/*! 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 65 An architect designs a theater w... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 65

An architect designs a theater with 15 seats in the first row, 18 in the second, 21 in the third, and so on. If the theater is to have a seating capacity of \(870,\) how many rows must the architect use in his design?

Short Answer

Expert verified
The architect must design 20 rows for the theater.

Step by step solution

01

Identify the Pattern

Look at the number of seats in each row to find a pattern. The seats in the rows form an arithmetic sequence: 15, 18, 21, ... where the first term, \(a_1\), is 15 and the common difference, \(d\), is 3.
02

Set Up the Formula

The formula for the sum of the first \(n\) terms of an arithmetic sequence is: \(S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d)\). We know \(S_n = 870\), \(a_1 = 15\), and \(d = 3\). Substitute these values into the formula: \(870 = \frac{n}{2} \cdot (30 + (n-1) \times 3)\).
03

Simplify the Expression

Simplify the expression obtained: \(870 = \frac{n}{2} \cdot (30 + 3n - 3)\). This simplifies further to \(870 = \frac{n}{2} (3n + 27)\). Multiply both sides by 2 to eliminate the fraction: \(1740 = n(3n + 27)\).
04

Solve the Quadratic Equation

Reorganize the equation: \(3n^2 + 27n - 1740 = 0\). Use the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = 27\), \(c = -1740\). Calculate the discriminant \(b^2 - 4ac = 27^2 - 4 \times 3 \times (-1740)\).
05

Calculate the Discriminant

Compute the discriminant: \(729 + 20880 = 21609\). Since the discriminant is a perfect square, we find \(\sqrt{21609} = 147\).
06

Apply the Quadratic Formula

Substitute into the quadratic formula: \(n = \frac{-27 \pm 147}{6}\). Calculate the two solutions: \(n = 20\) (as \(n = -29\) is not possible for the number of rows).
07

Confirm the Answer

Verify that using 20 rows gives a total of 870 seats by calculating the sum again using the arithmetic sequence sum formula.

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.

Sum of Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence. An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
To find the sum of an arithmetic series, we use the formula for the sum of the first \(n\) terms:
  • \( S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d) \)
In this formula:
  • \( S_n \) is the sum of the first \(n\) terms.
  • \( n \) is the number of terms.
  • \( a_1 \) is the first term of the sequence.
  • \( d \) is the common difference.
This formula helps calculate the total sum of all seats in the theater, given as 870. Understanding how to apply the formula can solve many practical problems involving sums of arithmetic sequences.
Quadratic Equation
Quadratic equations are essential in algebra and they come in the form \( ax^2 + bx + c = 0 \). Solving a quadratic equation requires finding the values of \( x \) that satisfy this equation.
In the theater seating problem, the expression simplifies to a quadratic form: \( 3n^2 + 27n - 1740 = 0 \).
To solve this, use the quadratic formula:
  • \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Here:
  • \( a = 3 \),
  • \( b = 27 \),
  • \( c = -1740 \).
The solution requires calculating the discriminant \( b^2 - 4ac \), which helps determine the nature of the roots. A positive discriminant, as seen here, indicates the presence of real and distinct solutions.
Common Difference
The common difference in an arithmetic sequence is the constant gap between consecutive terms. It dictates how the sequence progresses.
For the theater seating plan:
  • The first term \( a_1 \) is 15 (seats in the first row).
  • The common difference \( d \) is found as 3, since each subsequent row increases by three seats (18 鈥� 15 = 3).
Understanding the common difference is crucial as it directly impacts the calculation of the number of terms \( n \) needed to reach the seating capacity. It can help solve other problems with sequences by setting the pace of growth or decline in the series.
Arithmetic Series Formula
The arithmetic series formula finds vast applications beyond just calculating seats in a theater. It is vital for adding up any series of numbers that change consistently.This formula:
  • \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \)
allows quick summation of the series when you know:
  • \( a_1 \) as the starting point,
  • \( d \) as the common difference,
  • \( n \) the number of terms.
With the arithmetic series formula, real-world problems like determining how many rows of seats fit a set capacity become manageable. This formula provides a structured way to sum a sequence efficiently and is a key tool in many branches of science and engineering.

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

If \(a_{1}, a_{2}\) \(a_{3}, \ldots\) is a geometric sequence with common ratio \(r,\) show that the sequence $$\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \dots$$ is also a geometric sequence, and find the common ratio.

Which is larger, \((100 !)^{101}\) or (101!) \(^{100}\) ? [ Hint: Try factoring the expressions. Do they have any common factors?]

Factor using the Binomial Theorem. $$x^{8}+4 x^{6} y+6 x^{4} y^{2}+4 x^{2} y^{3}+y^{4}$$

The arithmetic mean (or average) of two numbers \(a\) and \(b\) is $$ m=\frac{a+b}{2} $$ Note that \(m\) is the same distance from \(a\) as from \(b,\) so \(a, m, b\) is an arithmetic sequence. In general, if \(m_{1}, m_{2}, \ldots, m_{k}\) are equally spaced between \(a\) and \(b\) so that $$ a, m_{1}, m_{2}, \dots, m_{k}, b $$ is an arithmetic sequence, then \(m_{1}, m_{2}, \ldots, m_{k}\) are called \(k\) arithmetic means between \(a\) and \(b\). (a) Insert two arithmetic means between 10 and 18 . (b) Insert three arithmetic means between 10 and 18 . (c) Suppose a doctor needs to increase a patient's dosage of a certain medicine from 100 mg to 300 mg per day in five equal steps. How many arithmetic means must be inserted between 100 and 300 to give the progression of daily doses, and what are these means?

All Cats Are Black? What is wrong with the following "proof" by mathematical induction that all cats are black? Let \(P(n)\) denote the statement "In any group of \(n\) cats, if one cat is black, then they are all black." Step 1 The statement is clearly true for \(n=1\) Step 2 Suppose that \(P(k)\) is true. We show that \(P(k+1)\) is true. Suppose we have a group of \(k+1\) cats, one of whom is black; call this cat "Tadpole." Remove some other cat (call it "Sparky") from the group. We are left with \(k\) cats, one of whom (Tadpole) is black, so by the induction hypothesis, all \(k\) of these are black. Now put Sparky back in the group and take out Tadpole. We again have a group of \(k\) cats, all of whomexcept possibly Sparky-are black. Then by the induction hypothesis, Sparky must be black too. So all \(k+1\) cats in the original group are black. Thus by induction \(P(n)\) is true for all \(n .\) since everyone has seen at least one black cat, it follows that all cats are black. PICTURE CANT COPY
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

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.