May 31, 2022 – Physicspupil

When $a=1$ , $b=2$ , $c=5$ , $d=6$ , and $m=0$ , give the value of

1. $ad^2+5bc$ 2. $a^2bc^2-3ad+a^4$ 3. $mabc+3a^2b^2c^2$ 4. $b^2cd-abc^2$ 5. $a^4b^3c+bcd$ 6. $ma^3+4cd-ad$ 7. $\displaystyle{\frac{ab^2}{2}+5c}$ 8. $\displaystyle{\frac{md}{3}+\frac{4cd}{b}}$ 9. $\displaystyle{\frac{ac}{d}+a}$ 10. $\displaystyle{\frac{a^2}{c}+3a^2b}$ 11. $\displaystyle{\frac{ab^2}{a}-\frac{a^5d}{b}}$ 12. $\displaystyle{\frac{c+d}{d}+\frac{d-c}{a}}$ 13. $\displaystyle{\frac{d^2-a^2b^2}{a^2b^2}}$ 14. $\displaystyle{\frac{mcd+3abc}{d^2-4b^2}}$ 15. $\displaystyle{\frac{a^2d^2-9b^2}{abcd}}$

_{J. R. Lux & R. S. Pieters. (1969). Exercises in Elementary Algebra}

Solution.

1.

$\begin{aligned} ad^2+5bc & = (1)(6)^2+5(2)(5) \\ & = 36 + 50 \\ & = 86 \\ \end{aligned}$

2.

$\begin{aligned} a^2bc^2-3ad+a^4 & = (1)^2(2)(5)^2-3(1)(6)+(1)^4 \\ & = 50-18+1 \\ & = 33 \\ \end{aligned}$

3.

$\begin{aligned} mabc+3a^2b^2c^2 & = (0)(1)(2)(5)+3(1)^2(2)^2(5)^2 \\ & = 0 + 3(4)(25) \\ & = 300 \\ \end{aligned}$

4.

$\begin{aligned} b^2cd-abc^2 & = (2)^2(5)(6)-(1)(2)(5)^2 \\ & = (4)(5)(6)-(2)(25) \\ & = 120-50 \\ & = 70 \\ \end{aligned}$

5.

$\begin{aligned} a^4b^3c+bcd & = (1)^4(2)^3(5)+(2)(5)(6) \\ & = 8(5) + 60 \\ & = 40 + 60 \\ & = 100 \\ \end{aligned}$

6.

$\begin{aligned} ma^3+4cd-ad & = (0)(1)^3+4(5)(6)-(1)(6) \\ & = 0 + 120 - 6 \\ & = 114 \\ \end{aligned}$

7.

$\begin{aligned} \frac{ab^2}{2} + 5c & = \frac{(1)(2)^2}{2}+5(5) \\ & = \frac{4}{2} + 25 \\ & = 2 + 25 \\ & = 27 \\ \end{aligned}$

8.

$\begin{aligned} \frac{md}{3}+\frac{4cd}{b} & = \frac{(0)(6)}{3}+\frac{4(5)(6)}{(2)} \\ & = 0 + \frac{120}{2} \\ & = 60 \\ \end{aligned}$

9.

$\begin{aligned} \frac{ac}{d}+a & = \frac{(1)(5)}{(6)} + (1) \\ & = \frac{5}{6} + 1 \\ & = 1\frac{5}{6} \\ \end{aligned}$

10.

$\begin{aligned} \frac{a^2}{c}+3a^2b & = \frac{(1)^2}{(5)}+3(1)^2(2) \\ & = \frac{1}{5}+6 \\ & = 6\frac{1}{5} \\ \end{aligned}$

11.

$\begin{aligned} \frac{ab^2}{a}-\frac{a^5d}{b} & = \frac{(1)(2)^2}{(1)}-\frac{(1)^5(6)}{(2)} \\ & = 4 - 3 \\ & = 1 \\ \end{aligned}$

12.

$\begin{aligned} \frac{c+d}{d}+\frac{d-c}{a} & = \frac{(5)+(6)}{(6)} + \frac{(6)-(5)}{(1)} \\ & = \frac{11}{6} + \frac{1}{1} \\ & = 1\frac{5}{6} + 1 \\ & = 2\frac{5}{6} \\ \end{aligned}$

13.

$\begin{aligned} \frac{d^2-a^2b^2}{a^2b^2} & = \frac{(6)^2-(1)^2(2)^2}{(1)^2(2)^2} \\ & = \frac{36-(1)(4)}{(1)(4)} \\ & = \frac{32}{4} \\ & = 8 \\ \end{aligned}$

14.

$\begin{aligned} \frac{mcd+3abc}{d^2-4b^2} & = \frac{(0)(5)(6)+3(1)(2)(5)}{(6)^2-4(2)^2} \\ & = \frac{0+30}{36-4(4)} \\ & = \frac{30}{36-16} \\ & = \frac{30}{20} \\ & = \frac{3}{2} \\ & = 1\frac{1}{2} \\ \end{aligned}$

15.

$\begin{aligned} \frac{a^2d^2-9b^2}{abcd} & = \frac{(1)^2(6)^2-9(2)^2}{(1)(2)(5)(6)} \\ & = \frac{(1)(36)-9(4)}{60} \\ & = \frac{36-36}{60} \\ & = \frac{0}{60} \\ & = 0 \\ \end{aligned}$

Boxes $A$ and $B$ are in contact on a horizontal, frictionless (i.e. $f=0$ ) surface, as shown in the Figure below. Box $A$ has mass $20.0\,\mathrm{kg}$ and box $B$ has mass $5.0\,\mathrm{kg}$ . A horizontal force of $100\,\mathrm{N}$ is exerted on box $A$ . What is the magnitude of the force that box $A$ exerts on box $B$ ?