Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

When the cube of a non-zero number \(y\) is subtracted from \(35,\) the result is equal to the result of dividing 216 by the cube of that number \(y .\) What is the sum of all the possible values of \(y ?\) a. \(\frac{5}{2}\) b. 5 c. 6 d. 10 e. 12

Short Answer

Expert verified
The sum of all possible values of \(y\) is 5.

Step by step solution

01

Set up the equation

Translate the problem statement into a mathematical equation. Given: When the cube of a non-zero number \(y\) is subtracted from \(35\), the result is equal to the result of dividing \(216\) by the cube of that number \(y\). Thus, the equation is: 35 - y^3 = \frac{216}{y^3}.
02

Eliminate the fraction

Multiply both sides of the equation by \(y^3\) to eliminate the fraction: y^3 (35 - y^3) = 216.
03

Simplify the equation

Distribute \(y^3\) on the left-hand side and simplify: 35y^3 - y^6 = 216 Rearrange to form a standard polynomial equation: y^6 - 35y^3 + 216 = 0.
04

Solve for \(z = y^3\)

Introduce a substitution \(z = y^3\) to turn the equation into a quadratic form: z^2 - 35z + 216 = 0.
05

Solve the quadratic equation

Solve for \(z\) using the quadratic formula: \[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = -35\), and \(c = 216\). \[ z = \frac{35 \pm \sqrt{35^2 - 4 \cdot 1 \cdot 216}}{2 \cdot 1} \] \[ z = \frac{35 \pm \sqrt{1225 - 864}}{2} \] \[ z = \frac{35 \pm \sqrt{361}}{2} \] \[ z = \frac{35 \pm 19}{2} \] Thus, the solutions are \(z = 27\) and \(z = 8\).
06

Determine the values of \(y\)

Since \(z = y^3\), find the corresponding values of \(y\): If \(z = 27\), then \(y^3 = 27\) implies \(y = 3\). If \(z = 8\), then \(y^3 = 8\) implies \(y = 2\).
07

Sum all possible values of \(y\)

Finally, find the sum of all possible values of \(y\): \(\text{Sum} = 3 + 2 = 5\).

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 Vaia!

Key Concepts

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

Cube of a Number
The cube of a number refers to raising a number to the third power. For a given number \(y\), its cube is represented as \(y^3\). This means multiplying the number by itself twice more: \(y \times y \times y = y^3\).

For example:
  • If \(y = 2\), then \(2^3 = 2 \times 2 \times 2 = 8\).
  • If \(y = 3\), then \(3^3 = 3 \times 3 \times 3 = 27\).

In the given problem, you need to solve an equation where a cube of a number plays a key role in both subtracting from 35 and dividing a constant.
Polynomial Equation
A polynomial equation involves variables raised to different powers with constant coefficients. They can range from simple (linear) to complex (higher-degree polynomials). The equation from our problem is an example of a polynomial equation:

When we arrived at
\[ y^6 - 35y^3 + 216 = 0 \], we had a polynomial equation of degree 6. These types of equations often have multiple solutions, as each root corresponds to a potential value of the variable.
Quadratic Formula
The quadratic formula is used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). The solutions for \(x\) can be found using the formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In the problem, substituting \(z = y^3\) converted our original polynomial to a quadratic form \(z^2 - 35z + 216 = 0\).

Here,
  • \(a = 1\)
  • \(b = -35\)
  • \(c = 216\)
. Using the quadratic formula, we found the solutions for \(z\) which helped us find the values for \(y\).
Substitution Method
The substitution method involves replacing one variable with another expression to simplify an equation. This is particularly helpful with polynomial equations that can be transformed into quadratic equations.

In this exercise, we introduced a substitution \(z = y^3\). This substitution recast our original sixth-degree polynomial into the quadratic equation \(z^2 - 35z + 216 = 0\).

Solving the quadratic equation for \(z\) then allowed us to back-substitute to find the values of \(y\). This technique is invaluable for transforming and solving higher-degree equations in mathematics.

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

There are 816 students in enrolled at a certain high school. Each of these students is taking at least one of the subjects economics, geography, and biology. The sum of the number of students taking exactly one of these subjects and the number of students taking all 3 of these subjects is 5 times the number of students taking exactly 2 of these subjects. The ratio of the number of students taking only the two subjects economics and geography to the number of students taking only the two subjects economics and biology to the number of students taking only the two subjects geography and biology is \(3: 6: 8 .\) How many of the students enrolled at this high school are taking only the two subjects geography and biology? a. 35 b. 42 c. 64 d. 136 e. 240

Two workers have different pay scales. Worker A receives \(\$ 50\) for any day worked plus \(\$ 15\) per hour. Worker \(B\) receives \(\$ 27\) per hour. Both workers may work for a fraction of an hour and be paid in proportion to their respective hourly rates. If worker A arrives at 9: 21 a.m. and receives the \(\$ 50\) upon arrival and Worker \(\mathrm{B}\) arrives at 10: 09 a.m. assuming both work continuously, at what time would their earnings be identical? a. \(11: 09 \mathrm{a.m}.\) b. \(12: 01 \mathrm{p.m}.\) c. \(2: 31 \mathrm{p.m}.\) d. \(3: 19 \mathrm{p.m}.\) e. \(3: 36 \mathrm{p.m}.\)

For all values of \(x, y,\) and \(z, x \diamond y \diamond z=x^{2}(y-1)(z+2)\). If \(a<0,\) which of the following shows \(R, S,\) and \(T\) arranged in order from least to greatest? \(R: 1 \diamond a \diamond 3\) \(S: 3 \diamond a \diamond 1\) \(T: a \diamond 3 \diamond 1\) a. \(R, S, T\) b. \(T, S, R\) c. \(R, T, S\) d. \(T, R, S\) e. \(S, R, T\)

A department of motor vehicles asks visitors to draw numbered tickets from a dispenser so that they can be served in order by number. Six friends have graduated from truck-driving school and go to the department to get commercial driving licenses. They draw tickets and find that their numbers are a set of evenly spaced integers with a range of 10 Which of the following could NOT be the sum of their numbers? a. 1,254 b. 1,428 c. 3,972 d. 4,316 e. 8,010

An ornithologist has studied a particular population of starlings and discovered that their population has increased by \(400 \%\) every ten years starting in \(1890 .\) If the initial population in 1890 was 256 birds, how large was the population of starlings in \(1970 ?\) a. 102,400 b. 10,000,000 c. 16,777,216 d. 20,000,000 e. 100,000,000

See all solutions

Recommended explanations on English Textbooks

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.

Sign-up for free