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Sagot :
Certainly! Let's work through the problem step-by-step to derive the correct inequality used to find the possible lengths of the base of the triangle.
1. Understanding the problem:
- The height [tex]\( h \)[/tex] of the triangle is 4 inches greater than twice its base [tex]\( x \)[/tex].
- The area of the triangle is no more than 168 square inches.
2. Expressing the height in terms of the base:
- Let [tex]\( x \)[/tex] be the base of the triangle.
- Then the height [tex]\( h \)[/tex] can be expressed as:
[tex]\[ h = 2x + 4 \][/tex]
3. Formula for the area of the triangle:
- The area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
- Substituting the expressions for base and height, we get:
[tex]\[ A = \frac{1}{2} \times x \times (2x + 4) \][/tex]
4. Given that the area is no more than 168 square inches:
- This translates to the inequality:
[tex]\[ \frac{1}{2} \times x \times (2x + 4) \leq 168 \][/tex]
5. Simplifying the inequality:
- First, multiply both sides of the inequality by 2 to get rid of the fraction:
[tex]\[ x \times (2x + 4) \leq 336 \][/tex]
- Distribute [tex]\( x \)[/tex] inside the parenthesis:
[tex]\[ 2x^2 + 4x \leq 336 \][/tex]
- Simplify by dividing all terms by 2:
[tex]\[ x^2 + 2x \leq 168 \][/tex]
6. Restating the inequality in standard form:
- This final form expresses a quadratic inequality:
[tex]\[ x (x + 2) \leq 168 \][/tex]
Thus, the correct inequality that can be used to find the possible lengths [tex]\( x \)[/tex] of the base of the triangle is:
[tex]\[ x (x + 2) \leq 168 \][/tex]
So the answer is:
[tex]\[ \boxed{x(x+2) \leq 168} \][/tex]
1. Understanding the problem:
- The height [tex]\( h \)[/tex] of the triangle is 4 inches greater than twice its base [tex]\( x \)[/tex].
- The area of the triangle is no more than 168 square inches.
2. Expressing the height in terms of the base:
- Let [tex]\( x \)[/tex] be the base of the triangle.
- Then the height [tex]\( h \)[/tex] can be expressed as:
[tex]\[ h = 2x + 4 \][/tex]
3. Formula for the area of the triangle:
- The area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
- Substituting the expressions for base and height, we get:
[tex]\[ A = \frac{1}{2} \times x \times (2x + 4) \][/tex]
4. Given that the area is no more than 168 square inches:
- This translates to the inequality:
[tex]\[ \frac{1}{2} \times x \times (2x + 4) \leq 168 \][/tex]
5. Simplifying the inequality:
- First, multiply both sides of the inequality by 2 to get rid of the fraction:
[tex]\[ x \times (2x + 4) \leq 336 \][/tex]
- Distribute [tex]\( x \)[/tex] inside the parenthesis:
[tex]\[ 2x^2 + 4x \leq 336 \][/tex]
- Simplify by dividing all terms by 2:
[tex]\[ x^2 + 2x \leq 168 \][/tex]
6. Restating the inequality in standard form:
- This final form expresses a quadratic inequality:
[tex]\[ x (x + 2) \leq 168 \][/tex]
Thus, the correct inequality that can be used to find the possible lengths [tex]\( x \)[/tex] of the base of the triangle is:
[tex]\[ x (x + 2) \leq 168 \][/tex]
So the answer is:
[tex]\[ \boxed{x(x+2) \leq 168} \][/tex]
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