Question

The lateral surface area \(S\) of a cone with a radius of 14 centimeters can be found using the formula $$S=\pi \cdot 14 \sqrt{14^{2}+h^{2}}$$ where \(h\) is the height (in centimeters) of the cone. a. Use unit analysis to check the units of the formula. b. Sketch the graph of the formula. c. Find the lateral surface area of a cone with a height of 30 centimeters.

Short answer

a) The units in the formula correctly yield an area in square centimeters, b) The graph of the formula is a part of a parabola increasing as height increases, c) The lateral surface area of the cone with a height of 30 cm is given by \(S =\pi \cdot 14 \sqrt{14^{2}+30^{2}}\) square centimeters.

Step-by-step solution

Unit analysis

For the unit analysis, notice that the radius is given in centimeters, and so is the height \(h\). In the formula \(S=\pi \cdot 14 \sqrt{14^{2}+h^{2}}\), squaring 14 cm and \(h\), results in areas (since \(cm^{2}\)). Taking the square root of an area turns it back into a one-dimensional measurement (cm). Thus, when this value is multiplied by \(14 cm\) again (unit of radius), the result is an area measured in square centimeters, so the unit analysis checks out.

Graph Sketching

To draw the graph of the formula, take \(S\) as the y-axis (dependent variable representing surface area) and \(h\) as the x-axis (independent variable representing the height of the cone). Recognize that due to the presence of the square root in the function, the graph shape will resemble a portion of a parabola. As the height increases, the surface area will also increase.

Compute the Lateral Surface Area

Substitute \(h = 30 cm\) into the formula for the lateral surface area to evaluate it. So \(S =\pi \cdot 14 \sqrt{14^{2}+30^{2}}\). Calculate the expression under the square root first, then take the square root, and finally multiply by both \(\pi\) and \(14\). This will give the lateral surface area of the cone.

Key concepts

Geometric Units Analysis

Understanding the units within a geometric formula is crucial for ensuring the accuracy of calculations and for making sure the results are in the appropriate form. In the formula for the lateral surface area of a cone, \(S=\pi \cdot r\sqrt{r^{2}+h^{2}}\), where \(r\) is the radius and \(h\) is the height, both are given in centimeters.

When performing unit analysis, squaring the radius and the height converts the units from centimeters to square centimeters, \(cm^2\), representing an area. The square root of the sum of these squares turns the area back into a linear measurement, specifically the slant height of the cone, which is once again in centimeters. Multiplying this by the radius (in cm) and by \(\pi\), which is unitless, the resultant unit is \(cm^2\), confirming that the formula calculates an area. This unit coherence is fundamental, especially when scaling figures or converting measurements, to maintain the integrity of the calculated values.

Graphing Algebraic Functions

Graphing is a powerful tool in visualizing the relationship between two variables in an algebraic function. In the context of the lateral surface area \(S\) of a cone in relation to \(h\), the height, graphing helps to illustrate how changes in height affect the surface area.

To graph the function \(S=\pi \cdot r\sqrt{r^{2}+h^{2}}\), one would plot \(h\) along the x-axis and \(S\) along the y-axis. Due to the square root term in the function, the graph will show the characteristic shape of a radical function, starting from the point that represents the radius of the base and rising continuously. The graph does not intersect with the y-axis, as there would be no cone without a height, and the rate of increase in the surface area slows down as the height increases. This graphical representation aids in better understanding the cone's geometry and the gradual change in the surface area as the cone gets taller.

Solving for Unknown Height

When dealing with geometric shapes such as cones, finding unknown measures is a common problem. Solving for an unknown height requires rearranging the formula and isolating the height variable. If the formula given is \(S=\pi \cdot r\sqrt{r^{2}+h^{2}}\), and \(S\) and \(r\) are known, one would need to perform algebraic manipulations to solve for \(h\).

Essentially, this involves first dividing both sides by \(\pi \cdot r\), then squaring both sides to remove the square root, followed by subtracting \(r^2\) from both sides, and finally taking the square root to find the value of the height \(h\). This process can be applied to solve for the unknown height in practical scenarios, such as determining the required height of a cone-shaped container to hold a certain volume, or in theoretical problems from geometry or calculus.

Surface Area Calculations

A fundamental concept in geometry is the calculation of surface areas of different shapes. For the cone, the lateral surface area is particularly important and is calculated using the formula \(S=\pi \cdot r\sqrt{r^{2}+h^{2}}\).

The lateral surface area includes the curved surface of the cone, but not the base. To find the lateral surface area for a given cone with specific measurements for radius and height, one must substitute those values into the formula. For example, to calculate the lateral surface area of a cone with a height of 30 centimeters and a radius of 14 centimeters, you substitute the values to get \(S = \pi \cdot 14 \sqrt{14^{2}+30^{2}}\), and carry out the operations step-by-step. Such calculations are vital in various practical applications, from determining the amount of material needed to cover a conical shape to calculating the exposed surface area for a heat or mass transfer in engineering problems.