Problem 70 - Elementary and Intermediate Algebra

Question

Use a calculator to find each function value. Round to the nearest ten- thousandth. See Example 5 and Using Your Calculator 70\. \(h(t)=\sqrt[3]{2.1 t+11}\) a. \(h(-0.4)\) b. \(h(15)\)

Step-by-Step Solution

Answer

a. 2.1699 b. 3.4674

Step 1: Understand the Function

The function given is \(h(t) = \sqrt[3]{2.1t + 11}\). This means that for any value of \(t\), you substitute it into the expression \(2.1t + 11\), and then take the cube root of the result.

Step 2: Calculate h(-0.4)

Substitute \(t = -0.4\) into the function:\[h(-0.4) = \sqrt[3]{2.1(-0.4) + 11}\]Calculate inside the cube root:\[2.1(-0.4) = -0.84\]Then, compute:\[-0.84 + 11 = 10.16\]Now, evaluate the cube root using a calculator:\[h(-0.4) = \sqrt[3]{10.16} \approx 2.1699\] (rounded to the nearest ten-thousandth).

Step 3: Calculate h(15)

Substitute \(t = 15\) into the function:\[h(15) = \sqrt[3]{2.1(15) + 11}\]Calculate inside the cube root:\[2.1(15) = 31.5\]Then, compute:\[31.5 + 11 = 42.5\]Now, evaluate the cube root using a calculator:\[h(15) = \sqrt[3]{42.5} \approx 3.4674\] (rounded to the nearest ten-thousandth).

Key Concepts

Function Evaluation

Function evaluation is a basic yet essential concept in mathematics, especially when dealing with expressions and equations. It involves substituting a given value for the variable in a function and then simplifying to find the output, which is also known as the function value. In our exercise example, we have a function defined as \(h(t) = \sqrt[3]{2.1t + 11}\). Every time a particular \(t\) value is provided, we substitute it into the function's formula—a process that requires accurate calculation skills. Here are key points to keep in mind:

• Identify the function and its variable(s). For instance, in this function, the variable is \(t\).
• Substitute the given value into the function. Replace every instance of the variable with this value, like substituting \(t=-0.4\) in the function to compute \(h(-0.4)\).
• Perform the necessary arithmetic operations—like multiplication or addition—inside the function before applying any roots or powers.
• The final step is to solve it completely to obtain the function's value. Use the proper mathematical tools, like calculators, to ensure accuracy.

Evaluating functions like \(h(t)\) helps build an understanding of how changes in input affect the output, which is a foundational concept in math and sciences.

Cube Roots

Cube roots refer to a special type of root where the result is a number that, when multiplied by itself three times, yields the original number. It's denoted by the symbol \(\sqrt[3]{...}\). Cube roots are essential in solving equations involving power of three and are commonly seen in algebra and higher-level math.Understanding cube roots requires a few key points:

• Cube root of a number \(x\) is \(y\) when \(y^3 = x\).
• Cube roots can be applied to both positive and negative numbers, unlike square roots, providing real number results.
• To solve for cube roots, you can use estimation, factoring, or most conveniently, calculators.

In our function example, \(h(t) = \sqrt[3]{2.1t + 11}\), you need to evaluate an expression by first simplifying the term inside the cube root and then using a calculator to accurately find the cube root. This kind of calculation illustrates how cube roots can transform and simplify the function evaluation process itself.

Calculator Use

Using a calculator efficiently is crucial for solving complex problems, especially those involving functions and roots. In the context of our exercise, a calculator helps in finding cube roots and rounding numbers accurately.Here’s how you can effectively use a calculator for function evaluation tasks:

• Enter the arithmetic expression as it appears in the problem. For example, input \(2.1 \times (-0.4) + 11\) directly for accuracy.
• Use the calculator's function for roots: most modern calculators have a dedicated function key for cube roots (sometimes marked as \(x\sqrt[3]{}\) or similar).
• After obtaining the cube root value, remember to round it according to the problem's requirements—in this case, to the nearest ten-thousandth.

Being proficient with a calculator simplifies the evaluation process and minimizes errors, allowing a student to focus on understanding the concepts rather than getting bogged down by complex arithmetic.