Problem 70 - Intermediate Algebra

Question

Use a calculator to find each function value. Round to the nearest ten- thousandth. See Example 5 and Using Your Calculator. $$ h(t)=\sqrt[3]{2.1 t+11} $$ a. \(h(-0.4)\) b. \(h(15)\)

Step-by-Step Solution

Answer

a. \( h(-0.4) \approx 2.1624 \), b. \( h(15) \approx 3.4949 \).

Step 1: Substitute and Calculate for h(-0.4)

First, substitute \( t = -0.4 \) into the function \( h(t) = \sqrt[3]{2.1t + 11} \). This gives us: \[ h(-0.4) = \sqrt[3]{2.1(-0.4) + 11} \] Calculate \( 2.1 \times -0.4 = -0.84 \), so the expression becomes \[ h(-0.4) = \sqrt[3]{-0.84 + 11} = \sqrt[3]{10.16} \]. Use a calculator to find the cube root of 10.16, which is approximately 2.1624 when rounded to the nearest ten-thousandth.

Step 2: Substitute and Calculate for h(15)

Next, substitute \( t = 15 \) into the function \( h(t) = \sqrt[3]{2.1t + 11} \). This gives us: \[ h(15) = \sqrt[3]{2.1(15) + 11} \] Calculate \( 2.1 \times 15 = 31.5 \), then \( 31.5 + 11 = 42.5 \). So, \[ h(15) = \sqrt[3]{42.5} \]. Use a calculator to find the cube root of 42.5, which is approximately 3.4949 when rounded to the nearest ten-thousandth.

Key Concepts

Function Evaluation

In algebra, the concept of function evaluation is fundamental and involves determining the output of a function for specific input values. This task requires that you understand how to substitute specific numbers into the function’s formula and perform the required calculations.

For example, given the function \( h(t) = \sqrt[3]{2.1t + 11} \), we are to evaluate \(h\) at two different points; namely, \(-0.4\) and \(15\). This essentially means replacing the variable \(t\) with these numbers and solving for \(h\).

• **Step-by-Step Process:** Start by substituting the given number for the variable in the function.
• Follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• After proper substitution, simplify the equation by performing the mathematical operations.
• Use a calculator for more complex calculations, especially when dealing with irrational numbers or roots.

By accurately evaluating functions, you can determine precise outcomes, which are vital in solving various algebraic problems.

Cube Root Calculation

Cube root calculation involves finding a number that, when multiplied by itself three times, returns the original number given under the root. Calculating a cube root can be challenging without technology, but understanding its basis is important.

For example, the expression \( \sqrt[3]{x} \) asks what number, when cubed, equals \(x\). In the given problem \( \sqrt[3]{10.16} \), we find the number that, when raised to the power of three, will result in 10.16. Calculators often have built-in functions to calculate cube roots, which simplifies this process.

Here are some points to consider when dealing with cube roots:

• **Nature of Cube Roots**: Unlike square roots, cube roots will yield real numbers for all real inputs because every real number has a real cube root.
• **Using Calculator**: Look for a 'cube root' or 'root' function; these are sometimes marked with symbols such as \( \sqrt[3]{} \) or use an exponent of \( \frac{1}{3} \) to represent cube roots.
• **Rounding**: When precision is needed, especially for non-integer outcomes, round your answer to a practical decimal place, like the ten-thousandth place in this problem.

Understanding cube roots deeply will aid in solving and simplifying more complex algebraic equations.

Using Calculator in Algebra

When engaging with algebraic concepts, utilizing a calculator can be a powerful tool, especially when the arithmetic becomes intricate or time-consuming.

A calculator becomes essential when dealing with calculations involving higher powers or roots, such as solving \( \sqrt[3]{42.5} \) in our evaluation of \( h(15) \). Here’s a concise guide on how best to use calculators in such scenarios:

• **Calculator Functions**: Ensure you are familiar with functions for basic operations, exponents, and roots. Many scientific calculators have these symbols clearly marked.
• **Accuracy and Precision**: Ensure you set your calculator to show results to the necessary decimal places. In our task, rounding to the nearest ten-thousandth was required.
• **Check Your Work**: Always double-check calculations, especially in step-by-step solutions, to ensure all intermediate results align with expectations.

Using calculators prudently simplifies complex algebraic expressions and helps confirm manual calculations, enhancing understanding and accuracy in reaching solutions.