Question

mr.gs quarter 1 exam--- good luck!!! sharing, no a.i. just h.i. solve the formula for the specified variable. 1) a = p(1 + nr) for n a) n = \\(\frac{a}{r}\\) b) n = \\(\frac{a - p}{pr}\\) c) n = \\(\frac{p - a}{pr}\\) d) n = \\(\frac{a - p}{pr}\\) solve the problem. 2) two cars leave a city and head in the same direction. after 3 hours, the faster car is 33 miles ahead of the slower car. the faster car has traveled 138 miles. find the speeds of the two cars. a) 48 mph and 59 mph b) 46 mph and 57 mph c) 110 mph and 121 mph d) 35 mph and 46 mph 3) if f(x) = 9x + 5 and g(x) = 2x²+3x - 7, find g(f(2)). a) 32 b) 68 c) 108 d) 1120 4) the width of a rectangle is x - \\(\frac{3}{4}\\) feet and its area is 8x³ + 14x² + 13x - 21 square feet. write a polynomial that represents the length of the rectangle. a) 8x² + 20x - 28 ft b) 8x² + 20x + 28 ft c) 8x² - 20x + 28 ft d) 8x² + 8x + 7 ft 5) a ladder is resting against a wall. the top of the ladder touches the wall at a height of 18 ft. find the length of the ladder if the length is 6 ft more than its distance from the wall. a) 30 ft b) 36 ft c) 18 ft d) 24 ft solve the polynomial inequality and graph the solution set on a number line. express the solution set in interval notation. 6) (x + 7)(x + 3)(x - 7) > 0 a) (7, ∞) b) (-7,-3) ∪ (7, ∞) c) (-∞,-3) d) (-∞,-7) ∪ (-3,7) perform the indicated operations and write the result in standard form. 7) (-6+\\(\sqrt{-4}\\))² a) 36 - 4i b) 32 - 24i c) 36 + 4i d) 40 - 24i

Explanation:

Step1: Solve $A = P(1+nr)$ for $n$

First, expand the right - hand side: $A = P + Pnr$. Then, subtract $P$ from both sides: $A - P=Pnr$. Finally, divide both sides by $Pr$ to get $n=\frac{A - P}{Pr}$.

Step2: Solve the speed problem

Let the speed of the slower car be $x$ mph and the speed of the faster car be $y$ mph. After 3 hours, the faster car has traveled 138 miles, so $y=\frac{138}{3}=46$ mph. The faster car is 33 miles ahead of the slower car after 3 hours. So the slower car has traveled $138 - 33 = 105$ miles in 3 hours, and its speed $x=\frac{105}{3}=35$ mph.

Step3: Find $g(f(2))$

First, find $f(2)$: $f(2)=9\times2 + 5=18 + 5=23$. Then, find $g(23)$: $g(x)=2x^{2}+3x - 7$, so $g(23)=2\times23^{2}+3\times23 - 7=2\times529+69 - 7=1058+69 - 7 = 1120$.

Step4: Find the length of the rectangle

The area of a rectangle $A = lw$. Given $A = 8x^{3}+14x^{2}+13x - 21$ and $w=x-\frac{3}{4}$. Using polynomial long - division or synthetic division (with $x=\frac{3}{4}$), we divide $8x^{3}+14x^{2}+13x - 21$ by $x-\frac{3}{4}$. The result is $8x^{2}+20x + 28$.

Step5: Find the length of the ladder

Let the distance from the wall be $x$ feet. Then the length of the ladder is $x + 6$ feet. Using the Pythagorean theorem $x^{2}+18^{2}=(x + 6)^{2}$. Expand: $x^{2}+324=x^{2}+12x + 36$. Subtract $x^{2}$ from both sides: $324=12x + 36$. Subtract 36 from both sides: $12x=288$, so $x = 24$ and the length of the ladder is $x + 6=30$ feet.

Step6: Solve the polynomial inequality $(x + 7)(x + 3)(x - 7)>0$

The critical points are $x=-7,x=-3,x = 7$. Test the intervals $(-\infty,-7),(-7,-3),(-3,7),(7,\infty)$. The solution is $(-7,-3)\cup(7,\infty)$.

Step7: Simplify $(-6+\sqrt{-4})^{2}$

First, rewrite $\sqrt{-4}=2i$. Then $(-6 + 2i)^{2}=(-6)^{2}+2\times(-6)\times2i+(2i)^{2}=36-24i-4=32-24i$.

Answer:

1. D. $n=\frac{A - P}{Pr}$
2. D. 35 mph and 46 mph
3. D. 1120
4. B. $8x^{2}+20x + 28$ ft
5. A. 30 ft
6. B. $(-7,-3)\cup(7,\infty)$
7. B. $32-24i$