Here are the step-by-step solutions for each problem on the worksheet.
1. Surface Area of a Rectangular Prism
To find the surface area, we need to calculate the area of all six faces of the box and add them together. The formula is:
$$Surface\ Area = 2(lw + lh + wh)$$
Given dimensions:
* Length ($l$) = $5\text{ cm}$
* Width ($w$) = $3\text{ cm}$
* Height ($h$) = $6\text{ cm}$
Step 1: Calculate the area of the three unique pairs of faces.
* Bottom and Top ($l \times w$): $5 \times 3 = 15\text{ cm}^2$
* Front and Back ($l \times h$): $5 \times 6 = 30\text{ cm}^2$
* Sides ($w \times h$): $3 \times 6 = 18\text{ cm}^2$
Step 2: Add these areas together.
$$15 + 30 + 18 = 63\text{ cm}^2$$
Step 3: Multiply by 2 (because there are two of each face).
$$63 \times 2 = 126\text{ cm}^2$$
Answer: $126\text{ cm}^2$
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2. Solve for $x$
This problem involves finding an exterior angle of a triangle. There is a helpful rule called the
Exterior Angle Theorem, which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
The remote interior angles are the ones inside the triangle that are not touching angle $x$. In this figure, those angles are $96^\circ$ and $32^\circ$.
Calculation:
$$x = 96 + 32$$
$$x = 128$$
*(Alternative Method: The angles inside a triangle add up to $180^\circ$. So, the third interior angle is $180 - 96 - 32 = 52^\circ$. Angles on a straight line add up to $180^\circ$, so $x = 180 - 52 = 128^\circ$.)*
Answer: $128$
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3. Evaluate the Expressions
Expression 1: $11^2$
This means $11$ multiplied by itself.
$$11 \times 11 = 121$$
Expression 2: $(-2)^5$
This means multiplying $-2$ by itself five times. Since the exponent is odd, the result will be negative.
$$(-2) \times (-2) \times (-2) \times (-2) \times (-2)$$
$$4 \times 4 \times (-2)$$
$$16 \times (-2) = -32$$
Expression 3: $\left(\frac{3}{2}\right)^2$
To square a fraction, you square the top number (numerator) and the bottom number (denominator) separately.
$$\frac{3^2}{2^2} = \frac{3 \times 3}{2 \times 2} = \frac{9}{4}$$
Answers:
* $121$
* $-32$
* $\frac{9}{4}$
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4. Coral Reef Growth
We need to find out how much the reef grows in total over 5 weeks.
Given:
* Growth per week = $0.19\text{ mm}$
* Time = $5\text{ weeks}$
Calculation:
Multiply the weekly growth by the number of weeks.
$$0.19 \times 5$$
Let's do the math:
$$19 \times 5 = 95$$
Since there are two decimal places in $0.19$, we move the decimal point two places to the left in the answer.
$$0.95$$
Answer: $0.95\text{ mm}$
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Final Answer:
1. $126\text{ cm}^2$
2. $128$
3. $121$, $-32$, $\frac{9}{4}$
4. $0.95\text{ mm}$
Parent Tip: Review the logic above to help your child master the concept of k12 worksheets math.
