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13 problems match “square roots”. Clear

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1. Positive x in a sequence of squares ●○○○○ 10 ptsThe numbers $1^2$, $x^2$, $3^2$ are consecutive terms of an arithmetic sequence. Find the positive value of $x$. Round to two decimal places.
2. Shortest side of a triangle ●○○○○ 10 ptsA triangle has vertices at coordinates $(1, 2)$, $(7, 10)$ and $(1, 12)$. What is the length of the shortest side of the triangle? Round to two decimal places.
3. Solve a quadratic with square roots ●○○○○ 10 ptsSolve $0=2x^2-18$ for $x$. Give both solutions.
4. Composition of two functions at points ●○○○○ 10 ptsIf $f(x) = x^2-2x+1$ and $g(x) = \sqrt{2x+1}$, what is the value of $f(g(4)) - g(f(3))$?
5. Side length from heptagon area ●○○○○ 10 ptsA regular heptagon has area $78.105$ square cm. Find its side length in cm, using the area formula $A \approx 3.633912444\,x^2$, where $x$ is the side length. Round to three decimal places.
6. Solve a quadratic with fractional coefficient ●○○○○ 10 ptsSolve $\dfrac{2}{3}x^2 - 4 = 12$ for $x$. Give both solutions, rounded to two decimal places.
7. Number of solutions of a trig-root equation ●●○○○ 20 ptsFind the number of real solutions $x$ of the equation $\sin\left(\frac{\pi\sqrt x}{4}\right)+\cos\left(\frac{\pi\sqrt{2-x}}{4}\right)=\sqrt 2$.
8. Product of cube diagonal lengths ●●○○○ 20 ptsA cube has side length 1. Find the product of the lengths of the diagonals of this cube, where a diagonal is a line segment between two vertices that is not an edge.
9. Limit of a square-root difference ●●○○○ 20 ptsFind $\lim_{n\to\infty} \sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)$. Give your answer as a decimal.
10. Limit of nested square roots ●●○○○ 20 ptsFind $\lim_{x\to\infty}\left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$.

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