/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 A tank has the shape of a cylind... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 35

A tank has the shape of a cylinder with hemispherical ends. If the cylindrical part is 100 centimeters long and has an outside diameter of 20 centimeters, about how much paint is required to coat the outside of the tank to a thickness of 1 millimeter?

Short Answer

Expert verified
Approximately 754 cubic centimeters of paint is required.

Step by step solution

01

Understanding the Shape

The tank consists of a cylindrical part with two hemispherical ends. The cylinder has length 100 cm and diameter 20 cm, which gives it a radius of 10 cm. Each hemisphere has the same radius as the cylinder, i.e., 10 cm.
02

Surface Area of the Cylinder

The surface area of the cylindrical part (excluding the ends) is calculated using the formula \(2\pi r h\), where \(r = 10\) cm and \(h = 100\) cm. Thus, the surface area of the cylinder is \(2\pi \times 10 \times 100 = 2000\pi\) square cm.
03

Surface Area of the Hemispherical Ends

Each hemisphere has a surface area of \(2\pi r^2\). Since there are two hemispheres, the total surface area is \(2 \times 2\pi \times 10^2 = 400\pi\) square cm.
04

Total Surface Area to be Painted

We sum the surface area of the cylinder and the hemispheres: \(2000\pi + 400\pi = 2400\pi\) square cm.
05

Convert Thickness from Millimeters to Centimeters

The thickness of the paint is 1 mm, which is equivalent to 0.1 cm.
06

Calculate Paint Volume Required

The volume of paint required is the product of the surface area and the thickness of the paint. \(Volume = 2400\pi \times 0.1 = 240\pi\) cubic cm.
07

Approximate the Volume of Paint

Using \(\pi \approx 3.1416\), the volume of paint is approximately \(240 \times 3.1416 = 753.98\) cubic cm, rounded to two decimal places.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Cylindrical Shapes
Cylindrical shapes are three-dimensional figures defined by their circular bases and uniform height.
A cylinder has two parallel circular bases connected by a curved surface. In our problem, the cylindrical part of the tank has a length of 100 cm and a radius, derived from half its diameter, of 10 cm.
To calculate the surface area of a cylindrical shape, excluding its top and bottom, you utilize the formula: \(2\pi rh\), where \(r\) is the radius and \(h\) is the height or length.
This formula essentially measures the lateral surface area, similar to how you would unroll the cylinder into a rectangle."},{
Exploring Hemispherical Ends
Hemispherical ends are half-shaped spheres that cap cylinders to create a streamlined shape, like that of a traditional fuel tank or submarine.
Each hemisphere shares the same radius as the cylinder it caps.
In this task, with the radius \(r\) being 10 cm, the formula for the surface area of a hemisphere is \(2\pi r^2\).
Given there are two hemispheres on the tank's ends, you need to double this calculation to account for both. Thus, the total hemispherical surface area arises from multiplying by two.
Calculation of Volume of Paint
After the surface areas are determined, the next step is addressing the thickness of the paint, which is given in millimeters in our scenario.
It's crucial to convert this measurement to centimeters for consistency with the rest of the calculations: 1 millimeter equals 0.1 centimeters.
The paint volume needed can be calculated by multiplying the total surface area by the paint鈥檚 thickness.
This step ensures that you account for the entire outer surface area to be covered by a uniform layer of paint.
Mathematical Problem Solving Approach
Working through mathematical problems like this requires a step-by-step approach that focuses on defining all variables and understanding the shapes involved.
In this exercise, breaking down the tank into its cylindrical and hemispherical components was crucial.
After identifying the tank parts, applying the correct geometric formulas facilitated the calculation of the surface areas needed for accurate paint volume estimation.
Patience and attention to detail鈥攍ike converting units and correctly approximating \(\pi\)鈥攁re key to arriving at a precise solution.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

, find dy/dx by logarithmic differentiation. $$ y=\frac{\left(x^{2}+3\right)^{2 / 3}(3 x+2)^{2}}{\sqrt{x+1}} $$

A revolving beacon light is located on an island and is 2 miles away from the nearest point \(P\) of the straight shoreline of the mainland. The beacon throws a spot of light that moves along the shoreline as the beacon revolves. If the speed of the spot of light on the shoreline is \(5 \pi\) miles per minute when the spot is 1 mile from \(P\), how fast is the beacon revolving?

Einstein's Special Theory of Relativity says that an object's mass \(m\) is related to its velocity \(v\) by the formula $$ m=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}=m_{0}\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2} $$ Here \(m_{0}\) is the rest mass and \(c\) is the speed of light. Use differentials to determine the percent increase in mass of an object when its velocity increases from \(0.9 c\) to \(0.92 c\).

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ f(x)=\sqrt{1-x^{2}} \text { at } a=0,[-1,1] $$

The period of a simple pendulum of length \(L\) feet is given by \(T=2 \pi \sqrt{L / g}\) seconds. We assume that \(g\), the acceleration due to gravity on (or very near) the surface of the earth, is 32 feet per second per second. If the pendulum is that of a clock that keeps good time when \(L=4\) feet, how much time will the clock gain in 24 hours if the length of the pendulum is decreased to \(3.97\) feet?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.