Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Sep 19, 2023

1. Formulate and rearrange formulas: a) A rectangle has a perimeter of $28 \mathrm{~cm}$ . Express its area in terms of a side length $x$ . b) For a rectangle, one side $x$ is twice the other side $y$ . Express the area in terms of $x$ . c) For a cube, express the volume $V$ in terms of the total edge length $K$ , and vice versa. d) For a cube, express the volume $V$ in terms of the surface area $O$ , and vice versa. e) For a cuboid with sides $a, b, c$ where $a=2b$ and $b=2c$ , provide a formula for the volume in terms of each variable.

STEP 1

Assumptions1. All the figures mentioned are geometrically perfect (i.e., all sides of a square are equal, all sides and angles of a rectangle are equal and opposite, etc.) . The units of measurements are consistent throughout the problem.

STEP 2

a) For a rectangle with a perimeter of28 cm and one side of length x, the other side will be $(28 -2x) /2$ . The area A of a rectangle is given by the product of its sides, so we can express A as a function of x $A(x) = x \cdot \frac{28 -2x}{2}$

STEP 3

b) For a rectangle where one side x is twice the length of the other side y, the area A can be expressed as a function of x $A(x) = x \cdot \frac{x}{2}$

STEP 4

c) For a cube, the total edge length K is12 times the side length a (since a cube has12 edges of equal length), and the volume V is the cube of the side length. So we can write $V(a) = a^3$ $K(a) =12a$ And inversely $a(V) = \sqrt[3]{V}$ $a(K) = \frac{K}{12}$

STEP 5

d) For a cube, the surface area O is times the square of the side length a (since a cube has faces, each of which is a square with area $a^2$ ), and the volume V is the cube of the side length. So we can write $V(a) = a^3$ $(a) =a^2$ And inversely $a(V) = \sqrt[3]{V}$ $a(O) = \sqrt{\frac{}{}}$

STEP 6

e) For a cuboid with edge lengths a, b, and c, where a =2b and b =2c, the volume V is the product of the edge lengths. So we can write $V(a) = a \cdot \frac{a}{2} \cdot \frac{a}{4}$ $V(b) =2b \cdot b \cdot \frac{b}{2}$ $V(c) =4c \cdot2c \cdot c$

Was this helpful?

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Contact Influencer program Policy Terms