In the given equation, $k$ is a constant and $y > k$. The solution to this equation is $41$. What is the value of $k$?
Solution: Rewrite the equation as $\frac{5}{\sqrt[6]{y-k}} = \sqrt[3]{y-k}$.
Cross-multiply: $5 = \sqrt[3]{y-k} \cdot \sqrt[6]{y-k}$.
Using fractional exponents: $5 = (y-k)^{1/3} \cdot (y-k)^{1/6} = (y-k)^{1/3 + 1/6} = (y-k)^{1/2}$.
$5 = \sqrt{y-k} \Rightarrow 25 = y - k$.
Given $y = 41$: $25 = 41 - k \Rightarrow k = 16$.
The correct answer is 16.
Question 2
$$4x^{2} - px + w = -63$$
In the given equation, $p$ and $w$ are integer constants. The equation has exactly one real solution. Which is NOT a possible value of $w$?
Solution: Rewrite the equation as $4x^2 - px + (w + 63) = 0$.
For exactly one real solution, the discriminant $D = b^2 - 4ac$ must be zero:
$(-p)^2 - 4(4)(w+63) = 0 \Rightarrow p^2 = 16(w+63)$.
Since $p$ is an integer, $16(w+63)$ must be a perfect square. This means $(w+63)$ must be a perfect square (e.g., $0, 1, 4, 9, 16...$).
Check values:
A) $-54 + 63 = 9$ (Perfect square)
B) $-47 + 63 = 16$ (Perfect square)
C) $-38 + 63 = 25$ (Perfect square)
D) $-30 + 63 = 33$ (NOT a perfect square)
The correct answer is D.
Question 3
An object's speed is increasing at a rate of $8.50$ meters per second squared. What is this rate, in miles per minute squared, rounded to the nearest tenth? (Use $1\text{ mile} = 1,609\text{ meters}$.)
Solution: Convert units using dimensional analysis:
$\frac{8.50 \text{ m}}{1 \text{ s}^2} \times \frac{1 \text{ mile}}{1609 \text{ m}} \times \left(\frac{60 \text{ s}}{1 \text{ min}}\right)^2$
$= \frac{8.50 \times 3600}{1609} \text{ miles/min}^2$
$= \frac{30600}{1609} \approx 19.018$.
Rounded to the nearest tenth, the value is $19.0$.
The correct answer is 19.0.
In the given equation, $k$ is a positive constant. The product of the solutions to the equation is $49$. What is the value of $k$?
Solution: For a quadratic $ax^2 + bx + c = 0$, the product of solutions is $c/a$.
Here $a = 2/7$ and $c = \sqrt{7k+4}$.
Product $= \frac{\sqrt{7k+4}}{2/7} = \frac{7\sqrt{7k+4}}{2} = 49$.
$7\sqrt{7k+4} = 98 \Rightarrow \sqrt{7k+4} = 14$.
$7k + 4 = 14^2 = 196$.
$7k = 192 \Rightarrow k = 192/7 \approx 27.42$.
The correct answer is 192/7 (or approx 27.4).
Question 5
Line $k$ contains the points $(-3, -40)$, $(v, 0)$, and $(5, 56)$. What is the value of $v$?
Solution: Since the points are on the same line, the slope between any two pairs must be equal.
Slope between $(-3, -40)$ and $(5, 56)$: $m = \frac{56 - (-40)}{5 - (-3)} = \frac{96}{8} = 12$.
Now find the slope between $(-3, -40)$ and $(v, 0)$: $12 = \frac{0 - (-40)}{v - (-3)} = \frac{40}{v + 3}$.
$12(v + 3) = 40 \Rightarrow 12v + 36 = 40 \Rightarrow 12v = 4 \Rightarrow v = 4/12 = 1/3$.
The correct answer is 1/3.