$, so our answer is $\boxed{\textbf{(B) }35}$.

See Also

1957 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 42 		Followed by
Problem 44
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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