Let $r$ be the radius & $h$ be the height of the cylinder having its total surface area $A$ (constant) since cylindrical container is closed at the top (circular) then its surface area (constant\fixed) is given as $$=\text{(area of lateral surface)}+2\text{(area of circular top/bottom)}$$$$A=2\pi rh+2\pi r^2$$ $$h=\frac{A-2\pi r^2}{2\pi r}=\frac{A}{2\pi r}-r\tag 1$$ Now, the volume of the cylinder $$V=\pi r^2h=\pi r^2\left(\frac{A}{2\pi r}-r\right)=\frac{A}{2}r-\pi r^3$$ differentiating $V$ w.r.t. $r$, we get $$\frac{dV}{dr}=\frac{A}{2}-3\pi r^2$$ $$\frac{d^2V}{dr^2}=-6\pi r<0\ \ (\forall\ \ r>0)$$ Hence, the volume is maximum, now, setting $\frac{dV}{dr}=0$ for maxima $$\frac{A}{2}-3\pi r^2=0\implies \color{red}{r}=\color{red}{\sqrt{\frac{A}{6\pi}}}$$ Setting value of $r$ in (1), we get $$\color{red}{h}=\frac{A}{2\pi\sqrt{\frac{A}{6\pi}}}-\sqrt{\frac{A}{6\pi}}=\left(\sqrt{\frac{3}{2}}-\frac{1}{\sqrt 6}\right)\sqrt{\frac{A}{\pi}}=\color{red}{\sqrt{\frac{2A}{3\pi}}}$$ Hence, the ratio of height $(h)$ to the radius $(r)$ is given as $$\color{}{\frac{h}{r}}=\frac{\sqrt{\frac{2A}{3\pi}}}{\sqrt{\frac{A}{6\pi}}}=\sqrt{\frac{12\pi A}{3\pi A}}=2$$ $$\bbox[5pt, border:2.5pt solid #FF0000]{\color{blue}{\frac{h}{r}=2}}$$
Check out our ratio calculator. Ratio - practice problems. Cylinder practice problems. Surface area to volume ratio calculator helps you determine the amount of surface an object has relative to its size. The concept of surface area to volume ratio, also denoted as SA/VOL or SA:V, is given considerable importance in the sciences (such as aerodynamics for instance), and this article will explore why. If you want to learn how to find volume and surface area of an object, you can use the volume calculator or the surface area calculator to learn how to find the volume and surface area of geometry shapes like cylinder, cube, cone, rectangular prism, or triangular prism, etc. Also, check out the body surface area calculator to find the surface area of a human body! Keep reading to learn answers to the following questions:
The surface area of an object or a body is the total area of all its exposed surfaces, i.e., SA is simply the outside area of an object. On the other hand, Volume refers to the amount of space occupied by the object; it can also be the amount of space inside of the object. For instance, let's say we have a cloth box. The box's surface area is a measure of the amount of wrapping paper required to cover the entire box precisely, while the volume of the box is a measure of the space inside the box which determines the number of clothes the box can carry.
Surface area to volume ratio is simply an object’s surface area divided by its volume. It gives the proportion of surface area per unit volume of the object (e.g., sphere, cylinder, etc.). Therefore, the formula to calculate surface area to volume ratio is: SA/VOL = surface area (x2) / volume (x3) SA/VOL = x-1,where x is the unit of measurement. Below is a table showing how to calculate the surface area to volume ratio of some common three-dimensional objects.
Since you've learned how to calculate the surface area to volume ratio, let's find out how to use the surface to volume ratio calculator:
The ratio of surface area to volume of an object is important in the sciences because it determines how fast matter and energy can be transferred within an object and between an object and its environment. Looking at the formulas given in the table above, you will find that when the length L of the cube or the radius R of the cylinder is doubled (or reduced by half), it does not lead to a proportionate increase (or decrease) in the value of the surface area and volume. This is because an increase or decrease in these parameters (length or radius) results in a greater increase (or decrease) in volume than the increase in surface area since the value of surface area is squared (x2) while that of volume is cubed (x3). As a result, the surface area to volume ratio is inversely proportional to the size of an object, given that length and radius determine the size. In other words, as the size of an object increases, its ratio of surface area to volume decreases; conversely, as the size of an object decreases, its ratio of surface area to volume increases. The implication of the surface area to volume ratio is that energy or matter can move faster in objects or organisms with a higher surface area to volume ratio than those with a lower surface area to volume ratio. The SA:V has significant implications in cell theory since cell surface area to volume ratio controls the success of its metabolic processes. Cells are small to allow substances like glucose and oxygen to move through diffusion and get rid of their waste. As the cell grows and the SA:V decreases, it may not be able to get these substances from one end of a cell to the next by diffusion as fast as it should, which slows down cell processes and growth. Thus, the cell surface area to volume ratio is improved through:
The principle also explains why sprinkled water evaporates faster than the same amount of water in a bucket or why granulated sugar dissolves faster than a sugar cube. Put simply: a higher surface area improves the reactivity of a process.
Surface area to volume ratio is the amount of surface area or total exposed area of a body relative to its volume or size. It is denoted as SA/VOL or SA:V.
The formula to calculate surface area to volume ratio is: SA/VOL = surface area (x2) / volume (x3) SA/VOL = x-1
The ratio of surface area to volume, or the surface area to volume ratio, is the amount of surface area or total exposed area of a body relative to its volume or size.
The surface area to volume ratio is important because it determines the rate of movement of materials or energy within a body and between a body and its environment. A high surface area to volume ratio means the body can swiftly transfer materials or energy because there is less space. In contrast, a low surface area to volume ratio means that the volume or size of the object is larger than the surface medium of transfer. Hence, it'll take a longer time for the materials or energy to reach their destination.
You can calculate your body surface area to volume ratio using the surface area to volume ratio formula or the simple-to-use body surface area calculator. |
What is the ratio of the surface area of the small cylinder to the surface area of the large cylinder and the ratio of volume if the ratio of radii is 3 7?
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