Given $$ H(t)=188\left(2.45^{-0.4 t}\right)+60 $$, evaluate each of the following. Round to the nearest hundredth if necessary. a) $$ H(0) $$ 136.73 Preview 136.73 Not quite. b) $$ H(1) $$ 191.37 Preview 191.37 Correct. Good Job! c) $$ H(2) $$ 222.9 Preview 222.9 Not quite. d) $$ H(10) $$ 248 Preview 248

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UpStudy AI · Accepted Answer

Evaluate the expression by following steps:
- step0: Evaluate:
  $$188\left(2.45^{-0.4t}\right)+60$$
- step1: Substitute:
  $$188\left(2.45^{-0.4\times 2}\right)+60$$
- step2: Multiply the numbers:
  $$188\left(2.45^{-0.8}\right)+60$$
- step3: Convert the expressions:
  $$188\left(\frac{49}{20}\right)^{-\frac{4}{5}}+60$$
- step4: Multiply the terms:
  $$\frac{376\sqrt[5]{245000}}{49}+60$$
- step5: Reduce fractions to a common denominator:
  $$\frac{376\sqrt[5]{245000}}{49}+\frac{60\times 49}{49}$$
- step6: Transform the expression:
  $$\frac{376\sqrt[5]{245000}+60\times 49}{49}$$
- step7: Multiply the numbers:
  $$\frac{376\sqrt[5]{245000}+2940}{49}$$
Substitute $$ t=10 $$ into the expression $$ 188\left(2.45^{-0.4 t}\right)+60 $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$188\left(2.45^{-0.4t}\right)+60$$
- step1: Substitute:
  $$188\left(2.45^{-0.4\times 10}\right)+60$$
- step2: Multiply the numbers:
  $$188\left(2.45^{-4}\right)+60$$
- step3: Convert the expressions:
  $$188\left(\frac{49}{20}\right)^{-4}+60$$
- step4: Multiply the terms:
  $$\frac{188\times 20^{4}}{49^{4}}+60$$
- step5: Reduce fractions to a common denominator:
  $$\frac{188\times 20^{4}}{49^{4}}+\frac{60\times 49^{4}}{49^{4}}$$
- step6: Transform the expression:
  $$\frac{188\times 20^{4}+60\times 49^{4}}{49^{4}}$$
Substitute $$ t=0 $$ into the expression $$ 188\left(2.45^{-0.4 t}\right)+60 $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$188\left(2.45^{-0.4t}\right)+60$$
- step1: Substitute:
  $$188\left(2.45^{-0.4\times 0}\right)+60$$
- step2: Multiply:
  $$188\left(2.45^{0}\right)+60$$
- step3: Calculate:
  $$188\times 1+60$$
- step4: Multiply:
  $$188+60$$
- step5: Add the numbers:
  $$248$$
Calculate or simplify the expression $$ 188\left(2.45^{-0.4 \cdot 2}\right)+60 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$188\times 2.45^{-0.4\times 2}+60$$
- step1: Multiply the numbers:
  $$188\times 2.45^{-0.8}+60$$
- step2: Convert the expressions:
  $$188\left(\frac{49}{20}\right)^{-\frac{4}{5}}+60$$
- step3: Multiply the terms:
  $$\frac{376\sqrt[5]{245000}}{49}+60$$
- step4: Reduce fractions to a common denominator:
  $$\frac{376\sqrt[5]{245000}}{49}+\frac{60\times 49}{49}$$
- step5: Transform the expression:
  $$\frac{376\sqrt[5]{245000}+60\times 49}{49}$$
- step6: Multiply the numbers:
  $$\frac{376\sqrt[5]{245000}+2940}{49}$$
Calculate or simplify the expression $$ 188\left(2.45^{-0.4 \cdot 10}\right)+60 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$188\times 2.45^{-0.4\times 10}+60$$
- step1: Multiply the numbers:
  $$188\times 2.45^{-4}+60$$
- step2: Convert the expressions:
  $$188\left(\frac{49}{20}\right)^{-4}+60$$
- step3: Multiply the terms:
  $$\frac{188\times 20^{4}}{49^{4}}+60$$
- step4: Reduce fractions to a common denominator:
  $$\frac{188\times 20^{4}}{49^{4}}+\frac{60\times 49^{4}}{49^{4}}$$
- step5: Transform the expression:
  $$\frac{188\times 20^{4}+60\times 49^{4}}{49^{4}}$$
Let's evaluate the function $$ H(t) = 188\left(2.45^{-0.4 t}\right) + 60 $$ step by step for each required value.

### a) Evaluate $$ H(0) $$

Substituting $$ t = 0 $$:
$$
H(0) = 188\left(2.45^{-0.4 \cdot 0}\right) + 60 = 188\left(2.45^{0}\right) + 60 = 188 \cdot 1 + 60 = 248
$$

### b) Evaluate $$ H(1) $$

This value has already been confirmed as correct:
$$
H(1) = 191.37
$$

### c) Evaluate $$ H(2) $$

Substituting $$ t = 2 $$:
$$
H(2) = 188\left(2.45^{-0.4 \cdot 2}\right) + 60
$$
Calculating $$ 2.45^{-0.8} $$:
$$
H(2) \approx 188 \cdot 0.151796 + 60 \approx 28.54 + 60 \approx 88.54
$$

### d) Evaluate $$ H(10) $$

Substituting $$ t = 10 $$:
$$
H(10) = 188\left(2.45^{-0.4 \cdot 10}\right) + 60
$$
Calculating $$ 2.45^{-4} $$:
$$
H(10) \approx 188 \cdot 0.065217 + 60 \approx 12.24 + 60 \approx 72.24
$$

### Summary of Results
- $$ H(0) = 248 $$
- $$ H(1) = 191.37 $$ (confirmed)
- $$ H(2) \approx 88.54 $$
- $$ H(10) \approx 72.24 $$

Please check the calculations for $$ H(2) $$ and $$ H(10) $$ as they differ from your expectations.
Here are the evaluated results: - $$ H(0) = 248 $$ - $$ H(1) = 191.37 $$ - $$ H(2) \approx 88.54 $$ - $$ H(10) \approx 72.24 $$

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