Skip to main content
Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 13th EditionCh. 5 - Systems and Matrices문제 68
6장, 문제 68

Solve each problem. Find the radius and height (to the nearest thousandth) of an open-ended cylinder with volume 50 in.3 and lateral surface area 65 in.2.
Diagram of an open cylinder with height h and radius r, alongside its rectangular lateral surface area labeled 2πr by h.

검증된 단계별 안내
1
Recall the formulas for the volume and lateral surface area of an open-ended cylinder. The volume \(V\) is given by \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. The lateral surface area \(A\) (without the top and bottom) is \(A = 2 \pi r h\).
Write down the system of equations using the given values: \(\pi r^2 h = 50\) and \(2 \pi r h = 65\).
From the lateral surface area equation \(2 \pi r h = 65\), solve for \(h\) in terms of \(r\): \(h = \frac{65}{2 \pi r}\).
Substitute the expression for \(h\) into the volume equation: \(\pi r^2 \left( \frac{65}{2 \pi r} \right) = 50\). Simplify this equation to get an equation in terms of \(r\) only.
Solve the resulting equation for \(r\), then use the value of \(r\) to find \(h\) using the expression \(h = \frac{65}{2 \pi r}\). Round both \(r\) and \(h\) to the nearest thousandth.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
6m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Volume of a Cylinder

The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height. This formula represents the amount of space inside the cylinder and is essential for relating the radius and height to the given volume.

Lateral Surface Area of an Open Cylinder

The lateral surface area of an open cylinder (without top and bottom) is given by A = 2πrh. This formula calculates the area of the curved surface, which helps establish a relationship between the radius and height based on the given surface area.
추천 영상:
6:19
Systems of Inequalities

Solving Systems of Nonlinear Equations

Finding the radius and height requires solving two equations simultaneously: one for volume and one for lateral surface area. This involves techniques such as substitution or elimination to solve nonlinear equations involving π, r, and h.
추천 영상:
5:48
Solving Systems of Equations - Substitution
관련 실천
교과서 질문

Graph the solution set of each system of inequalities.

ex−y≤1e^{x}-y\(\le\)1

x−2y≥4x-2y\(\ge\)4

576
views
교과서 질문

Solve each system. (Hint: In Exercises 69–72, let 1/x = t and 1/y = u.)

2/x + 1/y = 3/2

3/x - 1/y = 1

98
views
교과서 질문

Given A=[4−231],B=[510−237]A = \(\left\)[ \(\begin{matrix}\) 4 & -2 \\ 3 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 5 & 1 \\ 0 & -2 \\ 3 & 7 \(\end{matrix}\) \(\right\)] and C=[−541036]C = \(\left\)[ \(\begin{matrix}\) -5 & 4 & 1 \\ 0 & 3 & 6 \(\end{matrix}\) \(\right\)] , find each product, if possible. See Examples 5–7. BA

75
views
교과서 질문

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 1.5

x + 3y = 5

2x + 4y = 3

613
views
교과서 질문

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

5x+4y=105x + 4y = 10

5x+4y=105x + 4y = 10

640
views
교과서 질문

Find each product, if possible.

[−2−3−42−104−23][01412−132−2]\(\left\)[ \(\begin{matrix}\) -2 & -3 & -4 \\ 2 & -1 & 0 \\ 4 & -2 & 3 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 0 & 1 & 4 \\ 1 & 2 & -1 \\ 3 & 2 & -2 \(\end{matrix}\) \(\right\)]

78
views